Article Figures & Data
Figures
- Figure 1.
Graphical description of Bayes’ rule and the interaction between prior distributions and likelihood functions leading to the final posterior distribution. A, Bayes’ rule can be thought of as a reallocation of probability to the posterior after accounting for prior distributions and observed evidence. B, An example of a posterior generated from an inverse-gamma-distributed likelihood and a uniformly distributed prior. Uniform priors reflect the likelihood function, and thus the observed data with no redistribution of probability, making uniform distributions uninformative priors. However, care must be taken in using uniform distributions as observed data outside of prior bounds are mapped to zero probability. A second example of a posterior generated from an inverse-gamma-distributed likelihood and a Gaussian-distributed prior shows how a prior can be considered informative (right). In this posterior, the prior “shapes” the posterior to a greater extent than a uniform prior. Well-designed priors in neuroscientific data can thus shape posteriors away from responses that are not physiological and can be vetted in the statistical decision, data review, and postpublication review stages. Prior distributions with longer tails can handle extremes of observed data by mapping extreme events to low, but nonzero, representation in the posterior. Example B represents extremes of prior choices, with minimally informative priors often chosen to let the data “speak for itself” with little change to posterior from prior influence. C, Comparison of Bayesian and frequentist statistics. NSHT utilizes a parameterized likelihood function that describes the data-generating process. D, Decisions in NSHT are generally made based on quantification of the probability of observing test results as extreme as observed data assuming that the null hypothesis is true.
- Figure 2.
An illustrative example of how Bayesian inference might be used in an electrophysiological experiment. Prior distributions can be iteratively formulated by previous experiments drawn from both internal and published data. A, Observation of data distributions shows that both outcomes follow normal distributions, guiding the choice of a normal likelihood distribution. B, Examples of a minimally informative prior distribution (top) and a more informative prior distribution (bottom). Design of the prior distribution is guided by both expert knowledge by the investigators along side established published data. C, Incorporation of more informative prior distributions results in a similar decision rule as the less informative prior, but with better estimation of credible model parameters. D, The null hypothesis of a second hypothetical example demonstrating p-value sensitivity to stopping rules. E, Given the same data and sample size, stopping rules can fundamentally alter the sample distribution estimate and give conflicting p-value results.
- Figure 3.
Example data for Bayesian simple linear regression on population estimates of firing rate versus amplitude modulation depth stimuli. Bayesian regression models were applied to population single-unit firing rates elicited from IC with SAM tones. The goal of this model was to predict evoked firing rates from increases in SAM modulation depths. A, Density plot of observed firing rates versus SAM modulation depth. B, Histogram and KDE of the distribution of SAM-evoked IC firing rates across all modulation depths. Observation of the total distribution of results aids in choosing the likelihood distribution for inference. C, Schematic of amplitude-modulated stimuli.
- Figure 4.
Completed Bayesian inference quantifying linear relationships in evoked firing rate from increases in modulation depth. A, Scatter plot of observed firing rates versus SAM depth stimuli with fitted regression line estimates superimposed. Ninety-five percent HDI estimates of regression slopes are shown in orange, with the spread of lines encoding the 95th percentile of most likely slope values. B, Estimates of Bayesian linear regression parameters. Intercept term α was significantly above 0 (MAP = 3.1; 95%HDI does not overlap 0) which indicates basal firing rates above 0. Regression slope was small but significantly above 0 (MAP = 0.018; 95% HDI does not overlap 0), suggesting an increase in evoked firing rates with increased modulation depth. Error term ϵ was significantly above 0 (MAP = 0.74; 95% HDI does not overlap 0), suggesting some model deviation from observed data. However, error terms were considered small as ϵ MAP
- Figure 5.
Evaluation of MCMC chains can help diagnose ill-fitting distributions. A, Kernel density estimates of the marginal posteriors corresponding to each of the regression parameters of each MCMC trace. Qualitatively, chain distributions should appear similar to each other, suggesting good convergence to target distributions. B, Time-series plot of trace value versus sample number of marginal posteriors corresponding to each regression parameter. Qualitatively, good traces should have a “fuzzy caterpillar”-like shape, evident in all parameters of this model, indicative of good integration over the joint posterior distribution and effective sampling of the posterior.
