Extended Data Fig. 2: Evidence for or against a factor in a Bayesian ANOVA.

A Bayesian ANOVA is a form of model comparison. This figure illustrates how the Bayes factor can provide evidence for a simpler model by concentrating its predictions on a single parameter value. This example ANOVA determines whether or not the data D depend on the value of the factor Group by comparing the Null Model D=0*Group (left) against the Group Model D=β*Group, with a Cauchy prior on β (right). The top row illustrates the prior probability attributed to the different values of β under the two competing models. Note how both models include β = 0 as a possibility, but given that the probability values must integrate to 1 over the entire β space, for the Null Model p(β = 0) = 1 while for the Group Model, the probability is distributed across all plausible alternative values. The middle row shows the predicted t-values based on these priors, where t represents the difference between the data from the two groups as in Fig. 2. Note how these predictions are more peaked for the Null compared to the Group model. The bottom row compares the predicted probability of finding particular t-values under the two models, and shows how values close to zero (i.e., small or no difference between the groups) are predicted more often by the Null compared to the Group Model, while the opposite is true for large t-values. If conducting the experiment reveals a measured t-values close to zero, the Bayes Factor for including the factor Group would be substantially below 1, providing evidence for the absence of an effect of Group, while the inverse would be true for high t-values.