Archive for posterior likelihood ratio

dodging bullets, IEDs, and fingerprint detection at SimStat19

Posted in pictures, Statistics, University life with tags , , , , , , , , , , , , , , , , , on September 10, 2019 by xi'an

I attended a fairly interesting forensic science session at SimStat 2019 in Salzburg as it concentrated on evidence and measures of evidence rather than on strict applications of Bayesian methodology to forensic problems. Even though American administrations like the FBI or various police departments were involved. It was a highly coherent session and I had a pleasant discussion with some of the speakers after the session. For instance, my friend Alicia Carriquiry presented an approach to determined from images of bullets whether or not they have been fired from the same gun, leading to an interesting case for a point null hypothesis where the point null makes complete sense. The work has been published in Annals of Applied Statistics and is used in practice. The second talk by Danica Ommen on fiducial forensics on IED, asking whether or not copper wires used in the bombs are the same, which is another point null illustration. Which also set an interesting questioning on the dependence of the alternative prior on the distribution of material chosen as it is supposed to cover all possible origins for the disputed item. But more interestingly this talk launched into a discussion of making decision based on finite samplers and unknown parameters, not that specific to forensics, with a definitely surprising representation of the Bayes factor as an expected likelihood ratio which made me first reminiscent of Aitkin’s (1991) infamous posterior likelihood (!) before it dawned on me this was a form of bridge sampling identity where the likelihood ratio only involved parameters common to both models, making it an expression well-defined under both models. This identity could be generalised to the general case by considering a ratio of integrated likelihoods, the extreme case being the ratio equal to the Bayes factor itself. The following two talks by Larry Tang and Christopher Saunders were also focused on the likelihood ratio and their statistical estimates, debating the coherence of using a score function and presenting a functional ABC algorithm where the prior is a Dirichlet (functional) prior. Thus a definitely relevant session from a Bayesian perspective!


posterior likelihood ratio is back

Posted in Statistics, University life with tags , , , , , , , , , on June 10, 2014 by xi'an

“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.