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Anthony Lee - A general perspective on the Metropolis–Hastings kernel - Part 2
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 II: Metropolis--Hastings Markov chains with stopped processes as proposals (A. Lee)

We review briefly the framework from last week's talk, and use it to show that certain Markov chains that involve the simulation of a random number of auxiliary variables can be clearly validated. This is accomplished via "certificates" consisting of a probability measure on an extended state space, an involution and an acceptance function. As examples, we will cover a generalization of the basic version of the No U-turn Sampler from Hoffman and Gelman (2014), as well as multiple-try Metropolis and pseudo-marginal methods with adaptive numbers of proposals.

1. Recap: general result, involutions, densities.
2. Deterministic proposals and non-reversible Markov chains + HMC
3. Stopping times, stopped processes
4. Doubly infinite Markov chain proposals
5. NUTS-like kernels
6. Multiple try and pseudo-marginal schemes

Mar 24, 2021 05:00 PM in Paris

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Anthony Lee
Professor @University of Bristol
His research is mainly in the area of stochastic algorithms for approximating intractable quantities that arise in data analysis. Examples of such algorithms are Markov chain and Sequential Monte Carlo. He works on both theory and methodology, with a focus on algorithms that scale well in parallel and distributed computing environments. Research in this area is interdisciplinary, bringing together advances in applied probability, algorithms, architecture and statistics. He is also interested in computational and statistical trade-offs when conducting inference.