A Bayesian model of contextsensitive value attribution
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Abstract
Substantial evidence indicates that incentive value depends on an anticipation of rewards within a given context. However, the computations underlying this context sensitivity remain unknown. To address this question, we introduce a normative (Bayesian) account of how rewards map to incentive values. This assumes that the brain inverts a model of how rewards are generated. Key features of our account include (i) an influence of prior beliefs about the context in which rewards are delivered (weighted by their reliability in a Bayesoptimal fashion), (ii) the notion that incentive values correspond to precisionweighted prediction errors, (iii) and contextual information unfolding at different hierarchical levels. This formulation implies that incentive value is intrinsically contextdependent. We provide empirical support for this model by showing that incentive value is influenced by context variability and by hierarchically nested contexts. The perspective we introduce generates new empirical predictions that might help explaining psychopathologies, such as addiction.
https://doi.org/10.7554/eLife.16127.001Introduction
Choice preferences vary as a function of context, and recent studies have shed light on the processes underlying these contextual influences (Huber et al., 1982; Johnson and Busemeyer, 2005; Ludvig et al., 2013; Louie et al., 2013, 2014, 2015; Roe et al., 2011; Rigoli et al., 2016a, 2016b; Simonson and Tversky, 1992; Soltani et al., 2012; Stewart, 2009; Stewart et al., 2003; Summerfield and Tsetsos, 2012, 2015; Tsetsos et al., 2010, 2012, 2016; Tversky, 1972; Usher and McClelland, 2004; Vlaev et al., 2012). For example, in (Rigoli et al., 2016a, 2016b) participants performed a task where blocks of trials were associated with either a low or a highvalue context (with overlapping distributions). Choice behaviour was consistent with the hypothesis that the incentive values of identical rewards were larger in the low compared to the highvalue context. This and similar evidence suggests that, at least in some cases, contextual effects on choice behaviour are explained by an incentive value that reflects the relative value of rewards anticipated within a given context (Louie et al., 2013; Ludvig et al., 2013; Rigoli et al., 2016a; Stewart, 2009; Stewart et al., 2003).
However, the computational mechanisms underlying the context sensitive nature of incentive value remain unclear. A promising explanatory framework builds on the notion that the brain’s computations correspond to Bayesian inference and learning. Several empirical and theoretical arguments support a Bayesian inference as a general account of brain function (Chater et al., 2006; Clark, 2013; Dayan et al., 1995; Ernst, 2006; Friston, 2010). The application of similar principles to value learning and planning has inspired the notions of planning as inference and active inference (Botvinick and Toussaint, 2012; Friston et al., 2013, 2015; Pezzulo and Rigoli, 2011; Pezzulo et al., 2015; Solway and Botvinick, 2012). Here, we consider the possibility that the contextsensitive value is a product of Bayesian inference. This implies that incentive value will depend on expectation and uncertainty about rewards, conditioned upon contextual factors. If this is the case, we would expect to see choice behaviour change with any contextual variable that is an ancestor of rewards in the subject’s generative model of these rewards.
We refer to our account as a Bayesian model of contextsensitive value (BCV). Below, we introduce the model and compare it with previous accounts of contextual influence on incentive value and choice (Bushong et al., unpublished; Kőszegi and Rabin, 2006; Kőszegi and Szeidl, 2013; Louie et al., 2014, 2015). We then report data from two behavioural experiments where we analysed two key predictions of BCV.
Results
Bayesian account of contextsensitive value
The general framework of BCV is Bayesian, building on a proposal that the brain performs some form of Bayesian inference (Chater et al., 2006; Clark, 2013; Dayan et al., 1995; Ernst, 2006; Friston, 2010). This approach considers the brain to possess a generative model of the sensorium, comprising a set of random variables (i.e., hidden states or causes of sensory outcomes) and their causal links (i.e., probabilistic contingencies). The variables can be separated into hidden and observable variables; the former representing the latent causes of observations, and the latter representing sensory evidence or cues. Sensory evidence is conveyed by observable variables, and this evidence is combined with prior beliefs to produce a posterior belief about the (hidden) causes of observations. The application of this logic to perception is straightforward and has proved effective in explaining several empirical phenomena in perception (Chater et al., 2006; Clark, 2013; Dayan et al., 1995; Ernst, 2006; Friston, 2010). For instance, there is evidence for integrating different perceptual modalities (e.g., visual and tactile) in a manner consistent with Bayesian principles (Ernst, 2006).
We propose a Bayesian scheme for BCV that accommodates the influence of context on incentive value. BCV focuses on scenarios (i) where incentive value depends on contextual information (either represented by cues or by previous rewards) provided before options or rewards are presented, and (ii) where reward is defined by a single attribute (e.g., reward amount). To describe the basic principles of BCV, we adopt the formalism of Bayesian graphs (Bishop, 2006) where a generative model is described by nodes or circles, representing random variables (shaded and white circles refer to observed and nonobserved variables respectively), and arrows, representing causal relationships among variables. A simple generative model hypothesized by BCV is shown in Figure 1A, where C represents prior beliefs about the average reward expected in a given context. Formally, this corresponds to a (Gaussian) prior belief (with mean $\mu}_{c$ and variance $\sigma}_{c}^{2$ over the mean of a (Gaussian) distribution of reward options R (with variance $\sigma}_{R}^{2$). When R is observed, a posterior expectation about the context is obtained by application of Bayes rule (Bishop, 2006):
The crucial proposal we advance here is that the incentive value V(R) attributed to a certain reward option is embedded within this belief update process and corresponds to a precisionweighted prediction error (Friston, 2005); namely, to the difference between R and the expected reward μ_{C,} multiplied by a gain term, which depends on the variances of both reward and context (i.e., relative confidence or precision):
The notion that incentive value corresponds to precisionweighted prediction error links with ideas in other cognitive domains proposing that prior expectations are explained away and perception corresponds to (precisionweighted) residuals, or prediction errors (Blakemore et al., 1999; Brown et al., 2013; Friston, 2005; Rao and Ballard, 1999). For example, it has been proposed that our sense of agency emerges from explaining away actiondependent somatosensory predictions, and hence what is perceived as externally generated sensation corresponds to (precisionweighted) residuals of the sensory input (Brown et al., 2013).
Here, we propose that a similar mechanism is involved in attribution of incentive value. This implies two fundamental forms of contextual normalization. First, a subtractive normalization is exerted when $\mu}_{C$ is different from zero. For example, if we assign positive and negative numbers to rewards and punishments respectively, their corresponding incentive values may change in sign, depending on whether punishment (i.e., ${\mu}_{C}\text{}\text{}0$ ) or reward (i.e., ${\mu}_{C}\text{}\text{}0$ ) is expected on average within the context. This implies that small rewards can appear as losses in contexts where large rewards are expected. Second, a divisive normalization depends on the gain $\frac{{\sigma}_{C}^{2}}{{\sigma}_{C}^{2}+{\sigma}_{R}^{2}}$. This implies that the positive and negative value of profits (i.e.,$R\text{}\text{}{\mu}_{C}$) and losses (i.e.,$R\text{}\text{}{\mu}_{C}$) will be augmented or attenuated, depending upon the relative precision of prior beliefs about (prior confidence in) the context $({\sigma}_{C}^{2})$ and of sensory evidence about the reward option $({\sigma}_{R}^{2})$.
Equation two applies every time novel information about reward is provided, which is when a prediction error occurs. This happens (i) when a (primary or secondary) reward is delivered (or is not delivered when expected), which can be post choice as well as in other conditions (e.g., in classical conditioning paradigms, when a reward is delivered independent of action), and (ii) when one (or more) option is presented. The latter follows because an agent has an expectation about an option, which leads to a prediction error when the actual option is presented.
A key aspect of our proposal addresses how contextual variables are implemented within a generative model. One possibility, illustrated in Figure 1B, is a generative model that includes an observation O reporting information about context. This model assumes that a value C is drawn from a Gaussian distribution with mean ${\mu}_{C}=0$ and variance $\sigma}_{C}^{2$. A zero mean captures contextindependent information, as it implies that overall rewards (i.e.,$R\text{}\text{}0$) and punishments (i.e.,$R\text{}\text{}{\mu}_{C}$) will be attributed positive and negative incentive values respectively. The variance $\sigma}_{C}^{2$ reflects prior uncertainty about the hidden or latent context. A context observation O is sampled from a Gaussian distribution with mean f(C) and variance $\sigma}_{O}^{2$ (reflecting the reliability of the contextrelated cue). For simplicity, we assume that f(C) = C, though in general this can be any function (similar simplifications are assumed below). A reward R is observed after being sampled from a Gaussian distribution with mean C and variance $\sigma}_{R}^{2$ (reflecting uncertainty about the reward distribution).
We propose that agents form posterior beliefs about the context P(CO,R) using Bayesian belief updating – first accumulating contextual information by estimating P(CO), and then reward information to give P(CO,R). This sequential inference (c.f., evidence accumulation) is motivated by the fact that information about context is usually provided at an earlier time point than reward options. The mean of the posterior distribution P(CO) is:
And the posterior variance:
The mean of the posterior distribution P(CO,R) corresponds to:
Implying the following incentive value for reward:
This shows that, other things being equal, information about the context (reflected in the value of O) induces a subtractive value normalization.
A possible extension of this generative model is illustrated in Figure 1C where contexts are organized hierarchically (Pezzulo et al., 2015). Imagine evaluating the same dish in different restaurants (a lowlevel context) and in different neighbourhoods (a highlevel context). This example highlights the fact that some (highlevel) contexts are more generic, while other (lowlevel) contexts are more specific. Crucially, if a context exerted no impact on incentive value, one would expect that the dish would be equally attractive, irrespective of where it was experienced. If one context exerted an influence, we would expect, for example, that the incentive value of the dish depends on the restaurant and not on the neighbourhood. Finally, if both contextual levels are in play, one would predict that different incentive values would be attributed to the same dish as a function of both the restaurant and neighbourhood. Here, we examined the possibility that incentive value depends on generative models where contexts are nested hierarchically. A higherlevel contextual variable (e.g., the neighbourhood) is represented by a Gaussian distribution with mean $\mu}_{HC$ equal to zero and variance $\sigma}_{HC}^{2$, from which a value HC is sampled. Sensory evidence about HC is provided by HO, which is sampled from a Gaussian distribution with mean HC and variance $\sigma}_{HO}^{2$. A lowerlevel contextual variable (e.g., the restaurant) is represented by a (Gaussian) distribution with mean HC and variance $\sigma}_{LC}^{2$, from which a value LC is sampled. Sensory evidence about LC is provided and represented by LO which is sampled from a Gaussian distribution with mean LC and variance $\sigma}_{LO}^{2$. A reward option is obtained and sampled from a Gaussian distribution with mean LC and variance $\sigma}_{R}^{2$. We propose that agents infer the posterior distribution P(LCHO,LO,R) sequentially by estimating, in the order, P(HCHO), P(LCHO), P(LCHO,LO), and P(LCHO,LO,R). This produces an equation for incentive value with the following form (see Appendix for derivation):
Three normalization effects are implicit here. The first (${\tau}_{LO}LO$) is a subtractive normalization proportional to the value LO observed at the low contextual level. A second one (${\tau}_{HO}HO$) is a subtractive normalization proportional to the value HO observed at the high contextual level. The terms τ represent precisiondependent weights and describe the relative precision of the lowlevel ($\tau}_{LO$) and highlevel ($\tau}_{HO$) cues. Finally, a third factor (K) implements divisive normalization and depends on a gain term which includes reward variance (see Appendix).
In summary, this Bayesian formulation outlines a principled theoretical explanation for how we contextualise rewards based on prior expectation and uncertainty with potential deep hierarchical structure. The key role of uncertainty is reflected in the precisionweighting of the prediction errors (e.g., outcome or reward prediction errors).
The proposal advanced here has some similarities with classical theories of value, such as Expected Utility theory (von Neumann and Morgenstern, 1944) and Prospect theory (Kahneman and Tversky, 1979). For example, there are convergences between the influence of the average reward in BCV and the impact of wealth on marginal utility as postulated in Expected Utility theory. Similarities exist also between the role of the average reward in BCV and the status quo notion in Prospect theory, which distinguishes between loss and profit. Indeed, in BCV profits can be conceived in terms of values larger than the expected reward and losses as values smaller than the expected reward.
We see a more direct link between BCV and recent economic models which postulate that incentive value is adapted to the statistics of the expected reward distribution (Bushong et al., unpublished; Kőszegi and Rabin, 2006; Kőszegi and Szeidl, 2013), which in turn depends on prior experience within an environment (Stewart et al., 2006; Stewart, 2009). These theories can be broadly classified into those based on subtractive normalization, which assume that incentive value corresponds to the reward minus a reference value (Kőszegi and Rabin, 2006), and those based on divisive normalization, prescribing that incentive value corresponds to the reward divided (or multiplied; Kőszegi and Szeidl, 2013) by either the expected reward (Louie et al., 2014, 2015) or the range of an expected distribution of rewards (Bushong et al., unpublished; Kőszegi and Szeidl, 2013).
BCV differs in important ways from previous theories in its attempt to derive contextual normalization from normative assumptions of Bayesian statistics. This approach conceives incentive value as precisionweighted prediction error and implies two forms of contextual adaptation (Figure 2). First, as in some previous theories (Kőszegi and Rabin, 2006), subtractive normalization emerges as the expected reward is subtracted from the actual reward. Second, the gain term implements divisive normalization, an aspect similar to a recent model in which the range of the reward distribution (which is analogous – though not identical  to the gain term) divides the reward (Bushong et al., unpublished). These predictions are specific and distinguish BCV from other models. For instance, BCV predicts that divisive normalization derives from the gain term (i.e., reward variance) and not from the expected reward (Louie et al., 2014, 2015), and that the reward variance divides – and not multiplies (Kőszegi and Szeidl, 2013) – the prediction error. Importantly, these predictions are not ad hoc but derive necessarily from Bayesian assumptions.
Experiment one
Data from conditions where BCV is applicable, namely those involving a single attribute and where context depends on past options (and not simultaneously presented options), are relatively scarce. Here, empirical evidence has shown a subtractive normalization, whereby incentive values are rescaled to the expected reward (Kőszegi and Rabin, 2006; Ludvig et al., 2013; Rigoli et al., 2016a, 2016b). In addition, there is an absence of evidence for a divisive normalization exerted by the expected reward (i.e., where values are divided by the expected reward; Rigoli et al., 2016a, 2016b). Both findings are consistent with BCV. However, another key prediction of BCV relies on a divisive normalization dependent on reward variance (Figure 2C–D), though it remains unknown whether such variancedependent normalization actually occurs. Here, we present data from a behavioural experiment where we investigate this very question.
Participants performed a computerbased decisionmaking task (Figure 3) in which a monetary amount, changing trial by trial, was presented in the centre of the screen and participants had to choose whether to accept half of it for sure or select a 50–50 gamble between the full monetary amount and zero, a scenario where the sure option and gamble always carry the same expected value (EV). The task was organized in blocks, each associated with one of two contexts which determined the possible EVs associated with the block. These EVs were £3 and £4 for the lowvariance context, and £2, £3, £4 and £5 for the highvariance context. Note that average choice EV was equal across contexts (i.e., £3.5). Contexts were cued by the associated EVs displayed on the top of the screen in brackets. To ensure incentive compatibility, at the end of the experiment one single outcome was randomly selected and paid out to participants.
In the analyses we focused on choices common to both contexts, namely involving £3 and £4 EV. p<0.05 was used as significance criterion and trials with reaction times slower than 3 s (and faster than 400ms) excluded. For these choices, the average gambling probability did not differ from 50% (mean = 49; SD = 23; t(35) = −0.31, p<0.76). In addition, gambling probability was equivalent when comparing £4 and £3 choices (t(35) = −0.81, p =0.43). Across individuals, there was no correlation between the average gambling probability for £4 and £3 and the difference in gambling probability between these two EVs (Figure 4C–D; r(36) = −0.06, p =0.75). The latter result replicates previous findings (Rigoli et al., 2016a, 2016b) and supports the idea of a differentiation between an average gambling propensity and a preference to gamble with large or small EV as determinant of risk choice.
We investigated key predictions of BCV regarding a divisive normalization (or precision weighting) effect exerted by context, which is captured by the gain term $\frac{{\sigma}_{C}^{2}}{{\sigma}_{C}^{2}+{\sigma}_{R}^{2}}$ in equation 2. We can formalize our context manipulation by varying the reward variance $\sigma}_{R}^{2$, which is larger in the highvariance context, implying a smaller gain term. Given that £4 and £3 are larger and smaller than the context average (i.e., £3.5) respectively, we predicted that (at option presentation) they induce a positive and negative prediction error respectively. Because of the gain term, BCV predicts (Figure 2C–D) that £4 is attributed a larger (i.e., more positive) incentive value in the lowvariance context while £3 is attributed larger (i.e., less negative) incentive value in the highvariance context. Note that in our task there are two types of variance. The first refers to the variance of possible outcomes of the gamble (which is perfectly correlated with the EV of options, as in Rigoli et al., 2016), and is not the focus of our study. The second refers to variance across options (i.e., the variance characterizing the distribution of successive options), which is what we experimentally manipulate and investigate. In the model, this is reflected in and affects the gain term in equation 2.
First, we tested our central predictions by analysing raw choice data. Though we observed no overall difference in gambling across participants for £4 and £3 EV (see above), participants could be differentiated based on those who gambled more with £4 or £3 EV. In line with previous observations (Rigoli et al., 2016a, 2016b), we predicted that the impact of context on gambling depended on a subjectspecific propensity to gamble more with large or small choice EVs; in other words on whether a participant prefers to gamble for £3 compared to £4 EV or vice versa. Combining this prediction with BCV predictions (Figure 2C–D), participants who risked more with increasing EVs would be expected to gamble more for £4 and less for £3 in the lowvariance context (when £4 and £3 would be attributed larger and smaller incentive value respectively) compared to the highvariance context. On the contrary, participants who risked more with decreasing EVs would be expected to gamble less for £4 and more for £3 in the lowvariance context compared to the highvariance context. To examine these predictions we tested for an interaction, corresponding to the differential gambling percentage across contexts (lowvariance minus highvariance context) for £4 choices minus the differential gambling percentage across contexts (lowvariance minus highvariance context) for £3 choices. Across participants, the interaction term did not differ from zero (t(35) = 0.43, p =0.67) but, consistent with our hypothesis, it showed a significant correlation with the gambling probability for £4 minus £3 choices (Figure 4A,B,E,F; r(36) = 0.45, p = 0.005). Note that this result remains significant when using a Spearman correlation, which is less affected by outliers (rho = 0.51, p = 0.002).
Next, we adopted a modelbased approach to assess whether BCV explains choice data. Following equation 2, if the option EV is R, then its associated incentive value will be:
Where $\epsilon$ is an indicator of the high ($\epsilon =1$) or lowvariance context ($\epsilon =0$), $\mu}_{C$ indicates the contextual mean (and is equal to £3.5 for both contexts), and $\tau$ is a free parameter (bounded within the 0.1–10 range) which implements a gain term and captures divisive normalization of reward. To connect value adaptation to choice, we used a logistic regression model of gambling where the probability of choosing the gamble does not depend on the objective option EV, but on the associated incentive value (i.e., transformed by contextual normalization):
Where α is a valuerelated parameter which determines whether gambling increases (α > 0) or decreases (α < 0) with larger incentive value and ρ represents a gambling bias parameter.
We used likelihood ratio tests (see Materials and methods) to compare this model with simpler (i.e., reduced) models, where one or more parameters were set to zero. Model comparison favoured the full model (comparison with: random model: χ^{2}(108) = 3268, p<0.001; model with α: = 2502, p<0.001; model with ρ: χ^{2}(72) = 1526, p<0.001; model with α and ρ: (36) = 704, p<0.001). In addition, the model predicts that the free parameter τ is smaller than one or log(τ) is less than zero. Consistent with BCV and variancedependent normalization, the mean of log(τ) was significantly smaller than zero (t(35) = −2.81, p = 0.008).
We next compared the full model with an alternative model without a subtractive normalization component (as postulated by BCV, where the expected reward is subtracted to the actual reward); namely, where the incentive value was equal to $V\left(R\right)={\tau}^{\epsilon}R$. The latter model derives from previous accounts of contextsensitive value (Bushong et al., unpublished; Kőszegi and Szeidl, 2013). We compared the negative loglikelihood of the two models (given that they had an equal number of free parameters) and found a smaller score for the model with the subtractive component (difference in loglikelihood: 254). We tested this difference by performing a Chisquare test with one degree of freedom (treating the model predicted by BCV as having an additional parameter). This test was significant (χ^{2}(1) = 508, p<0.001), meaning that the model implementing both subtractive and divisive normalization (derived from BCV and described by equation 8) fits the data better.
Finally, we used the (selected) model and subjectspecific parameter estimates of the last analysis to generate simulated choice behaviour and performed behavioural analyses on the ensuing data (data were simulated 100 times and the average statistics are reported). Consistent with real data, the full model replicated the lack of correlation between average gambling (for £4 and £3 choices) and the difference in gambling for £4 and £3 choices (average r(36) = −0.06, p = 0.73), while a correlation emerged when data were simulated using a model without the gambling bias parameter ρ (average r(36) = 0.51, p = 0.001). Moreover, and again consistent with empirical data, the full model replicated the correlation between gambling for £4 minus £3 choices and the contextEV interaction effect term (average r(36) = 0.48, p = 0.003), a result not obtained when data were simulated using a model without the valuefunction parameter α (average r(36) = 0.01, p = 0.95) or without the context parameter τ (average r(36) = 0.09, p = 0.60).
Collectively, these analyses validate the proposal of a divisive normalization component dependent on a gain term (in turn dependent on reward variance) consistent with BCV.
Experiment two
Another key prediction of BCV is that the generative model can reflect contexts organized hierarchically and that incentive value and choice are adapted to contextual information available at different hierarchical levels (Figure 2E–F). So far, there has been a focus on nonhierarchical settings (Kőszegi and Rabin, 2006; Ludvig et al., 2013; Rigoli et al., 2016a, 2016b), and therefore whether adaptation combines influence from context at multiple hierarchical levels remains unknown. Here, we present data from a behavioural experiment where we investigated this question.
Participants played a computerbased task (Figure 5) where, on each trial, two rectangles representing two decks of cards appeared. Each card was associated with a monetary reward, and the average card reward for each deck was displayed in brackets on the deck. The decks were coloured; one in grey and the other in blue. A card was pseudorandomly drawn from the blue deck and the corresponding monetary reward was presented in the middle of the screen. Participants had to choose between half of the monetary reward for sure and a gamble between the full reward and a zero outcome, each with 50% chance. Note that, as in experiment one, the two options carry the same EV. After making a choice, the outcome was then shown. To ensure incentive compatibility, at the end of the experiment, one outcome was randomly selected and paid out to participants.
The deck selected by the computer alternated pseudorandomly over blocks. In addition, two sets of decks alternated over longer blocks in a pseudorandom way. The first deckset (lowvalue deckset) comprised decks associated with an average of £5 and £7, the second (highvalue deckset) comprised decks associated with an average of £7 and £9. The cards of the £5 deck were associated with £3, £5 and £7; the cards of the £7 deck were associated with £5, £7 and £9; the cards of the £9 deck were associated with £7, £9 and £11. The aim of this experimental paradigm was to manipulate context at two hierarchical levels, a lowlevel associated with decks, and a highlevel associated with decksets. Note that the rewards overlapped between contexts at both levels; namely, across decks and decksets. In relation to decks, £5 and £7 cards were common to both decks in the lowvalue deckset, and £7 and £9 cards were common to both decks in the highvalue deckset. If this level of context exerted an influence, it should elicit changes in choice consistent with BCV when comparing choices based upon the same reward across decks. In relation to decksets, the deck associated with £7 average was present in both sets. If the deckset level exerted an influence, this would elicit changes in choice consistent with context sensitive values during the presentation of the £7 deck.
The average gambling percentage did not differ from fifty percent across subjects (mean = 54; SD = 16; t(31) = 1.48, p = 0.12). P<0.05 was used as significance criterion and trials with reaction times slower than 3 s (and faster than 400 ms) excluded. We assessed the impact of option EV on choice using a logistic regression model, where EV was included as regressor. The associated regression coefficient was not significantly different from zero across participants (t(31) = 1.67, p = 0.11). We then investigated the relationship between the average propensity to gamble and the effect of EV on gambling but did not find any correlation (Figure 7A; r(32) = 0.23, p = 0.21). This result again replicates previous studies using a similar paradigm (Rigoli et al., 2016a, 2016b), and highlights two different determinants of risk attitude; one linked with a baseline gambling propensity and the other linked with a preference to gamble with large or small reward amounts.
Though across participants, we observed no overall effect of option EV on gambling, participants could be differentiated based on those who showed a positive or negative effect of option EV on gambling. Based on previous findings (Rigoli et al., 2016a, 2016b), we hypothesized that context sensitive value would predispose participants who preferred to gamble for large EVs to gamble more when EVs were larger relative to contextual expectations, namely in lower value contexts (Figure 6). Similarly, we expected participants who preferred to gamble for small EVs to gamble more when EVs were smaller relative to contextual expectations, namely, in higher value contexts. We investigated this hypothesis both at the level of decks and decksets (Figure 6).
At the level of decks, we computed – for each deckset – the difference in gambling between lower and higher value decks for rewards common to both decks (corresponding to £5 and £7 in the lowvalue deckset, and to £7 and £9 in the highvalue deckset). The mean of these two differences correlated across subjects with the effect of EV on gambling (i.e., the associated regression coefficient of the logistic regression model; Figure 7C; r(32) = 0.55, p = 0.001), consistent with a contextualisation of reward by decks at the lower contextual level. At the level of decksets, we computed the difference in gambling between the low and high value deckset for the £7 deck (common to both decksets). This difference correlated across subjects with the effect of EV on gambling (i.e., the associated regression coefficient of the logistic regression model; Figure 7E; r(32) = 0.42, p = 0.018), consistent with a context sensitive value effect of the higher contextual level.
