Bayes Factors and Posterior Estimation: Two Sides of the Very Same Coin

Equivalence testing for linear regression.

“…For example, with equal prior model probabilities, a BF 10 equal to 0.20 indicates that the null model is five times more likely than the alternative model. Going forward, we suppose that the equal prior model probabilities (Pr(Model 0) = Pr(Model 1) = 0.5) are always assumed, as is often (implicitly) done in practice; but see Tendeiro and Kiers (2019) and Campbell and Gustafson (2023) for discussion of this practice. Bayesian methods require one to define appropriate prior distributions for all model parameters (Consonni & Veronese, 2008).…”

Discussion of “Specifying prior distributions in reliability applications”

“…Once the posterior probabilities (2) have been computed, a summary of evidence provided by the observed data in favor of model $$$ {M}_0 $$$ is given by the Bayes factor $$$ {B}_{01} $$$ (see, e.g., Kass and Raftery 3 and Campbell and Gustafson 4 ), which is defined as the ratio of the posterior odds of $$$ {M}_0 $$$ to its prior odds: $$$$ {B}_{01}=\frac{\Pr \left\{{M}_0\mid \mathrm{data}\right\}/\Pr \left\{{M}_1\mid \mathrm{data}\right\}}{\Pr \left\{{M}_0\right\}/\Pr \left\{{M}_1\right\}}. $$$$ When the models $$$ {M}_0 $$$ and $$$ {M}_1 $$$ are equally prior probable, so that $$$ \Pr \left\{{M}_0\right\}=\Pr \left\{{M}_1\right\}=0.5 $$$, then the Bayes factor is equal to the posterior odds $$$ \Pr \left\{{M}_0\mid \mathrm{data}\right\}/\Pr \left\{{M}_1\mid \mathrm{data}\right\} $$$.…”