Using the past to estimate sensory uncertainty
- Psychology Department, Durham University, United Kingdom
- Department of Psychiatry and Psychotherapy, University of Tübingen, Germany
- Department of Psychology, Friedrich-Alexander University Erlangen-Nuernberg, Germany
- Centre for Computational Neuroscience and Cognitive Robotics, University of Birmingham, United Kingdom
- Max Planck Institute for Intelligent Systems, Germany
- European Molecular Biology Laboratory, Genome Biology Unit, Germany
- Division of Computational Genomics and Systems Genetics, German Cancer Research Center (DKFZ), Heidelberg, Germany, Germany
- Donders Institute for Brain, Cognition and Behaviour, Radboud University, Netherlands
Figures
Figure 1 with 1 supplement
Audiovisual localization paradigm and Bayesian causal inference model for learning visual reliability.
(A) Visual (V) signals (cloud of 20 bright dots) were presented every 200 ms for 32 ms. The cloud’s location mean was temporally independently resampled from five possible locations (−10°, −5°, 0°, …
Figure 1—figure supplement 1
Generative model for the Bayesian learner.
The Bayesian Causal Inference model explicitly models whether auditory and visual signals are generated by one common (C = 1) or two independent sources (C = 2) (for further details see Körding et …
Figure 2 with 1 supplement
Time course of visual noise and relative auditory weights for continuous sequences of visual noise.
The visual noise (i.e. STD of the cloud of dots, right ordinate) and the relative auditory weights (mean across participants ± SEM, left ordinate) are displayed as a function of time. The STD of the …
Figure 2—figure supplement 1
Time course of the relative auditory weights for continuous sequences of visual noise when controlling for location of the cloud of dots in the previous trial.
Relative auditory weights (mean across participants ± SEM, left ordinate) and visual noise (i.e. STD of the cloud of dots, right ordinate) are displayed as a function of time as shown in Figure 2 of …
Figure 3
Observers’ relative auditory weights for continuous sequences of visual noise.
Relative auditory weights wA of the 1st (solid) and the flipped 2nd half (dashed) of a period (binned into 20 bins) plotted as a function of the normalized time in the sinusoidal (red), the RW1 …
Figure 4
Observed and predicted relative auditory weights for continuous sequences of visual noise.
Relative auditory weights wA of the 1st (solid) and the flipped 2nd half (dashed) of a period (binned into 20 bins) plotted as a function of the normalized time in the sinusoidal (red), the RW1 …
Figure 5 with 2 supplements
Time course of visual noise and relative auditory weights for sinusoidal sequence with intermittent jumps in visual noise (N = 18).
(A) The visual noise (i.e. STD of the cloud of dots, right ordinate) is displayed as a function of time. Each cycle included one abrupt increase and decrease in visual noise. The sequence of visual …
Figure 5—figure supplement 1
Time course of relative auditory weights and visual noise for the sinusoidal sequence with intermittent jumps in visual noise for the exponential and instantaneous learning models.
Relative auditory weights wA,bin (mean across participants) of the 1st (solid) and the flipped 2nd half (dashed) of a period (binned into 15 time bins) plotted as a function of the time in the …
Figure 5—figure supplement 2
Time course of relative auditory weights and root mean squared error of the computational models before and after the jumps in the sinusoidal sequence with intermittent jumps.
(A) Relative auditory weights wA (mean across participants) shown as a function of time around the up-jumps (left panel) and the down-jumps (right panel) for observers’ behavior, the instantaneous, …
Figure 6
Time course of the relative auditory weights, the standard deviation (STD) of the visual cloud and the STD of the visual uncertainty estimates.
(A) Relative auditory weights wA of the 1st (solid) and the flipped 2nd half (dashed) of a period (binned into 15 bins) plotted as a function of the time in the sinusoidal sequence. Relative …
Appendix 2—figure 1
Generative model, for one (C = 1) or two sources (C = 2).
Appendix 2—figure 2
Approximation of theta using Laplace approximation.
Appendix 2—figure 3
Comparing variational Bayes approximation with a numerical discretised grid approximation.