We next investigated whether BCV explains the context effects implicit in these results. Since, in our task, contexts are organized hierarchically (i.e., decks and decksets are associated with high and low levels – in the sense that the set determines the possible decks), we refer to the generative model shown in Figure 1C, where incentive value is described by equation 7. Recall that there are three normalization terms under this model (see equation 7): first a subtractive term (${\tau}_{Lo}LO$) proportional to the value LO observed at the low contextual level, second a subtractive term (${\tau}_{Ho}HO$) proportional to the value HO observed at the high contextual level, and third a divisive factor (K) dependent on precision. In our task, the average contextual reward was manipulated, enabling us to examine the subtractive terms outlined in equation 7. We did this by treating the contextual averages as observations of the true underlying contexts of the task (in equation 7, HO and LO would be associated with deckset and deck respectively). However, since reward variance (which enters the divisive normalization factor K) was not manipulated, this task is not suitable to quantify an effect of precisionweighting – and therefore we omitted this factor from the model of our empirical data. Specifically, based on equation 7 and omitting K, if the option EV is R, then its associated incentive value will be:
Where χ_{LO} indicates the average option EV for the deck (for £9 deck: χ_{LO} = 4.5; for £7 deck: χ_{L}_{O }= 3.5; for £5 deck: χ_{LO} = 2.5), χ_{HO}indicates the average option EV for the deckset (highvalue deckset: χ_{HO} = 4; lowvalue deckset: χ_{HO} = 3) τ_{LO} is a free parameter that mediates contextual effects at the deck level, and τ_{HO} is a free parameter that mediates contextual effects at the deckset level (see Materials and methods).
To connect value adaptation to choice, we used a logistic regression model as in equation 9 where α is a valuerelated parameter which determines whether gambling increases (α > 0) or decreases (α < 0) with larger incentive value and ρ represents a gambling bias parameter. We used a likelihood ratio test to compare this model with simpler (reduced) models where one or more parameters were set to zero. The full model was favoured (comparison with: random model: χ^{2}(128) = 2959, p<0.001; model with α: χ^{2}(96) = 1530, p<0.001; model with ρ: χ^{2}(96) = 1622, p<0.001; model with α and ρ: χ^{2}(64) = 262, p<0.001; model with α, ρ and τ_{LO}(32) = 134, p<0.001; model with α, ρ and τ_{HO}: χ^{2}(32) = 58, p<0.001). In addition, consistent with context sensitivity at both hierarchical levels, the contextrelated free parameters of the full model were significantly larger than zero (Figure 6; τ_{LO}: t(31) = 4.55, p<0.001; τ_{HO} t(31) =2.67, p = 0.012).
We next compared the full model with an alternative model where the context parameters (capturing the influence of the reward expected within a context at multiple hierarchical levels) divided the reward rather than being subtracted from the reward; in other words where the incentive value corresponds to:
The latter model derives from previous accounts of context sensitivity (Louie et al., 2014, 2015). We compared the negative loglikelihood of the two models (given they had an equal number of free parameters) and found a smaller score for the model with subtractive normalization (difference in loglikelihood for divisive minus subtractive model; full models: 39; for models with α, ρ and τ_{LO}: 13; for models with α, ρ and τ_{LH}: 25). This difference was tested statistically by performing a Chisquare test with one degree of freedom (treating the model predicted by BCV as having an additional parameter). This test was significant (full models: χ^{2}(1) = 78, p<0.001; for models with α, ρ and τ_{LO}: χ^{2}(1) = 26, p<0.001; for models with α, ρ and τ_{LH}: χ^{2}(1) = 50, p<0.001), meaning that the model implementing subtractive normalization (consistent with BCV) is a better explanation for the data.
Finally, we used the (selected) model and subjectspecific parameter estimates of the last analysis to generate simulated choice behaviour and performed behavioural analyses on the ensuing data (data were simulated 100 times and the average statistics are reported). As with real data, our modelling replicated the absence of correlation between average gambling and the effect of reward on gambling (i.e., the slope of the logistic regression; see above) (Figure 7B; average r(32) = 0.06, p = 0.76), while a correlation emerged when the data were simulated using a model without the gambling bias parameter ρ (average r(32) = 0.85, p<0.001). Moreover, consistent with empirical data, the full model replicated the correlation between (i) the effect of EV on gambling and the difference in gambling across decks for choices common to both decks of a deckset (combining both decksets; Figure 7D; average r(32) = 0.58, p<0.001), (ii) the effect of EV on gambling and the difference across decksets in gambling for the £7 deck (common to both decksets; Figure 7F; average r(32) = 0.57, p=0.001). These correlations were not replicated when data were simulated using a model without the valuerelated parameter α (first correlation: average r(32) = 0.09, p = 0.62; second correlation: average r(32) = =0.02, p = 0.89). Furthermore, the first correlation was not replicated when using a model without the parameter τ_{LO} (average r(32) = −0.15, p = 0.41) and the second correlation was not obtained when using a model without the parameter τ_{HO}(average r(32) = −0.03, p = 0.87).
Collectively, these analyses show subtractive normalization exerted by contextual effects at multiple hierarchical levels consistent with predictions from BCV.
Discussion
We propose a Bayesian scheme (BCV) as a model of contextual influences on incentive value attribution. BCV is based on Bayesian inference principles and on generative models of reward. Adopting two novel experimental designs, we provide behavioural evidence that supports two key predictions of BCV, namely that value attribution is affected by reward variance (which exerts divisive normalization) and by hierarchically organized contexts.
Our account is motivated by normative principles of Bayesian statistics – and fits within a Bayesian brain hypothesis framework (Chater et al., 2006; Clark, 2013; Dayan et al., 1995; Ernst, 2006; Friston, 2010). As such, it provides a principled account of decision making under uncertainty. In particular, it accommodates expectation and uncertainty that may have a deep hierarchical structure, as in real world situations. Bayesian schemes are based on a formal and a clear definition of the imperatives that motivate cognitive processes, which are conceived in terms of inference. This allows BCV to establish a link with Bayesian perspectives in other domains of cognitive neuroscience, helping unifying perspectives on brain functioning.
Our proposal is closely linked to the framework of planning as inference and active inference (Botvinick and Toussaint, 2012; Friston et al., 2013, 2015; Pezzulo and Rigoli, 2011; Pezzulo et al., 2015; Solway and Botvinick, 2012). This recasts decisionmaking and planning – usually understood in terms of value or utility maximization – as a form Bayesian inference, and hence can provide a unifying inferential account of perception and action. Hierarchical implementations of active inference schemes have been proposed previously, and the notion of hierarchicallyorganized contexts fits comfortably within these schemes (Pezzulo et al., 2015). BCV extends this framework by focusing on the determinants of incentive value, conceived as precisionweighted prediction error based on (potentially hierarchical) contextual expectations.
BCV postulates that the two fundamental determinants of incentive value are prediction error and precision. A prediction error is determined by the difference between the observed and expected reward which, in BCV, derives from integrating different expectations under contextual uncertainty. Gain depends on (relative) precision or confidence– and ensures that the prediction error is normalised and (Bayes) optimally weighted in relation to uncertainty about both context and reward. In brief, only precise prediction errors have an effect on expectations higher in the hierarchy during Bayesian belief updating. BCV predicts that precision exerts an influence in two ways. First, at the highest hierarchical levels, precision determines the optimal integration of multiple contextual representations–as it mandates that contexts characterized by a high precision (greater reliability) exert more influence on reward expectancy. For instance, if we assume that subjects have very precise beliefs about the lowlevel context (e.g., the deck), then the effect of the highlevel (e.g., the deck set) will disappear. Formally, this is because in the hierarchical model the lowlevel context constitutes a Markov blanket for the posterior expectation about the reward option (Bishop, 2006). In other words, the effect of the highlevel context tells us that if subjects are using a hierarchical model, there must be posterior uncertainty about the lowlevel context. Heuristically, even though they can see which deck they are currently playing with, they still nuance their expectations about this deck based upon the deckset from which it came. Second, at the lowest hierarchical level, the precision determines the gain assigned to the prediction error and hence is a direct determinant of incentive value. In our first experiment, we show evidence consistent with the latter expression of precision.
The central role attributed to precisionweighted prediction error is consistent with Bayesian models in other domains, and speaks to the idea of common computational principles in the brain. In fact, one central idea of many influential Bayesian proposals is that, when sensory inputs are presented, predictions are explained away and the resulting perception corresponds to (precisionweighted) prediction errors. For instance, predictive coding models are based on evidence that activity in certain brain regions responsible for perception reflects prediction errors and not raw sensory inputs (Friston, 2005; Rao and Ballard, 1999). Moreover, it has been proposed that our sense of agency depends on explaining away somatosensory predictions associated with motor commands, with unexplained sensations alone (i.e., residuals) attributed to external forces (Brown et al., 2013). A similar view characterizes active inference schemes, which assume that action is not steered by stimuli per se but by the (precisionweighted) prediction error elicited by those stimuli, expressing the extent to which they depart from expectation in a meaningful way (Friston et al., 2013, 2015).
Within BCV contextual representations can be hierarchically organized, with high levels characterized by more general conditions. Because reward options are descendants of all hierarchical levels, any context can exert an influence on incentive value, insofar as these levels determine the reward expected in a certain condition. Specifically, BCV can integrate–in a Bayes optimal way–contextindependent beliefs about the reward distribution with contextsensitive beliefs unfolding at multiple hierarchical levels. The possibility that subjects use hierarchical generative models is in fact supported by our empirical findings. Our results are consistent with the idea that rewards have larger incentive values when both high and lowlevel contexts are characterized by reward distributions with a smaller average. This indicates that information about more specific (e.g., the deck) and more general (e.g., the deckset) contexts are integrated to determine incentive values and choice behaviour.
A hierarchical nesting might explain why contextual effects observed in psychological experiments are usually substantial but not extreme. In other words, it is unlikely that 10 p will be attributed the same value as £100, even when the contextual manipulation may appear to induce an equivalence between the two quantities. This can be explained by contextual effects from the highest hierarchical level (e.g., that reward options have, in general, a prior expectation of zero). This supraordinate level can be conceived as representing a contextindependent distribution of rewards that may derive from the overall statistics of our environment (Stewart et al., 2006, 2009) and/or from innate prior beliefs about the distribution of incentives (Rigoli et al., 2016c, 2016d).
The proposal that incentive value corresponds to (precisionweighted) reward prediction error should not be confounded with the idea that it corresponds to the posterior reward expectation. Though both possibilities derive from Bayesian principles, they make opposite predictions about the role of prior reward expectancy. While the posterior reward expectation hypothesis predicts the larger value with larger prior reward expectancy, our data show larger value with smaller prior expectancy, consistent with the prediction error hypothesis presented here (see also Rigoli et al., 2016a, 2016b).
With reference to the three levels of analysis (i.e., computational, algorithmic and implementation) proposed by Marr (1982), BCV speaks to the computational level as it focuses on normative principles (implicit in optimal Bayesian inference) proposed to explain value and choice adaptation. In addition, BCV also has implications for the other levels and there are now several biologicallyplausible accounts of how Bayesian inference might be implemented in the brain (e.g., Doya et al., 2007; Friston, 2005; Hennequin et al., 2014; Knill and Pouget, 2004), where some accounts consider neuronal circuits (and generative models) characterized by a hierarchical organization (e.g., Friston, 2005). BCV fits comfortably within these biologicallyplausible accounts. Consistent with BCV are findings that several brain regions show a response to reward that adapts to both expected reward (e.g., in signalling reward prediction error) and reward range (Bermudez and Schultz, 2010; Cox and Kable, 2014; Louie et al., 2011; PadoaSchioppa, 2009; PadoaSchioppa and Assad, 2008; Park et al., 2012; Rigoli et al., 2016a; Tobler et al., 2005 Kobayashi et al., 2010; Tremblay and Schultz, 1999). Recent evidence for an association between neural and choice adaptation is also in line with BCV (Rigoli et al., 2016a). However, key neurobiological predictions of BCV, including the specific neural mechanisms that realize choice adaptation as well as the implementation of hierarchical generative models of reward, await further investigation.
Theoretical work has indicated that adaptation of neuronal responses is consistent with efficient coding, whereby the signalling of a finite pool of neurons – with a finite dynamic range of responses – can be optimally tuned to the statistics of stimuli in the environment, so as to maximize discriminability among the stimuli (Carandini and Heeger, 2012; Louie et al., 2015; Rangel and Clithero, 2012; Summerfield and Tsetsos, 2015.) This idea has now been extended to reward processing. Here proposals diverge as to the prediction of whether adaptive neuronal coding determines either stability or adaptation in choice behaviour (Louie et al., 2015; PadoaSchioppa and Rustichini, 2014; Rangel and Clithero, 2012; Summerfield and Tsetsos, 2015). In addressing this, BCV (similar to other Bayesian inference schemes for continuous variables) implements adaptive coding, because precisionweighted prediction error postulated to be signalled by valueprocessing neurons is a normalized quantity (Doya et al., 2007). Also in line with previous accounts (Louie et al., 2015; Rangel and Clithero, 2012; Summerfield and Tsetsos, 2015), BCV proposes that this normalized signal corresponds to incentive value, and that adaptive coding in the brain should be reflected in a behavioural choice adaptation. In other words, BCV implies that, as well as neural signalling, behaviour itself is tuned to the statistics of the incentives, so as to maximize discriminability among these incentives.
