posterior likelihood ratio is back

“The PLR turns out to be a natural Bayesian measure of evidence of the studied hypotheses.”

Isabelle Smith and André Ferrari just arXived a paper on the posterior distribution of the likelihood ratio. This is in line with Murray Aitkin’s notion of considering the likelihood ratio

f(x|\theta_0) / f(x|\theta)

as a prior quantity, when contemplating the null hypothesis that θ is equal to θ0. (Also advanced by Alan Birnbaum and Arthur Dempster.) A concept we criticised (rather strongly) in our Statistics and Risk Modelling paper with Andrew Gelman and Judith Rousseau.  The arguments found in the current paper in defence of the posterior likelihood ratio are quite similar to Aitkin’s:

  • defined for (some) improper priors;
  • invariant under observation or parameter transforms;
  • more informative than tthe posterior mean of the posterior likelihood ratio, not-so-incidentally equal to the Bayes factor;
  • avoiding using the posterior mean for an asymmetric posterior distribution;
  • achieving some degree of reconciliation between Bayesian and frequentist perspectives, e.g. by being equal to some p-values;
  • easily computed by MCMC means (if need be).

One generalisation found in the paper handles the case of composite versus composite hypotheses, of the form

\int\mathbb{I}\left( p(x|\theta_1)<p(x|\theta_0)\right)\pi(\text{d}\theta_1|x)\pi(\text{d}\theta_0|x)

which brings back an earlier criticism I raised (in Edinburgh, at ICMS, where as one-of-those-coincidences, I read this paper!), namely that using the product of the marginals rather than the joint posterior is no more a standard Bayesian practice than using the data in a prior quantity. And leads to multiple uses of the data. Hence, having already delivered my perspective on this approach in the past, I do not feel the urge to “raise the flag” once again about a paper that is otherwise well-documented and mathematically rich.


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