## Carlin and Chib (1995) for fixed dimension problems

Yesterday, I was part of a (public) thesis committee at the Université Pierre et Marie Curie, in down-town Paris. After a bit of a search for the defence room (as the campus is still undergoing a massive asbestos clean-up, 20 years after it started…!), I listened to Florian Maire delivering his talk on an array of work in computational statistics ranging from the theoretical (Peskun ordering) to the methodological (Monte Carlo online EM) to the applied (unsupervised learning of classes shapes via deformable templates). The implementation of the online EM algorithm involved the use of pseudo-priors à la Carlin and Chib (1995), even though the setting was a fixed-dimension one, in order to fight the difficulty of exploring the space of templates by a regular Gibbs sampler. (As usual, the design of the pseudo-priors was crucial to the success of the method.) The thesis also included a recent work with Randal Douc and Jimmy Olsson on ranking inhomogeneous Markov kernels of the type

$P \circ Q \circ P \circ Q \circ ...$

against alternatives with components (P’,Q’). The authors were able to characterise minimal conditions for a Peskun-ordering domination on the components to transfer to the combination. Quite an interesting piece of work for a PhD thesis!

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