We highlight shortcomings of the model, though the framework itself may be fruitful in addressing some of these shortcomings. First, our focus is on scenarios where the incentive value depends on contextual information (either represented by cues or by previous rewards) provided before reward delivery. Another form of context effect on incentive value is induced by options that are simultaneously available (Louie et al., 2013, 2015). Further theoretical work is needed to link BCV and this form of influence, though a sequential inferential process (e.g. Bayesian belief updating), similar to the process described here (possibly linked to attention) might be involved in simultaneous contextual effects. Second, our focus has been on conditions where reward is defined by a single attribute (e.g., reward amount). Contextual influences (e.g., the decoy effect) can emerge when multiple dimensions need to be evaluated and integrated, as investigated by multiattribute theories (Huber et al., 1982; Johnson and Busemeyer, 2005; Roe et al., 2011; Simonson and Tversky, 1992; Soltani et al., 2012; Tsetsos et al., 2010, 2012, 2016; Tversky, 1972; Usher and McClelland, 2004). BCV can in principle be extended to these scenarios, for instance connecting to a body of work on multisensory integration using Bayesian principles (Ernst, 2006). This would provide an opportunity to model attentional processes determining an optimal weighting of different attributes based on their importance and reliability. Third, our current formulation assumes that the model parameters are given, while these parameters need to be learned in the first place. Questions about the mechanisms that might underpin learning of generative models adopted for Bayesian inference are still largely open, though substantial contributions exist particularly in the context of structure learning (Acuna and Schrater, 2009; Behrens et al., 2007; Collins and Frank, 2013; Courville et al., 2006; FitzGerald et al., 2014; Gershman and Niv, 2010; Mathys et al., 2011).
Here, we have assumed that variables of the generative model are Gaussian. This allows us to present the model in a simple and clear way, as posterior beliefs can be inferred analytically with relatively simple equations as adopted in standard decisionmaking schemes (Rescorla and Wagner, 1972). Though Gaussian assumptions are probably an oversimplification, with appropriate adjustments BCV can be extended to generative models with nonGaussian variables (Jazayeri and Shadlen, 2010). Indeed, the arguments behind BCV can be applied to any variables with an exponential distribution. However, the key idea (tested in our experiments – and here derived from Gaussian assumptions) that reward average and variance elicit subtractive and divisive normalization, respectively, is quite general and can also be applied, for instance, to uniform (and, in general, nonskewed) distributions.
Finally, there are questions related to psychopathology that can be fruitfully formulated in terms of BCV, for example addiction. Consider the consequences of drug misuse, including the development of tolerance (i.e., the need of increased dosages to obtain the same effects as those obtained previously) and the lack of satisfaction when engaging in activities that were pleasurable before the development of addiction. BCV interprets these effects in terms of increases in expected reward (following drug misuse) that decreases the incentive value of rewards, including the drug itself and other motivational stimuli. A similar explanation has been proposed by classical homeostatic theories, where ingestion of the drug is conceived in terms of a means to reestablish a biological set point (e.g., expressed in baseline activity of dopamine neurons), coupled with the fact that the repeated drug misuse raises this set point. BCV formalise and extend the setpoint model. First, it relaxes the homeostatic assumption because incentive value depends on a reference point (i.e., the reward average), but does not correspond to distance from a set point as in homeostatic schemes. Indeed, setpoint models predict that drug consumption always reduces a negative affect state by reestablishing a set point. Conversely, BCV suggests that drug consumption can decrease a negative state (when the drugassociated outcome is worse than expected) but also induce a positive affect (when the drugassociated outcome is better than expected), a prediction more consistent with empirical evidence (Robinson and Berridge, 2000). Second, leveraging a Bayesian framework, BCV assigns a crucial role to reward uncertainty, above and beyond a role assigned to expected reward. For instance, increased uncertainty over prior reward beliefs may boost the magnitude of the (positive) prediction error elicited by drug consumption, hence enhancing individual predisposition to drug addiction.
In summary, we introduce a normative Bayesian model to explain the influence of contexts on incentive values. Key features of this account include an explicit generative model of reward and the assumption that incentive value corresponds to precisionweighted prediction error. This formulation implies that incentive value is intrinsically contextdependent. We tested key predictions of the model in two human experiments and show choice behaviour consistent with an adapting incentive value based on reward variance and on an average reward expected after integrating contexts at two hierarchical levels, one more general and the other more specific. An important consideration is that expression of context effect, though apparently irrational, can derive from Bayes (optimal) inference. Indeed, if incentive values are (precisionweighted) prediction errors, they are necessarily contextdependent, and this dependency can be described under a Bayes optimal scheme. We argue that this approach could be useful in generating new empirical predictions and in explaining phenomena in psychopathologies characterized by dysfunctional value attribution, such as addiction.
Materials and methods
Participants
36 healthy righthanded adults (19 females; 20–40 age range; mean age 26) participated in experiment one, and 32 healthy righthanded adults (18 females, aged 20–40, mean age 27) participated in experiment two. All participants had normal or correctedtonormal vision. None had a history of head injury, a diagnosis of any neurological or psychiatric condition, or was currently on medication affecting the central nervous system. The first experiment was conducted at the Wellcome Trust Centre for Neuroimaging at the University College London and was approved by the University College of London Research Ethics Committee. The second experiment was conducted at the Institute of Psychiatry, Psychology & Neuroscience at the King’s College of London and was approved by the King’s College of London Research Ethics Committee. All participants provided written informed consent and were paid for participating.
Experimental paradigm and procedure
Experiment one
Participants performed a computerbased decisionmaking task lasting approximately 30 min. A monetary amount (referred as trial amount), changing trial by trial, was presented in the centre of the screen and participants had to choose whether to accept half of it for sure (pressing a left button) or select a gamble (pressing a right button). The outcomes of this choice were either zero or the full monetary amount, each with equal probability, ensuring the sure option and gamble always had the same EV. The task was organized in blocks, each associated with one of two contexts which determined the possible EVs associated with the block. These EVs were £3 and £4 for the lowvariance context, and £2, £3, £4 and £5 for the highvariance context. Note that average choice EV was equal across contexts (i.e., £3.5). Contexts were cued by the associated EVs displayed on the top of the screen in brackets. Before a new block started, the statement 'New set' appeared for two seconds, followed by the contextual cue for two seconds. Next, the trial amount of the first trial was displayed followed, after the subject made a choice, by the outcome shown for one second. The possible contextual cue remained on the screen during an inter trial interval that lasted one and a half seconds. Participants had three seconds to make their choices; otherwise the statement 'too late' appeared and they received a zero outcome amount.
Eight blocks were presented, alternating between a low and highvariance context with the order counterbalanced across subjects. The former blocks comprised 40 trials each and the latter blocks comprised 80 trials, in such a way that the EVs common to both contexts (i.e., £3 and £4) were shown an equal amount of time in the two contexts. The order of trial amounts and outcomes were pseudorandomized. At the end of the experiment, one outcome was randomly selected among those received and added to an initial participation payment of £5. Before the task, participants were fully instructed both on task contingencies and payment rules.
Experiment two
Participants played a computerbased task lasting approximately 40 min. On each of the 480 trials, two rectangles representing two decks of cards appeared, one on the top and the other on the bottom of the screen. Each card was associated with a monetary amount, and the average amount of each deck was displayed in brackets upon the deck. The decks were coloured: one in gray and the other in blue and were shown for 1.5 s. On each trial a card was pseudorandomly drawn from the blue deck and the corresponding monetary amount was presented in the middle of the screen. Participants had to choose between half of the monetary amount for sure (pressing a right button) and a gamble between the full amount and a zero outcome (pressing a left button), each with 50% chance. After choosing, the outcome appeared for one second and a new trial started immediately. If no response occurred before three seconds, a statement 'too late' was presented for one second, resulting in a zero outcome.
Among the two decks shown on the screen, the selected deck (coloured in blue) alternated pseudorandomly over blocks (each including 5 trials). In addition, at some points during the task, the decks were replaced by new decks. Two sets of decks alternated over blocks of 20 trials in a pseudorandom way. The first deckset (lowvalue deckset) comprised decks returning £5 and £7 on average, the second deckset (highvalue deckset) comprised decks returning £7 and £9 on average. The cards of the £5 deck could be associated with £3, £5 and £7; the cards of the £7 deck could be associated with £5, £7 and £9; the cards of the £9 deck could be associated with £7, £9 and £11. When a new deck was selected (i.e., it was coloured in blue), decks were shown for 2.5 s before the card amount appeared; when a new deckset appeared, decks were shown for 4.5 s before the card amount appeared. At the end of the experiment, one of the outcomes was randomly selected by the computer, added to an initial payment of £5 and the total amount was paid to participants. The participants were fully instructed about task rules and about the way payment was carried out prior to task performance.
Behavioural modelling
The free parameters of the models were estimated separately for each subject using fminsearchbnd function of the Optimization toolbox in Matlab. Parameters were constrained within the following ranges: −5 and 5 for α (in both experiments), −10 and 10 for ρ (in both experiments), 0.1 and 10 for τ (in experiment one), −1 and 1 for τ_{LC} and τ_{HC} (in experiment two). In addition, to minimize the effect of biased outlier estimates, Gaussian priors with mean zero (Daw, 2011) were used for estimation of log(τ) (in experiment one) and estimation of τ_{LC} and τ_{HC} (in experiment two). Starting values for parameter estimation was zero for all parameters, except for the context parameter τ in experiment one for which it was one. Distributions of estimated parameters are reported in Figure 4—figure supplements 1 (experiment one) and Figure 7—figure supplements 1 (experiment two), and show no evidence of outliers (i.e., scores larger or smaller than 3 SD compared to the mean) for any of the parameters.
For each model, the loglikelihood of the choice data given the best fitting parameters (estimated by the method described above) was computed subject by subject and summed across subjects. We compared the full model with nested models, namely where one or more parameters were fixed to zero. To do this, we used the standard approach of the likelihoodratio test (Casella and Berger, 2002; Daw, 2011), which allows for a comparison of nested models. This is based on the fact that the difference in negative loglikelihood times two (2d) between a nested and a more complex model follows a chisquare distribution, where the number of degrees of freedom is equal to the number of additional parameters of the more complex model. A chisquare test can be performed to estimate the probability that the observed 2d is due to chance under the null hypothesis that data are generated by the nested model, allowing acceptance or rejection of the null hypothesis.
Appendix
We derive equation 7 from the generative model shown in Figure 1C. A higherlevel contextual variable (e.g., the neighbourhood) is represented by a Gaussian distribution with mean $\mu}_{HC$ equal to zero and variance $\sigma}_{HC}^{2$, from which a value HC is sampled. Sensory evidence about HC is provided and represented by HO which is sampled from a Gaussian distribution with mean HC and variance $\sigma}_{HO}^{2$. A lowerlevel contextual variable (e.g., the restaurant) is represented by a (Gaussian) distribution with mean HC and variance $\sigma}_{LC}^{2$, from which a value LC is sampled. Sensory evidence about LC is provided and represented by LO which is sampled from a Gaussian distribution with mean LC and variance $\sigma}_{LO}^{2$ A reward is obtained and sampled from a Gaussian distribution with mean LC and variance $\sigma}_{R}^{2$. We propose that agents infer the posterior distribution P(LCHO,LO,R) sequentially by estimating, in the order, P(HCHO), P(LCHO), P(LCHO,LO), and P(LCHO,LO,R). The posterior mean of P(HCHO) is:
And the posterior variance:
The posterior mean of P(LCHO) is equal to ${\mu}_{HCHO}({\mu}_{LCHO}={\mu}_{HCHO})$ , while the posterior variance is:
The posterior mean of P(LCHO,PO) is:
And the posterior variance:
Finally, the posterior mean of P(LCHO,LO,R) is:
Implying (with few algebraic transformations) the following incentive value for the reward:
This equation implements three normalization factors: (i) a subtractive normalization factor $\left(\frac{{\sigma}_{LCHO}^{2}}{{\sigma}_{LCHO}^{2}+{\sigma}_{LO}^{2}}LO\right)$ proportional to the value LO observed at the low contextual level, (ii) a subtractive normalization factor $\left(\frac{{\sigma}_{LO}^{2}}{{\sigma}_{LCHO}^{2}+{\sigma}_{LO}^{2}}\frac{{\sigma}_{HC}^{2}}{{\sigma}_{HC}^{2}+{\sigma}_{HO}^{2}}HO\right)$ proportional to the value HO observed at the high contextual level, (iii) a divisive normalization factor $\left(\frac{{\sigma}_{LCHO\phantom{\rule{0ex}{0ex}},LO}^{2}}{{\sigma}_{LCHO\phantom{\rule{0ex}{0ex}},LO}^{2}+{\sigma}_{R}^{2}}\right)$ that captures the weighting dependent on the (relative) reward variance. If we define the three factors as and and K respectively, we obtain equation 7.
References

