## the demise of the Bayes factor

Posted in Books, Kids, Statistics, Travel, University life with tags , , , , , , , , on December 8, 2014 by xi'an

With Kaniav Kamary, Kerrie Mengersen, and Judith Rousseau, we have just arXived (and submitted) a paper entitled “Testing hypotheses via a mixture model”. (We actually presented some earlier version of this work in Cancũn, Vienna, and Gainesville, so you may have heard of it already.) The notion we advocate in this paper is to replace the posterior probability of a model or an hypothesis with the posterior distribution of the weights of a mixture of the models under comparison. That is, given two models under comparison,

$\mathfrak{M}_1:x\sim f_1(x|\theta_1) \text{ versus } \mathfrak{M}_2:x\sim f_2(x|\theta_2)$

we propose to estimate the (artificial) mixture model

$\mathfrak{M}_{\alpha}:x\sim\alpha f_1(x|\theta_1) + (1-\alpha) f_2(x|\theta_2)$

and in particular derive the posterior distribution of α. One may object that the mixture model is neither of the two models under comparison but this is the case at the boundary, i.e., when α=0,1. Thus, if we use prior distributions on α that favour the neighbourhoods of 0 and 1, we should be able to see the posterior concentrate near 0 or 1, depending on which model is true. And indeed this is the case: for any given Beta prior on α, we observe a higher and higher concentration at the right boundary as the sample size increases. And establish a convergence result to this effect. Furthermore, the mixture approach offers numerous advantages, among which [verbatim from the paper]: