Step 2 of mash uses Bayes’ theorem to compute the posterior distribution for each effect given the condition-by-condition results and the fitted prior (eq. 1). These posterior distributions yield improved effect estimates—posterior means and standard deviations—that account for sparsity and correlations among effects. We use these estimates to define quantitative measures of effect sharing between any two conditions: “sharing by sign” (effects have the same sign), and “sharing by magnitude” (effects have similar magnitude—here defined to be within a factor of 2, although other thresholds could be used; see Supplementary Note). The posterior distributions also yield a condition-specific measure of significance for each effect, the “local false sign rate”, or lfsr16, that is analogous to a false discovery rate, but more stringent because it requires true discoveries to be not only non-zero, but also correctly signed. Finally, mash also computes, for each unit, a Bayes Factor that summarizes the overall significance of the unit—i.e., overall evidence for a non-zero effect in any condition.