1
Structure learning in human sequential decisionmakingPLOS Computational Biology 6:1001003–1001008.https://doi.org/10.1371/journal.pcbi.1001003

2
Learning the value of information in an uncertain worldNature Neuroscience 10:1214–1221.https://doi.org/10.1038/nn1954

3
Reward magnitude coding in primate amygdala neuronsJournal of Neurophysiology 104:3424–3432.https://doi.org/10.1152/jn.00540.2010
 4

5
Spatiotemporal prediction modulates the perception of selfproduced stimuliJournal of Cognitive Neuroscience 11:551–559.https://doi.org/10.1162/089892999563607

6
Planning as inferenceTrends in Cognitive Sciences 16:485–488.https://doi.org/10.1016/j.tics.2012.08.006

7
Active inference, sensory attenuation and illusionsCognitive Processing 14:411–427.https://doi.org/10.1007/s1033901305713

8
Normalization as a canonical neural computationNature Reviews Neuroscience 13:51–62.https://doi.org/10.1038/nrn3136
 9

10
Probabilistic models of cognition: conceptual foundationsTrends in Cognitive Sciences 10:287–291.https://doi.org/10.1016/j.tics.2006.05.007
 11

12
Whatever next? Predictive brains, situated agents, and the future of cognitive scienceThe Behavioral and Brain Sciences 36:181–204.https://doi.org/10.1017/S0140525X12000477

13
Cognitive control over learning: creating, clustering, and generalizing taskset structurePsychological Review 120:190–229.https://doi.org/10.1037/a0030852

14
Bayesian theories of conditioning in a changing worldTrends in Cognitive Sciences 10:294–300.https://doi.org/10.1016/j.tics.2006.05.004

15
BOLD subjective value signals exhibit robust range adaptationJournal of Neuroscience 34:16533–16543.https://doi.org/10.1523/JNEUROSCI.392714.2014

16
Trialbytrial data analysis using computational modelsDecision Making, Affect, and Learning: Attention and Performance XXIII 23:3–38.https://doi.org/10.1093/acprof:oso/9780199600434.003.0001
 17

18
Bayesian Brain: Probabilistic Approach to Neural Coding and LearningCambridge, MA: MIT Press.

19
Human Body Perception From the Inside Outp 105–p 131, A Bayesian view on multimodal cue integration, Human Body Perception From the Inside Out.

20
Model averaging, optimal inference, and habit formationFrontiers in Human Neuroscience 8:457.https://doi.org/10.3389/fnhum.2014.00457

21
Active inference and epistemic valueCognitive Neuroscience 6:187–214.https://doi.org/10.1080/17588928.2015.1020053

22
The anatomy of choice: active inference and agencyFrontiers in Human Neuroscience 7:598.https://doi.org/10.3389/fnhum.2013.00598

23
A theory of cortical responsesPhilosophical Transactions of the Royal Society B: Biological Sciences 360:815–836.https://doi.org/10.1098/rstb.2005.1622

24
The freeenergy principle: a unified brain theory?Nature Reviews Neuroscience 11:127–138.https://doi.org/10.1038/nrn2787

25
Learning latent structure: carving nature at its jointsCurrent Opinion in Neurobiology 20:251–256.https://doi.org/10.1016/j.conb.2010.02.008

26
Advances in Neural Information Processing Systems2240–2248., Fast samplingbased inference in balanced neuronal networks, Advances in Neural Information Processing Systems.

27
Adding asymmetrically dominated alternatives: violations of regularity and the similarity hypothesisJournal of Consumer Research 9:90–98.https://doi.org/10.1086/208899

28
Temporal context calibrates interval timingNature Neuroscience 13:1020–1026.https://doi.org/10.1038/nn.2590

29
A dynamic, stochastic, computational model of preference reversal phenomenaPsychological Review 112:841–861.https://doi.org/10.1037/0033295X.112.4.841

30
Prospect theory: An analysis of decision under riskEconometrica 47:263–291.https://doi.org/10.2307/1914185

31
The Bayesian brain: the role of uncertainty in neural coding and computationTrends in Neurosciences 27:712–719.https://doi.org/10.1016/j.tins.2004.10.007

32
Adaptation of reward sensitivity in orbitofrontal neuronsJournal of Neuroscience 30:534–544.https://doi.org/10.1523/JNEUROSCI.400909.2010

33
A model of referencedependent preferencesThe Quarterly Journal of Economics 121:1133–1165.

34
A model of focusing in economic choiceThe Quarterly Journal of Economics 128:53–104.https://doi.org/10.1093/qje/qjs049

35
Adaptive neural coding: from biological to behavioral decisionmakingCurrent Opinion in Behavioral Sciences 5:91–99.https://doi.org/10.1016/j.cobeha.2015.08.008

36
Reward valuebased gain control: divisive normalization in parietal cortexJournal of Neuroscience 31:10627–10639.https://doi.org/10.1523/JNEUROSCI.123711.2011

37
Normalization is a general neural mechanism for contextdependent decision makingProceedings of the National Academy of Sciences of the United States of America 110:6139–6144.https://doi.org/10.1073/pnas.1217854110

38
Dynamic divisive normalization predicts timevarying value coding in decisionrelated circuitsJournal of Neuroscience 34:16046–16057.https://doi.org/10.1523/JNEUROSCI.285114.2014

39
Extreme Outcomes Sway Risky Decisions from ExperienceJournal of Behavioral Decision Making 27:146–156.https://doi.org/10.1002/bdm.1792

40
Vision: A Computational Investigation Into the Human Representation and Processing of Visual InformationNew York: Freeman.

41
A bayesian foundation for individual learning under uncertaintyFrontiers in Human Neuroscience 5:39.https://doi.org/10.3389/fnhum.2011.00039
 42

43
Rational attention and adaptive coding: A puzzle and a solutionAmerican Economic Review, 104, 10.1257/aer.104.5.507.

44
Rangeadapting representation of economic value in the orbitofrontal cortexJournal of Neuroscience 29:14004–14014.https://doi.org/10.1523/JNEUROSCI.375109.2009

45
Adaptive coding of reward prediction errors is gated by striatal couplingProceedings of the National Academy of Sciences of the United States of America 109:4285–4289.https://doi.org/10.1073/pnas.1119969109

46
Active Inference, homeostatic regulation and adaptive behavioural controlProgress in Neurobiology 134:17–35.https://doi.org/10.1016/j.pneurobio.2015.09.001

47
The value of foresight: how prospection affects decisionmakingFrontiers in Neuroscience 5:79.https://doi.org/10.3389/fnins.2011.00079

48
Value normalization in decision making: theory and evidenceCurrent Opinion in Neurobiology 22:970–981.https://doi.org/10.1016/j.conb.2012.07.011
 49

50
Classical Conditioning II: Current Research and Theory64–99, A theory of Pavlovian conditioning: Variations in the effectiveness of reinforcement and nonreinforcement, Classical Conditioning II: Current Research and Theory.
 51

52
Dopamine Increases a ValueIndependent Gambling PropensityNeuropsychopharmacology, 10.1038/npp.2016.68.
 53

54
The dopaminergic midbrain mediates an effect of average reward on pavlovian vigorJournal of Cognitive Neuroscience 13:1–15.https://doi.org/10.1162/jocn_a_00972

55
The psychology and neurobiology of addiction: an incentivesensitization viewAddiction 95 Suppl 2:S91–S117.
 56

57
Choice in Context: Tradeoff Contrast and Extremeness AversionJournal of Marketing Research 29:281.https://doi.org/10.2307/3172740

58
A rangenormalization model of contextdependent choice: a new model and evidencePLOS Computational Biology 8:e1002607.https://doi.org/10.1371/journal.pcbi.1002607

59
Goaldirected decision making as probabilistic inference: a computational framework and potential neural correlatesPsychological Review, 119, 10.1037/a0026435.
 60

61
Prospect relativity: How choice options influence decision under riskJournal of Experimental Psychology: General 132:23–46.https://doi.org/10.1037/00963445.132.1.23

62
Decision by sampling: The role of the decision environment in risky choiceThe Quarterly Journal of Experimental Psychology 62:1041–1062.https://doi.org/10.1080/17470210902747112
 63

64
Do humans make good decisions?Trends in Cognitive Sciences 19:27–34.https://doi.org/10.1016/j.tics.2014.11.005
 65
 66

67
Salience driven value integration explains decision biases and preference reversalProceedings of the National Academy of Sciences of United States of America 109:9659–9664.https://doi.org/10.1073/pnas.1119569109

68
Economic irrationality is optimal during noisy decision makingProceedings of the National Academy of Sciences of United States of America. 113:3102–3107.https://doi.org/10.1073/pnas.1519157113

69
Preference reversal in multiattribute choicePsychological Review, 117, 10.1037/a0020580.

70
Elimination by aspects: A theory of choicePsychological Review 79:281–299.https://doi.org/10.1037/h0032955

71
Loss aversion and inhibition in dynamical models of multialternative choicePsychological Review, 111, 10.1037/0033295X.111.3.757.

