Figures and data
Introduction.
A: Each of N animals produce a cell-count from a total of R brain regions of interest. Cell-count data is typically undersampled with N ≪ R. Scientists analyse the brain sections from experiment for positive signals. Here an example section is shown where teal points mark cells expressing the immediate early gene c-Fos. The final cell-count is equal to the sum of these individual items (sagittal brain map taken from the Allen mouse brain atlas: https://mouse.brain-map.org. B: Partial pooling is a hierarchical structure that jointly models observations from some shared population distribution. It is a spectrum that depends on the value of the population variance τ. When τ = 0 there is no variation in the population and each individual observation is modeled as a conditionally independent estimate of some fixed population mean θ (complete pooling). As τ tends to infinity observations do not combine inferential strength but inform an independent estimate γi (no pooling). In between the two extremes combine. Each observation can contribute to the population estimate while simultaneously supporting a local one to effectively model the variance in the data. Observed quantities have been highlighted with a thicker stroke in the graphical model. C: An example of partial pooling on simulated count data. As the population standard deviation increases on the x-axis the individual estimates exp(γi) trace a path from a completely pooled estimate to an unpooled estimate. Circular points give the raw data values. Parameters are exponentiated because the outcomes are Poisson and so parameters are fit on the log scale.
Parameter table for the hierarchical model.
Methods.
A table of partial pooling behaviour for different likelihood and prior combinations. Rows are the two prior choices for the population distribution, and columns the two distributions for the data. Within each cell the expectation of the marginal posterior p(exp(γi)|θ, τ, y) is plotted as a function of τ. The solid black line is the expectation of the marginal posterior p(θ|τ, y) with one standard deviation highlighted in grey. Top left: Combining a normal prior for the population with a Poisson likelihood is unsatisfactory in the presence of a zero observation. The zero observations influence the population mean in an extreme way owing to their high importance under the Poisson likelihood. Bottom left: By changing to a horseshoe prior the problematic zero observations can escape the regularisation machinery. However, regularisation of the estimates with positive observations is much less impactful. Top right: A zero-inflated Poisson likelihood accounts for the zero observations with an added process, reducing the burden on the population estimate to compromise between extreme values. Bottom right: No model.
Recognition memory circuit.
Schematic of the recognition memory network adapted from (31). Bold arrows show the assumed two-way connection between the medial prefrontal cortex and the hippocampus facilitated by the NRe. Colours highlight the HPC (red), MPC (blue) and specific areas of the rhinal cortex (yellow). The NRe was lesioned in the experiment.
Results—Case study 1.
log2-fold differences for each surgery type. The 95% Bayesian HDI is given in green, and the 95% confidence interval calculated from a Welch’s t-test in orange. Horizontal lines within the intervals mark the posterior mean of the Bayesian results, and the raw data means in the t-test case. The x-axis is ordered in terms of decreasing p-value from the significance test and ticks have been color-paired with the nodes in the recognition memory circuit diagram Figure 3. Black ticks are not present in the circuit because they are the control regions in the experiment. A: log2-fold differences between sham-familiar (SF) and sham-novel (SN) groups. B: log2-fold differences between lesion-familiar (LF) and lesion-novel (LN) groups.
Results—Case study 2.
log2 fold differences in GFP positive cells between mouse genotypes, heterozygous (HET) and knockout (KO), for each of the fifty recorded brain regions spread across two rows. The 95% Bayesian HDI is given in purple and pink for the Bayesian horseshoe and zero-inflated model. The 95% confidence interval calculated from a Welch’s t-test in orange. Horizontal lines within the intervals mark the posterior mean of the Bayesian results and the data estimate for the t-test. The x-axis is ordered in terms of decreasing p-value from the significance test.
Zero count observations.
On the left under data: boxplots with medians and interquartile ranges for the raw data for two example brain regions. The shape of each point pairs left and right hemisphere readings in each of the five animals. For both example regions there is a heterozygous animal with zero readings in both hemispheres. On the right under inference: HDIs and confidence intervals are plotted. The Bayesian estimates are not strongly influenced by the zero-valued observations and have means close to the data median. This explains the advantage of the Bayesian results over the confidence interval.
Diagnostics – Poisson.
Standard Poisson model, case study – 1.
Diagnostics – Horseshoe.
Horseshoe model, case study – 2.
Diagnostics – ZIPoisson.
Zero-inflated Poisson, case study – 2.
PPC – Poisson.
Posterior predictive check for the standard Poisson model in case study – 1. A: The proportion of zeroes in the data matches the proportion of zeroes in posterior predictive samples. This proportion is zero. B: The distribution of standard deviations computed over a number of posterior predictive datasets (histogram), aligns with the standard deviation of the data.
PPC – Horseshoe.
Horseshoe model, case study – 2. Posterior predictive check for the standard horseshoe model in case study – 2. A: The proportion of zeroes in the data is larger than those found in posterior predictive datasets. This makes sense, because the likelihood is still a Poisson distribution. B: The distribution of standard deviations computed over a number of posterior predictive datasets (histogram), aligns with the standard deviation of the data.
PPC – ZIPoisson.
Zero-inflated Poisson, case study – 2. A: The proportion of zeroes in the data matches the proportion of zeroes in posterior predictive samples. B: The distribution of standard deviations computed over a number of posterior predictive datasets (histogram), aligns with the standard deviation of the data.
Horseshoe densities.
A: Conditional posterior. B: MCMC pair plots. Divergent samples are coloured in pink, non-divergent in blue.
Modified horseshoe densities.
A: The conditional posterior 
