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Christophe Andrieu - A general perspective on the Metropolis–Hastings kernel - Part 1
Since its inception the Metropolis–Hastings kernel has been applied in sophisticated ways to address ever more challenging and diverse sampling problems. Its success stems from the flexibility brought by the fact that its verification and sampling implementation rest on a local “detailed balance” condition, as opposed to a global condition in the form of a typically intractable integral equation. While checking the local condition is routine in the simplest scenarios, this proves much more difficult for complicated applications involving auxiliary structures and variables.

The aim of these two presentations is an attempt to bring together ideas making verification of correctness of complex Markov chain Monte Carlo kernels a purely mechanical or algebraic exercise, while at the same time enabling simpler and unambiguous communication of complex ideas. This is also an opportunity to present new algorithms arising from, it is hoped, the gained clarity.

Part I: the Metropolis-Hastings triptych, densities and some examples (C. Andrieu)

We first briefly review the abstract framework needed, together with notation, with a particular focus on the bare bone requirements for a MH update to be correct. An attempt is made to convince the audience that the use of general, yet basic, measure theoretic concepts brings simplicity and clarity to the arguments. The usefulness of this point of view is illustrated on various example, ranging from very simple to moderately complicated example.

1. Motivating example
2. General result, involutions and densities
3. Simple examples
4. Deterministic proposals and non-reversible Markov chains
5. Delayed rejection, slices, and the extra chance
7. Discrete time Event chain algorithms
8. MHAAR: MH with averaged acceptance ratios

Mar 17, 2021 05:00 PM in Paris

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Speakers

Christophe Andrieu
Professor @University of Bristol
Christophe Andrieu is professor of statistical science at the University of Bristol. He gained his PhD in 1998 at Paris XV, in Signal Processing. After a 3 year postdoc in the Signal Processing group in Cambridge he joined Bristol in 2001.