72
Does the brain calculate value?Trends in Cognitive Sciences 15:546–554.https://doi.org/10.1016/j.tics.2011.09.008
 73
Decision letter

Sam GershmanReviewing Editor; Harvard University, United States
In the interests of transparency, eLife includes the editorial decision letter and accompanying author responses. A lightly edited version of the letter sent to the authors after peer review is shown, indicating the most substantive concerns; minor comments are not usually included.
Thank you for submitting your article "A Bayesian model of contextsensitive value attribution" for consideration by eLife. Your article has been favorably evaluated by Sabine Kastner (Senior Editor) and three reviewers, one of whom, Sam Gershman (Reviewer #1), also served as Guest Reviewing Editor.
The reviewers have discussed the reviews with one another and the Reviewing Editor has drafted this decision to help you prepare a revised submission.
Summary:
This work develops a Bayesian theory of context effects in decision making, and tests it in a behavioral experiment. The proposed model goes beyond previous efforts by introducing multiple levels of hierarchy. The reviewers agree that this is an interesting and important topic, but have concerns about the analysis methodology, presentation of the model/results, adequacy of the experimental data, and relation to previous work.
Essential revisions:
1) Analysis methodology: please address the points raised by Reviewers 1 and 3 concerning statistical analyses and model comparison.
2) Presentation of the model/results: all the reviewers suggested ways in which the presentation could be improved, including clarification of modeling details and task description.
3) Adequacy of experimental data: Reviewer 2 raised important concerns about whether the reported experiment provides a strong test of the theory (particularly the precisionweighting component), and Reviewer 3 pointed out a number of confounds in the design. The reviewers are in agreement that the paper should include new experimental data that addresses the confounds and ideally also provides a stronger test of the theory.
4) Relation to previous work: Reviewer 1 points out several other theoretical frameworks for certain context effects. Please address these in the revision.
Reviewer #1:
1) I didn't see what would seem to be the most obvious analysis of the choice data, namely comparing the LV vs. HV bars (as well as 5 vs. 9 bars) in Figure 4B. The authors report correlations but shouldn't the model predict a difference in means as in Figure 2?
2) The simulation details were unclear to me. Did the authors simulate a synthetic dataset many times and then average the correlations, or did they just run a single simulation? I think the former procedure is better justified, since we don't know whether the results found with the simulations are idiosyncratic or reliable.
3) Some researchers, such as Rangel & Clithero (2012) and Louie, Glimcher & Webb (2015), have drawn connections between context effects and efficient coding, appealing to the idea that divisive normalization is a mechanism for removing statistical redundancies. A Bayesian theory like BCV enables but does not require efficient coding; a given distribution could be coded with varying degrees of efficiency. However, BCV also appears to make some claims about mechanistic implementation which might be relevant to the question of efficiency. How do the authors see the relationship between these theories and BCV? More generally, the authors could deepen their contribution by considering the realization of BCV across Marr's levels.
4) Showing that BCV can account for earlier value normalization results would also bolster the theory. In addition, there is a rich literature on context/decoy effects that would be relevant to at least mention here. Relatedly, work in economics has studied the idea that reference points depend on expectations (e.g., Koszegi & Rabin, 2006), and there have been a number of important recent papers on range effects and relativistic choice processes (e.g., Bordalo et al., 2012; Cunningham, 2013,; Bushong, Rabin & Schwartzstein, 2015). It would be illuminating to better understand how these frameworks relate to BCV.
Reviewer #2:
My major reservation about the paper is that the human data offers only partial confirmation for the Bayesian context model. In its full specification (in the Appendix), the BCV model predicts that incentive value will depend on a divisive scaling term (implementing precision weighting dependent on relative reward variance) and subtractive prediction error terms (low and high context reward predictions, weighted by their relative contextual cue variances). However, the effects of precision – which are a key element of Bayesian approaches – are untested in the analysis of the experimental data. As the authors state in the paper, reward variance is not manipulated and the divisive term (K) cannot be examined. In addition, the subtractive terms should also be precisionweighted by terms relating posterior and prior variances (denoted by tau_LO and tau_HO). However, the results only speak to overall context effects, essentially asking whether there is an overall effect of low and high context on gambling (population regression effects in Figure 5, and the chisquare tests showing positive tau_LO and tau_HO parameters).
The issue is that adaptive effects to average rewards are well known (for example, successive contrast effects in the animal literature and reference point models like prospect theory); without validating the precisiondependent predictions of their model, I'm not sure that the authors can convincingly argue that BCV is a more appropriate model – particularly as other models are not tested. Given the experimental setup, testing the divisive weighting term is not possible in this dataset; however, can the authors make any predictions about not just the significance but the relative magnitude of the weighting factors tau_LO and tau_HO (predicted and fit to data)?
Reviewer #3:
1) Behavioural effects of reward context on incentive value are already wellestablished and even the specific paradigm used here has already been published by the same authors. Is there any new, surprising behavioural effect that follows from the new model?
2) The Bayesian model presented in the manuscript is not formally compared to other wellestablished models that may similarly account for the behavioural effects. The authors should show with formal model comparisons that their model outperforms other classic (nonBayesian) models commonly employed to model context effects on valuebased choice.
3) The model is not biologically realistic. This is not always a problem; in fact, there are many elegant demonstrations that Bayesian frameworks can account for optimal performance in various domains better than other accounts. However, in the specific context of rewardguided decisionmaking, it is unclear why and how a Bayesian framework should apply, and to what degree it is more consistent with behaviour and the underlying neural computations. Please provide a lot more information on how this model may be implemented by neural computations. In particular, it would help if there was any empirical evidence for the hierarchical representation of reward context.
4) The Abstract claims that the model "generates new empirical predictions and may help explain important phenomena in psychopathologies such as addiction." I found the corresponding text in the Discussion rather vague. Please provide explicit predictions for specific experimental effects that follow from this model and please explain much more concretely which important phenomena in psychopathologies are explained by it.
5) The manuscript claims in several places that a reward's incentive value corresponds to the (precisionweighted) prediction error. This is misleading. By definition, the incentive value is the property of a stimulus/expected reward that triggers approach behaviour and choice of the corresponding option. This representation must therefore be computed before the choice is taken and the reward is obtained. The prediction error, by contrast, is the deviation of the reward obtained as a consequence of the choice from the reward expected prior to the choice. This postchoice representation can therefore not be the incentive value guiding choice. The authors need to clarify their terminology and ensure that they remain consistent with established definitions in the literature.
In addition to the conceptual points listed above, the manuscript also has shortcomings with respect to methodology and results presentation that will need to be addressed:
6) The behavioural task was not designed to allow proper tests of the full model. Some of these problems are listed by the authors themselves and have led to adaptations of the model so that it could be fit to the data. For instance, reward variance in the different decks is not varied, is heavily constrained (there are only 3 different reward values per deck), and is perfectly correlated with average reward magnitude. To properly test whether the precision of reward prediction errors established by the different contexts really plays an important role, the authors should fit the full model to datasets with contexts that differ substantially in their reward variance and that disentangle reward magnitude from reward variance. Moreover, the contexts should be associated with a lot more than just three possible reward values so that the form of the expected reward distributions can be properly approximated (see below).
7) The model specification does not match the environment established by the behavioural task. The decks were associated with 3 equiprobable values per deck. Therefore, "smart" subjects would employ a flat discrete expectation of the three possible values within the given context. In contrast, the modelling solution is based on continuous (Gaussian) distributions that are not restricted by the bounds imposed by each context. There are two problems with this: (1) If we assume that the subjects indeed optimally integrate all information, then a flat prior belief bounded by the context's minimum and maximum reward would be accurate. This cannot be modelled by the presented specification. (2) Even if subjects employed continuous Gaussian priors to model reward expectations, such distributions would probably not be narrow enough with respect to the context bounds (i.e., the priors would wrongly lead to expectations of rewards that are outside of the bounded scale). These problems will probably be evident if the authors report the values of the latent variables after fitting the model to the empirical data. Please include such a table to allow the reader to inspect this issue.
In my view, if the authors really wanted to maintain a Bayesian optimal observer model, then they should examine how the prior (potentially flat, but can also be modelled if the authors like) is combined with the likelihood of the actual numeric representation to obtain a posterior estimate that should naturally occur within the actual numeric bounds of the context (for an example on how to formally deal with Bayesian problems of this kind, i.e. bounded contexts, see Jazayeri and Shadlen 2010, Nature Neuroscience). This formal specification can then be expanded to the interesting contextual hierarchical framework that the authors propose in their study.
8) The authors rescaled the contextual averages and reward values to perform their model fits. Why? A correctly specified model should be able to take as inputs the actual values of the contexts and rewards of their behavioural paradigm (which are all singledigit numbers after all). This would help to assess the model's explanatory power.
9) The authors perform model comparison by summing loglikelihoods across participants. I find the selection of this approach for model comparison surprising, given that several of the coauthors have pushed the use of precise Bayesian model selection methods that properly account for the complexity and variability of the model fits across trials and participants. The authors should employ such methods and should provide values quantifying the quality of the model fits after penalizing for model complexity.
10) The behavioural task is described in a fashion that makes it hard to replicate. For instance, is it true that on every trial, a card was drawn from the blue deck? If so, how were the different deck contexts varied across the different colours? Please make sure the task described in sufficient detail so that another person could program it.
https://doi.org/10.7554/eLife.16127.014Author response
Essential revisions:
1) Analysis methodology: please address the points raised by Reviewers 1 and 3 concerning statistical analyses and model comparison.
We believe that all the points concerning statistical analyses and model comparison have been addressed satisfactorily. Please see specific replies to reviewers below.
2) Presentation of the model/results: all the reviewers suggested ways in which the presentation could be improved, including clarification of modeling details and task description.
We are grateful for the feedback from reviewers on this aspect, and we have clarified the presentation of the model and task description (see replies below).
3) Adequacy of experimental data: Reviewer 2 raised important concerns about whether the reported experiment provides a strong test of the theory (particularly the precisionweighting component), and Reviewer 3 pointed out a number of confounds in the design. The reviewers are in agreement that the paper should include new experimental data that addresses the confounds and ideally also provides a stronger test of the theory.
Thanks for the feedback on this point. We have clarified why we believe that the experiment is adequately designed to investigate a key prediction of BCV in relation to contextual influences at multiple hierarchical levels. However, we also agree that the previous experiment is not suitable to test other key predictions of the theory; most importantly the effect of precision as suggested by reviewers.
Investigating the role of precision is something we already had in mind when we had completed this study. Indeed, we already had run an experiment investigating the role of context precision in value adaptation. As the reviewers have highlighted this point we are delighted to be able to include these new data in our revision. These data provide supportive evidence for a role of context precision, consistent with BCV. As pointed out by the reviewers, such role is a key feature of Bayesian schemes and is something which specifically supports BCV.
In sum, we now report a new experiment (along the lines suggested by the reviewers) investigating the role of reward variance (experiment one) together with the previous experiment investigating the role of hierarchical contexts (experiment two).
4) Relation to previous work: Reviewer 1 points out several other theoretical frameworks for certain context effects. Please address these in the revision.
Thanks for feedback on this point. We now link our work more extensively with other theoretical frameworks (and we compare explicitly predictions from different frameworks; see replies below).
Reviewer #1:
1) I didn't see what would seem to be the most obvious analysis of the choice data, namely comparing the LV vs. HV bars (as well as 5 vs. 9 bars) in Figure 4B. The authors report correlations but shouldn't the model predict a difference in means as in Figure 2?
Thanks for the feedback on this point, and we agree this needs to be explained better. Figure 2 shows predictions about incentive value V(R) postulated by BCV in the different conditions of our experiments. Crucially, the predicted V(R) does not correspond to the predicted gambling proportion, as (based on previous findings; Rigoli et al., 2016a; 2016b) we expected the gambling proportion to depend on V(R) in a different way for different participants. Specifically, we predicted that the incentive value determines choice behaviour according to a logistic regression P(gambling) = sigmoid(α V(R)+ρ), where the slope α is positive or negative for different participants. Given this prediction and the adaptation processes postulated by BCV: if we consider two contexts with common trial EVs, we expected that participants with positive α would gamble more for common EVs when these are associated with larger V(R), while we expected that participants with negative α would gamble more for common EVs when these are associated with smaller V(R). To test these predictions we correlated α (i.e., the slope of the logistic regression) with the difference in gambling for common EVs. Note that these predictions do not imply an overall difference in mean gambling across contexts, as the gambling difference across contexts is expected to differ when comparing participants with positive and negative α. This issue is now clarified in analysis of experiment one:
“Though we observed no overall difference in gambling across participants for £4 and £3 EV (see above), participants could be differentiated based on those who gambled more with £4 or £3 EV. […] To examine these predictions we tested for an interaction, corresponding to the differential gambling percentage across contexts (lowvariance minus highvariance context) for £4 choices minus the differential gambling percentage across contexts (lowvariance minus highvariance context) for £3 choices.”