Top row: Example visual stimuli over eight subsequent trials. Middle row: The distribution of estimated sample variance, with no learning over trials. Bottom row: The distribution of _V;t for the …
Tables
Table 1
Analyses of the temporal asymmetry of the relative auditory weights across the four sequences of visual noise using repeated measures ANOVAs with the factors sequence part (1st vs. flipped 2nd half), bin and jump position (only for the sinusoidal sequences with intermittent jumps).
| Effect | F | df1 | df2 | p | Partial η2 | |
|---|---|---|---|---|---|---|
| Sinusoid | Part | 12.162 | 1 | 24 | 0.002 | 0.336 |
| Bin | 92.007 | 3.108 | 74.584 | <0.001 | 0.793 | |
| PartXBin | 2.167 | 2.942 | 70.617 | 0.101 | 0.083 | |
| RW1 | Part | 14.129 | 1 | 32 | 0.001 | 0.306 |
| Bin | 76.055 | 4.911 | 157.151 | <0.001 | 0.704 | |
| PartXBin | 1.225 | 4.874 | 155.971 | 0.300 | 0.037 | |
| RW2 | Part | 2.884 | 1 | 18 | 0.107 | 0.138 |
| Bin | 60.142 | 3.304 | 59.467 | <0.001 | 0.770 | |
| PartXBin | 3.385 | 4.603 | 82.849 | 0.010 | 0.158 | |
| Sinusoid with intermittent jumps | Jump | 28.306 | 2 | 34 | <0.001 | 0.625 |
| Part | 24.824 | 1 | 17 | <0.001 | 0.594 | |
| Bin | 76.476 | 1.873 | 31.839 | <0.001 | 0.818 | |
| JumpXPart | 0.300 | 2 | 34 | 0.743 | 0.017 | |
| JumpXBin | 8.383 | 3.309 | 56.247 | <0.001 | 0.330 | |
| PartXBin | 1.641 | 3.248 | 55.222 | 0.187 | 0.088 | |
| JumpXPartXBin | 0.640 | 5.716 | 97.175 | 0.690 | 0.036 |
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Note: The factor bin comprised nine levels in the first three and seven levels in the fourth sequence. In this sequence, the factor Jump comprised three levels. If Mauchly tests indicated significant deviations from sphericity (p<0.05), we report Greenhouse-Geisser corrected degrees of freedom and p values.
Table 2
Model parameters (median), absolute WAIC and relative.
ΔWAIC values for the three candidate models in the four sequences of visual noise.
| Sequence | Model | σA | Pcommon | σ0 | or γ | WAIC | ΔWAIC |
|---|---|---|---|---|---|---|---|
| Sinusoid | Instantaneous learner | 5.56 | 0.63 | 8.95 | - | 81931.2 | 109.9 |
| Bayesian learner | 5.64 | 0.65 | 9.03 | κ: 7.37 | 81821.3 | 0 | |
| Exponential discounting | 5.62 | 0.64 | 9.02 | γ: 0.23 | 81866.9 | 45.6 | |
| RW1 | Instantaneous learner | 6.30 | 0.69 | 8.46 | - | 110051.2 | 89.0 |
| Bayesian learner | 6.29 | 0.72 | 8.68 | κ: 8.06 | 109962.2 | 0 | |
| Exponential discounting | 6.26 | 0.70 | 8.75 | γ: 0.33 | 109929.9 | −32.3 | |
| RW2 | Instantaneous learner | 6.36 | 0.72 | 10.79 | - | 62576.4 | 201.3 |
| Bayesian learner | 6.49 | 0.78 | 10.9 | κ: 6.7 | 62375.2 | 0 | |
| Exponential discounting | 6.46 | 0.73 | 11.0 | γ: 0.25 | 62421.5 | 46.3 | |
| Sinusoid with intermittent jumps | Instantaneous learner | 6.38 | 0.65 | 8.19 | - | 83891.4 | 94.9 |
| Bayesian learner | 6.45 | 0.68 | 8.26 | κ: 6.13 | 83796.5 | 0 | |
| Exponential discounting | 6.43 | 0.67 | 8.20 | γ: 0.24 | 83798.1 | 1.64 |
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Note: WAIC values were computed for each participant and summed across participants. A low WAIC indicates a better model. ΔWAIC is relative to the WAIC of the Bayesian learner.
Additional files
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Supplementary file 1
Seven tables showing results of additional analyses.
- https://cdn.elifesciences.org/articles/54172/elife-54172-supp1-v2.docx
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Transparent reporting form
- https://cdn.elifesciences.org/articles/54172/elife-54172-transrepform-v2.docx