- Figure 6.
Bayesian implementations of t tests provide descriptive quantification of differences in group means, standard deviations, and effect sizes drawn directly from observed data. A, Schematic of the basal ganglia–thalamocortical computational model. Modeled DBS stimulation was performed in the subthalamic nucleus with LFP recordings made from the globus pallidus internus. All simulations were performed assuming dopamine-depleted conditions. B, Example LFP power spectral densities from no stimulation condition (green), subthreshold DBS stimulation (0.01 mA, 100 µs pulse widths, 130 Hz, orange), and effective DBS stimulation (1.8 mA, 240 µs pulse widths, 130 Hz, blue). C, Total β-band power in no stimulation and DBS stimulation was calculated (n = 2,000 repetitions). Group comparison by BEST shows significant reduction in mean β-band power as evidenced by group mean difference 95% HDIs not including 0 (MAP
- Figure 7.
Example of Bayesian multilinear regression incorporating a hierarchical structure. A, In this experiment, rodents were implanted with fiber-optic arrays into the auditory thalamus and planar recording arrays into the auditory cortex. Single-unit responses were recorded from INS stimuli with applied energy and interstimulus intervals varied to derive dose–response curves. Figure was drawn using BioRender under publication license (www.biorender.com). B, Hierarchical schematic of Bayesian multilinear regression. Hierarchical structures are advantageous in accounting for within- and between-subject variability or for repeated measures designs. C, Resulting parameter distributions from dose–response models. Energy was a significant contributor to maximum firing rate, with increasing laser energy resulting in increased maximum firing rate, as determined by the 95% HDI of the laser energy term
- Figure 8.
An example of Bayesian inference using ANOVA-like models. A, General schematic of BANOVA/BANCOVA models. Traditional ANOVAs have two key assumptions: normality of group data and homogeneity of variance. Normality of group data is imposed in BANOVA-like models as normal distributions around group parameters with homogeneity of variance encoded as a uniform distribution around posterior variance term
Tables
Bayesian Frequentist Philosophy of probability Quantification of uncertainty Frequency of events Data inference P(hypothesis|data) “data driven” P(data|hypothesis) Decision rules HDI, credible interval, ROPE, distribution of observed data distributions, Bayes’ factors P-value, confidence interval, Type I/II error control Model parameters Distributions formed around uncertainty in observed data Fixed but unknown values inferred from estimators (maximum likelihood/least squares) Data distributions Any distribution Parametric models require data to follow normal distributions. Nonparametric methods are limited Requires explicit declaration of prior distribution Yes No Computational complexity Higher Lower Advantages Inference backed by evidence from observed data, a complete probabilistic description of experimental data, inference less dependent on sample size. Error control through prior and posterior predictive checks Robust error control (with proper experimental design), quick and easy to implement Model R ELPD DSE St Log Var 5 1 −5,337.48 0.00 ST Log Var 100 2 −5,337.62 0.420867 ST Log Var 0.5 3 −5,337.76 0.409773 St Log Var 25 4 −5,338.15 0.492297 ST Log Var 10 5 −5,338.18 0.300197 ST Log Var 1 6 −5,338.26 0.331152 N Log Var 10 7 −5,340.60 3.308668 N Log Var 1 8 −5,341.09 3.293779 N Log Var 5 9 −5,341.16 3.296273 N Log Var 0.5 10 −5,342.46 3.300550 ST Semilog Var 1 11 −5,466.76 15.845916 St Semilog Var 5 12 −5,467.12 15.856552 ST Semilog Var 10 13 −5,467.15 15.895646 ST Semilog Var 0.5 14 −5,467.18 15.866405 ST Var 1 15 −15,336.31 79.406629 ST Var 0.5 16 −15,355.67 80.415787 St Var 5 17 −15,355.67 80.415787 N Var 10 18 −16,119.11 82.384329 N Var 1 19 −16,132.23 83.549811 N Var 0.5 20 −16,154.55 84.262219
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