In analysis of experiment two:
“Though across participants we observed no overall effect of option EV on gambling, participants could be differentiated based on those who showed a positive or negative effect of option EV on gambling. […] We investigated this hypothesis both at the level of decks and decksets (Figure 6).”
And in the caption of Figure 6:
“Gambling proportion for different card amounts and different context conditions, separately for (A) participants showing a negative effect of card monetary amount (i.e., the slope of the logistic regression model with card amount as predictor) on gambling (n = 19) and (B) participants showing a positive effect of card amount on gambling (n = 13). […] These data are consistent with our hypotheses; namely (i) with predictions arising from BCV (Figure 2D) which implies subtractive normalization of incentive value at both hierarchical levels and (ii) with the prediction (derived from previous observations; Rigoli et al., 2016a; 2016b) that the influence of incentive value on gambling proportion depends on the individual preference to gamble with large or small card amounts.”
2) The simulation details were unclear to me. Did the authors simulate a synthetic dataset many times and then average the correlations, or did they just run a single simulation? I think the former procedure is better justified, since we don't know whether the results found with the simulations are idiosyncratic or reliable.
The referee is correct and in the initial submission of the manuscript a single simulation was employed, but we agree that a better method would entail repeating several simulations and considering the average correlation from these simulations. We have followed this suggestion and this method is now adopted in the new manuscript, where 100 simulations where considered and the average correlation reported. This is now clarified:
“Finally, we used the (selected) model and subjectspecific parameter estimates of the last analysis to generate simulated choice behaviour and performed behavioural analyses on the ensuing data (data were simulated 100 times and the average statistics are reported).”
3) Some researchers, such as Rangel & Clithero (2012) and Louie, Glimcher & Webb (2015), have drawn connections between context effects and efficient coding, appealing to the idea that divisive normalization is a mechanism for removing statistical redundancies. A Bayesian theory like BCV enables but does not require efficient coding; a given distribution could be coded with varying degrees of efficiency. However, BCV also appears to make some claims about mechanistic implementation which might be relevant to the question of efficiency. How do the authors see the relationship between these theories and BCV? More generally, the authors could deepen their contribution by considering the realization of BCV across Marr's levels.
Thanks for this suggestion for greater elaboration on this point. We agree a crucial aspect of any theory of value adaptation needs to address implications for efficient coding. This is now discussed here, together with a more general framing of the theory within the Marr’s model, as proposed by the reviewer:
“With reference to the three levels of analysis (i.e., computational, algorithmic and implementation) proposed by Marr (1982), BCV speaks to the computational level as it focuses on normative principles (implicit in optimal Bayesian inference) proposed to explain value and choice adaptation. […] In other words, BCV implies that, as well as neural signalling, behaviour itself is tuned to the statistics of the incentives, so as to maximize discriminability among these incentives.”
4) Showing that BCV can account for earlier value normalization results would also bolster the theory. In addition, there is a rich literature on context/decoy effects that would be relevant to at least mention here. Relatedly, work in economics has studied the idea that reference points depend on expectations (e.g., Koszegi & Rabin, 2006), and there have been a number of important recent papers on range effects and relativistic choice processes (e.g., Bordalo et al., 2012; Cunningham, 2013; Bushong, Rabin & Schwartzstein, 2015). It would be illuminating to better understand how these frameworks relate to BCV.
We agree it is relevant to discuss the theory as it relates to previous models mentioned by the reviewer (and other classical theories). We now do this:
“The proposal advanced here has some similarities with classical theories of value, such as Expected Utility theory (von Neumann & Morgenstern, 1944) and Prospect theory (Kahneman & Tversky, 1979). […] For instance, BCV predicts that divisive normalization derives from the gain term (i.e., reward variance) and not from the expected reward (Louie et al., 2014; 2015), and that the reward variance divides – and not multiplies (Kőszegi & Szeidl, 2014) – the prediction error. Importantly, these predictions are not ad hoc but derive necessarily from Bayesian assumptions.”
In addition, we agree it is important to discuss the theory in light of earlier empirical results. As also indicated by the reviewer, most empirical studies have focused on effects elicited by options simultaneously available during choice and where multiple attributes are presented (e.g. the decoy effect). On the contrary, BCV focuses on context effects elicited by expectations before options are presented and where a single attribute is salient. Data on this domain are relatively scarce, and are discussed in the introduction to the new experiment we now report (experiment one in the new manuscript):
“Data from conditions where BCV is applicable, namely those involving a single attribute and where context depends on past options (and not simultaneously presented options), are relatively scarce. […] Here, we present data from a behavioural experiment where we investigate this very question.”
We also mention explicitly other forms of context effect emerging in conditions where BCV is not applicable at the moment, like the decoy effect. We acknowledge this as a shortcoming of BCV, though we argue that similar principles might be valid in these conditions too:
“We highlight shortcomings of the model, though the framework itself may be fruitful in addressing some of these shortcomings. […] This would provide an opportunity to model attentional processes determining an optimal weighting of different attributes based on their importance and reliability.”
Reviewer #2:
My major reservation about the paper is that the human data offers only partial confirmation for the Bayesian context model. In its full specification (in the Appendix), the BCV model predicts that incentive value will depend on a divisive scaling term (implementing precision weighting dependent on relative reward variance) and subtractive prediction error terms (low and high context reward predictions, weighted by their relative contextual cue variances). However, the effects of precision – which are a key element of Bayesian approaches – are untested in the analysis of the experimental data. As the authors state in the paper, reward variance is not manipulated and the divisive term (K) cannot be examined. In addition, the subtractive terms should also be precisionweighted by terms relating posterior and prior variances (denoted by tau_LO and tau_HO). However, the results only speak to overall context effects, essentially asking whether there is an overall effect of low and high context on gambling (population regression effects in Figure 5, and the chisquare tests showing positive tau_LO and tau_HO parameters).
The issue is that adaptive effects to average rewards are well known (for example, successive contrast effects in the animal literature and reference point models like prospect theory); without validating the precisiondependent predictions of their model, I'm not sure that the authors can convincingly argue that BCV is a more appropriate model – particularly as other models are not tested. Given the experimental setup, testing the divisive weighting term is not possible in this dataset; however, can the authors make any predictions about not just the significance but the relative magnitude of the weighting factors tau_LO and tau_HO (predicted and fit to data)?
We are grateful for this suggestion, and agree on the importance of demonstrating an actual effect of precision. Such a test is something we already had in mind when preparing this manuscript, and hence, before obtaining these reviews, we had run (and analysed data from) a novel experiment investigating the role of context precision in value adaptation. We are grateful that the reviewer has highlighted the importance of this point – and now we present data from this new experiment which addresses this very point (experiment one in the new manuscript). In brief, this provides evidence for context precision consistent with BCV. As pointed out by the reviewer, such a role is a key feature of Bayesian schemes and is an observation which supports BCV over alternative theories (e.g., those based on reference point adaptation alone such as Prospect theory).
The question about the comparison between factors tau_LO and tau_HO is interesting, though we believe that our task is not suitable to address this question. An experiment aimed at investigating this question will require an explicit manipulation of the salience of information regarding different hierarchical levels (e.g., manipulating the relative “perceptual noise” of this information), while in our task it is unclear whether (and which) level is more salient.
Reviewer #3:
1) Behavioural effects of reward context on incentive value are already wellestablished and even the specific paradigm used here has already been published by the same authors. Is there any new, surprising behavioural effect that follows from the new model?
We agree this should be made clearer. Two novel and important empirical predictions generated by BCV are: (i) the fact that hierarchically organized contexts can exert adaptation on incentive value and choice, (ii) the variance of the reward context can exert adaptation on incentive value and choice. We investigate both predictions in the current version of the manuscript. The first prediction is tested in the behavioural study reported also in initial submission version of the manuscript. In the present version, we now report a new experiment where we also test the second prediction. These issues are now clarified in the Discussion:
“We propose a Bayesian scheme (BCV) as a model of contextual influences on incentive value attribution. BCV is based on Bayesian inference principles and on generative models of reward. Adopting two novel experimental designs, we provide behavioural evidence that supports two key predictions of BCV, namely that value attribution is affected by reward variance (which exerts divisive normalization) and by hierarchically organized contexts.”
2) The Bayesian model presented in the manuscript is not formally compared to other wellestablished models that may similarly account for the behavioural effects. The authors should show with formal model comparisons that their model outperforms other classic (nonBayesian) models commonly employed to model context effects on valuebased choice.
Thanks for highlighting this point. We agree that the comparison with other theories should be explained better and formally tested. We now do this in the new version of the manuscript. The connections with other theories are explained:
“The proposal advanced here has some similarities with classical theories of value, such as Expected Utility theory (von Neumann & Morgenstern, 1944) and Prospect theory (Kahneman & Tversky, 1979). […] Importantly, these predictions are not ad hoc but derive necessarily from Bayesian assumptions.”
In addition, we perform formal model comparison to compare BCV with predictions derived from alternative theories. This is done for experiment one:
“We next compared the full model with an alternative model without a subtractive normalization component (as postulated by BCV, where the expected reward is subtracted to the actual reward); namely, where the incentive value was equal to $V\left(R\right)={\tau}^{\epsilon}R$. […] This test was significant (χ^{2}(1) = 508, p < 0.001), meaning that the model implementing both subtractive and divisive normalization (derived from BCV and described by equation 8) fits the data better.”
And for experiment two:
“We next compared the full model with an alternative model where the context parameters (capturing the influence of the reward expected within a context at multiple hierarchical levels) divided the reward rather than being subtracted from the reward; in other words where the incentive value corresponds to:
[…] This test was significant (full models: χ^{2}(1) = 78, p < 0.001; for models with α, ρ and ${\tau}_{LO}$: (1) = 26, p < 0.001; for models with α, ρ and τ_{LH}: (1) = 50, p < 0.001), meaning that the model implementing subtractive normalization (consistent with BCV) is a better explanation for the data.”
3) The model is not biologically realistic. This is not always a problem; in fact, there are many elegant demonstrations that Bayesian frameworks can account for optimal performance in various domains better than other accounts. However, in the specific context of rewardguided decisionmaking, it is unclear why and how a Bayesian framework should apply, and to what degree it is more consistent with behaviour and the underlying neural computations. Please provide a lot more information on how this model may be implemented by neural computations. In particular, it would help if there was any empirical evidence for the hierarchical representation of reward context.
Thanks for highlighting this point. We agree it is important to clarify the link between BCV and brain function. We stress that there are several existing proposals as to how Bayesian inference schemes (and hence BCV) might be implemented in the brain (e.g., Doya et al., 2007; Friston, 2005; Hennequin et al., 2014; Knill & Pouget, 2004). Therefore we consider that BCV is relevant for ideas regarding neural implementation level, and is equally biologically plausible as other Bayesian inference schemes.
Several neural observations are consistent with BCV, such as the observation that a reward signal in several brain regions is adapted to a reference point and to reward range (Bermudez & Schultz, 2010; Cox & Kable, 2014; Louie et al., 2011; PadoaSchioppa, 2009; PadoaSchioppa & Assad, 2008; Park et al., 2012; Rigoli et al., 2016a; Tobler et al., 2005 Kobayashi et al., 2010; Tremblay & Schultz, 1999), and like a recent finding of a link between neural and choice adaptation (Rigoli et al., 2016). Yet there are important aspects of BCV which will need to be probed at the neural level. For instance, as pointed out by the reviewer, a test of an hypothesis of a hierarchical organization of reward context in the brain is lacking. However, we believe that the fact that aspects of the model have not yet been tested represents a strength (rather than a weakness) of the theory, as it makes novel and precise predictions. We now refer to these issues in the following text:
“With reference to the three levels of analysis (i.e., computational, algorithmic and implementation) proposed by Marr (1982), BCV speaks to the computational level as it focuses on normative principles (implicit in optimal Bayesian inference) proposed to explain value and choice adaptation. […] In other words, BCV implies that, as well as neural signalling, behaviour itself is tuned to the statistics of the incentives, so as to maximize discriminability among these incentives.”
4) The Abstract claims that the model "generates new empirical predictions and may help explain important phenomena in psychopathologies such as addiction." I found the corresponding text in the Discussion rather vague. Please provide explicit predictions for specific experimental effects that follow from this model and please explain much more concretely which important phenomena in psychopathologies are explained by it.
We agree this was not clear. Two empirical predictions arising from the theory are: (i) hierarchically organized contexts exert adaptation of incentive value and choice, (ii) variance of the reward context can exert adaptation of incentive value and choice. We investigate both predictions here, the second with new experimental data. The first prediction was tested in the previous version of the manuscript. In this revised manuscript, we report an experiment where the second prediction is tested. The specific predictions of the model are now presented in detail in Figure 2 and are discussed here in relation with other models:
“The proposal advanced here has some similarities with classical theories of value, such as Expected Utility theory (von Neumann & Morgenstern, 1944) and Prospect theory (Kahneman & Tversky, 1979). […] Importantly, these predictions are not ad hoc but derive necessarily from Bayesian assumptions.”
In relation with the link with psychopathology, we focus on addiction. BCV may be useful to understand the computational mechanisms underlying the development of addiction and tolerance. In addition, BCV may be useful also to clarify the computational mechanisms representing an individual predisposition to drug abuse, for instance dependent on increased uncertainty of prior belief on reward. This is clarified:
“Finally, there are questions related to psychopathology that can be fruitfully formulated in terms of BCV, for example addiction. […] For instance, increased uncertainty over prior reward beliefs may boost the magnitude of the (positive) prediction error elicited by drug consumption, hence enhancing individual predisposition to drug addiction.”
5) The manuscript claims in several places that a reward's incentive value corresponds to the (precisionweighted) prediction error. This is misleading. By definition, the incentive value is the property of a stimulus/expected reward that triggers approach behaviour and choice of the corresponding option. This representation must therefore be computed before the choice is taken and the reward is obtained. The prediction error, by contrast, is the deviation of the reward obtained as a consequence of the choice from the reward expected prior to the choice. This postchoice representation can therefore not be the incentive value guiding choice. The authors need to clarify their terminology and ensure that they remain consistent with established definitions in the literature.
We apologise for the confusion and agree on a need for clarity. BCV is applicable every time novel information about reward is provided, which is when a prediction error occurs. This happens: (i) when a (primary or secondary) reward is delivered (or is not delivered when expected), which can be post choice but also in other conditions (e.g., in classical conditioning paradigms), when a reward is delivered independent of action) (ii) when one (or more) option is presented, which is the condition exploited in our experiments. This is because an agent has expectancies about the value of an option, which can be confirmed or not when options are presented, leading to prediction errors. We agree that, to avoid confusion, this needs to be clarified. We stress that the idea that a prediction error occurs when options are presented is not uncommon in the reward literature, as in standard computational temporal difference algorithms (Sutton & Barto, 1998). This is now clarified:
“Equation two applies every time novel information about reward is provided, which is when a prediction error occurs. […] The latter follows because an agent has an expectation about an option, which leads to a prediction error when the actual option is presented.”
In addition to the conceptual points listed above, the manuscript also has shortcomings with respect to methodology and results presentation that will need to be addressed:
6) The behavioural task was not designed to allow proper tests of the full model. Some of these problems are listed by the authors themselves and have led to adaptations of the model so that it could be fit to the data. For instance, reward variance in the different decks is not varied, is heavily constrained (there are only 3 different reward values per deck), and is perfectly correlated with average reward magnitude. To properly test whether the precision of reward prediction errors established by the different contexts really plays an important role, the authors should fit the full model to datasets with contexts that differ substantially in their reward variance and that disentangle reward magnitude from reward variance. Moreover, the contexts should be associated with a lot more than just three possible reward values so that the form of the expected reward distributions can be properly approximated (see below).
We appreciate this suggestion, and agree that investigating the effect of precision is a fundamental test for BCV. Such test is something we already had in mind, and had run a novel experiment prior to submission of the original manuscript. This experiment is relevant as in it we investigate the role of context precision in value adaptation. We now include these data (experiment one in the new manuscript) which provide evidence for a role of context precision consistent with BCV. As pointed out by the reviewer, such role is a key feature of Bayesian schemes and supports BCV over alternative accounts.
To avoid confusion, we now clarify in the novel paradigm that there are two types of variance. The first refers to the variance of possible outcomes of the gamble (which is perfectly correlated with the EV of options as in a previous experiment and as in (Rigoli et al., 2016a; 2016b), and is not the focus of the experiment. The second refers to the variance across options (i.e., the variance characterizing the distribution of seriallyprovided options), which is what we manipulate and is reflected in and affects the gain term of equation one and two. This is clarified:
“Note that in our task there are two types of variance. The first refers to the variance of possible outcomes of the gamble (which is perfectly correlated with the EV of options, as in Rigoli et al., 2016), and is not the focus of our study. The second refers to variance across options (i.e., the variance characterizing the distribution of successive options), which is what we experimentally manipulate and investigate. In the model, this is reflected in and affects the gain term in equation 2.”
7) The model specification does not match the environment established by the behavioural task. The decks were associated with 3 equiprobable values per deck. Therefore, "smart" subjects would employ a flat discrete expectation of the three possible values within the given context. In contrast, the modelling solution is based on continuous (Gaussian) distributions that are not restricted by the bounds imposed by each context. There are two problems with this: (1) If we assume that the subjects indeed optimally integrate all information, then a flat prior belief bounded by the context's minimum and maximum reward would be accurate. This cannot be modelled by the presented specification. (2) Even if subjects employed continuous Gaussian priors to model reward expectations, such distributions would probably not be narrow enough with respect to the context bounds (i.e., the priors would wrongly lead to expectations of rewards that are outside of the bounded scale). These problems will probably be evident if the authors report the values of the latent variables after fitting the model to the empirical data. Please include such a table to allow the reader to inspect this issue.
In my view, if the authors really wanted to maintain a Bayesian optimal observer model, then they should examine how the prior (potentially flat, but can also be modelled if the authors like) is combined with the likelihood of the actual numeric representation to obtain a posterior estimate that should naturally occur within the actual numeric bounds of the context (for an example on how to formally deal with Bayesian problems of this kind, i.e. bounded contexts, see Jazayeri and Shadlen 2010, Nature Neuroscience). This formal specification can then be expanded to the interesting contextual hierarchical framework that the authors propose in their study.
We understand the reviewer’s concerns and agree. It is true that real agents may build nonGaussian representations, especially in conditions where the true distributions are nonGaussian. However, the advantage of using Gaussian assumptions is that posterior probability can be estimated analytically with equations which are simple and clear, and resemble common formulations widely used in decisionmaking and motivation research (e.g., RescorlaWagner update rule). Therefore, we would like to stick with Gaussian variables when presenting the model, which we believe is a better approach in making the model clear and transparent. Also on the question of fitting the model to choice data, we believe that Gaussian assumptions should not be problematic, as the equations used are more general. In other words, these equations can be derived from Gaussian assumptions but are valid also for other distributions such as uniform distributions (in general, nonskewed distributions).
Please consider equation 8:
This simply prescribes a subtractive normalization to the average reward $(R{\mu}_{C})$, and a divisive normalization elicited by reward variance, captured by a quantity τ (a free parameter bounded between 0.1 and 10) in the lowvariance context (where ε = 1 indicates a low variance and ε = 0 indicates highvariance context). This equation is valid also for other distributions, such as uniform distributions (in general, nonskewed distributions). In addition, we did not impose a bound a priori on the parameter τ between 0 and 1, as would be predicted by BCV. This because we tested whether the τ estimated from choice data was between 0 and 1.
Similarly, please consider equation 10:
This prescribes subtractive normalization exerted by the average reward expected at two hierarchical levels. Again this equation is general and valid also for other distributions, such as uniform distributions (in general, nonskewed distributions). As above, we did not impose a bound a priori on the parameter τ to be positive, as would be predicted by BCV. This is because we tested whether the τ estimated from choice data was positive.
Overall, our goal was to (i) derive equations using Gaussian assumptions, but in a general form potentially consistent also with other nonskewed distributions (e.g., uniform) (ii) fit these equations to choice with unconstrained context parameters, (iii) test whether the estimated context parameters where consistent with predictions.
The problem of context boundaries is also relevant. In general, in Bayesian approaches, it is often useful to model participants’ beliefs in continuous space, even in contexts which are “objectively” bounded. For instance, in models of normalization, incentive values go beyond the “objective” context boundaries almost by definition. Therefore we would expect that if one assumes bounded variables within the generative model, we would not obtain normalization. This we believe represents a fundamental argument for using continuous and not bounded variables in the context of value normalization. This is clarified in the Discussion:
“Here, we have assumed that variables of the generative model are Gaussian. This allows us to present the model in a simple and clear way, as posterior beliefs can be inferred analytically with relatively simple equations as adopted in standard decisionmaking schemes (Rescorla & Wagner, 1972). […] However, the key idea (tested in our experiments – and here derived from Gaussian assumptions) that reward average and variance elicit subtractive and divisive normalization, respectively, is quite general and can also be applied, for instance, to uniform (and, in general, nonskewed) distributions.”
8) The authors rescaled the contextual averages and reward values to perform their model fits. Why? A correctly specified model should be able to take as inputs the actual values of the contexts and rewards of their behavioural paradigm (which are all singledigit numbers after all). This would help to assess the model's explanatory power.
Thanks for pointing this out. We now follow the reviewer’s advice and use non rescaled contextual averages:
“Where ${\chi}_{LO}$ indicates the average option EV for the deck (for £9 deck: ${\chi}_{LO}=4.5;$for £7 deck: ${\chi}_{LO}=3.5$; for £5 deck: ${\chi}_{LO}=2.5$), ${\chi}_{HO}$ indicates the average option EV for the deckset (highvalue deckset: ${\chi}_{HO}=4$; lowvalue deckset: ${\chi}_{HO}=3$), τ_{LO }is a free parameter that mediates contextual effects at the deck level, and ${\tau}_{HO}$ is a free parameter that mediates contextual effects at the deckset level (see methods).”
9) The authors perform model comparison by summing loglikelihoods across participants. I find the selection of this approach for model comparison surprising, given that several of the coauthors have pushed the use of precise Bayesian model selection methods that properly account for the complexity and variability of the model fits across trials and participants. The authors should employ such methods and should provide values quantifying the quality of the model fits after penalizing for model complexity.
Thanks for highlighting this point. It is true that in the previous version we estimated the parameters without strong constraints. This is potentially problematic because this method may lead to outlier estimates which are biased, and is a potential problem for our findings in relation with modelcomparison and in relation with withinsubjects ttests over the context parameters. To address this possibility, we now adopt a Gaussian prior with mean zero (Daw, 2011) during estimation of the context parameters. This allows extreme parameter estimates only if they improve substantially the negative loglikelihood, hence avoiding possible biased extreme outliers. Note in addition that a prior mean of zero does not bias the ttests since we tested the parameters against zero. For a sanitycheck, in supplemental materials we now report frequencies of the parameter estimates (for experiment one: Figure 4—figure supplement 1; for experiment two: Figure 7—figure supplement 1). We can see that outlier estimates (i.e., scores larger or smaller than 3 SD compared to the mean) are absent.
For model comparison, we considered the probability of the data given the model and given the parameters estimated in this way. We used the likelihood ratio test because our modelcomparison involves nested models. This approach is a standard and wellestablished method for modelcomparison involving nested models (Casella & Berger, 2002; Daw, 2011). Penalization for model complexity is implicit in this test since it is based on the null hypothesis that data are generated with the simpler model, and hence the more complex model is accepted only if evidence for the data given the simpler model is below significance threshold. In addition, different levels of complexity are distinguished and penalized in a different way, thanks to the fact that the degrees of freedom of the likelihood ratio test (capturing how “strong” the evidence against the simper model needs to be to refuse the null hypothesis) increase with the number of extra parameters. The details regarding the method adopted for model fitting and model comparison are explained in detail:
“The free parameters of the models were estimated separately for each subject using fminsearchbnd function of the Optimization toolbox in Matlab. […] A chisquare test can be performed to estimate the probability that the observed 2d is due to chance under the null hypothesis that data are generated by the nested model, allowing acceptance or rejection of the null hypothesis.”
10) The behavioural task is described in a fashion that makes it hard to replicate. For instance, is it true that on every trial, a card was drawn from the blue deck? If so, how were the different deck contexts varied across the different colours? Please make sure the task described in sufficient detail so that another person could program it.
Thank you – this is now clarified in the caption of Figure 5:
“The decks were coloured one in blue and the other in grey, indicating the selected and unselected deck respectively. Among these two decks shown on the screen, the selected deck (coloured in blue) alternated pseudorandomly over blocks (each including 5 trials). In addition, two sets of decks alternated over longer blocks (20 trials) in a pseudorandom way”.
And in the main text:
“Among the two decks shown on the screen, the selected deck (coloured in blue) alternated pseudorandomly over blocks (each including 5 trials). In addition, at some points during the task, decks were replaced by new decks. Two sets of decks alternated over blocks of 20 trials in a pseudorandom way.”
https://doi.org/10.7554/eLife.16127.015Article and author information
Author details
Funding
Wellcome Trust (098362/Z/12/Z)
 Francesco Rigoli
 Karl J Friston
 Raymond J Dolan
European Research Council
 Sukhwinder S Shergill
NIHRBRC
 Cristina Martinelli
The funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication.
Acknowledgements
This work was supported by the Wellcome Trust (Ray Dolan Senior Investigator Award 098362/Z/12/Z) and the Max Planck Society. The Wellcome Trust Centre for Neuroimaging is supported by core funding from the Wellcome Trust 091593/Z/10/Z. SSS is supported by a consolidator award from the European Research Council. C.M is funded by the NIHRBRC at South London and Maudsley NHS Foundation Trust and Institute of Psychiatry, Psychology and Neuroscience King’s College London via a research studentship.
Ethics
Human subjects: Experiment one was approved by the University College London Research Ethics Committee. Experiment two was approved by the King's College of London Research Ethics Committee. All participants provided written informed consent and were paid for participating.
Reviewing Editor
 Sam Gershman, Harvard University, United States
Publication history
 Received: March 16, 2016
 Accepted: June 16, 2016
 Accepted Manuscript published: June 21, 2016 (version 1)
 Version of Record published: July 18, 2016 (version 2)
Copyright
© 2016, Rigoli et al.
This article is distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use and redistribution provided that the original author and source are credited.
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