Peskun–Tierney ordering for Markovian Monte Carlo: Beyond the reversible scenario

Historically time-reversibility of the transitions or processes underpinning Markov chain Monte Carlo methods (MCMC) has played a key role in their development, while the self-adjointness of associated operators together with the use of classical functional analysis techniques on Hilbert spaces have led to powerful and practically successful tools to characterise and compare their performance. Similar results for algorithms relying on nonreversible Markov processes are scarce. We show that for a type of nonreversible Monte Carlo Markov chains and processes, of current or renewed interest in the physics and statistical literatures, it is possible to develop comparison results which closely mirror those available in the reversible scenario. We show that these results shed light on earlier literature, proving some conjectures and strengthening some earlier results

On extended state-space constructions for monte carlo methods

This thesis develops computationally efficient methodology in two areas. Firstly, we consider a particularly challenging class of discretely observed continuous-time point-process models. For these, we analyse and improve an existing filtering algorithm based on sequential Monte Carlo (smc) methods. To estimate the static parameters in such models, we devise novel particle Gibbs samplers. One of these exploits a sophisticated non-entred parametrisation whose benefits in a Markov chain Monte Carlo (mcmc) context have previously been limited by the lack of blockwise updates for the latent point process. We apply this algorithm to a Lévy-driven stochastic volatility model. Secondly, we devise novel Monte Carlo methods – based around pseudo-marginal and conditional smc approaches – for performing optimisation in latent-variable models and more generally. To ease the explanation of the wide range of techniques employed in this work, we describe a generic importance-sampling framework which admits virtually all Monte Carlo methods, including smc and mcmc methods, as special cases. Indeed, hierarchical combinations of different Monte Carlo schemes such as smc within mcmc or smc within smc can be justified as repeated applications of this framework

Stability of Noisy Metropolis-Hastings

Pseudo-marginal Markov chain Monte Carlo methods for sampling from intractable distributions have gained recent interest and have been theoretically studied in considerable depth. Their main appeal is that they are exact, in the sense that they target marginally the correct invariant distribution. However, the pseudo-marginal Markov chain can exhibit poor mixing and slow convergence towards its target. As an alternative, a subtly different Markov chain can be simulated, where better mixing is possible but the exactness property is sacrificed. This is the noisy algorithm, initially conceptualised as Monte Carlo within Metropolis (MCWM), which has also been studied but to a lesser extent. The present article provides a further characterisation of the noisy algorithm, with a focus on fundamental stability properties like positive recurrence and geometric ergodicity. Sufficient conditions for inheriting geometric ergodicity from a standard Metropolis-Hastings chain are given, as well as convergence of the invariant distribution towards the true target distribution

A general perspective on the Metropolis-Hastings kernel

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 rests 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. Our aim is to develop a framework making establishing correctness of complex Markov chain Monte Carlo kernels a purely mechanical or algebraic exercise, while making communication of ideas simpler and unambiguous by allowing a stronger focus on essential features -- a choice of embedding distribution, an involution and occasionally an acceptance function -- rather than the induced, boilerplate structure of the kernels that often tends to obscure what is important. This framework can also be used to validate kernels that do not satisfy detailed balance, i.e. which are not reversible, but a modified version thereof

Analysis of two-component Gibbs samplers using the theory of two projections

The theory of two projections is utilized to study two-component Gibbs samplers. Through this theory, previously intractable problems regarding the asymptotic variances of two-component Gibbs samplers are reduced to elementary matrix algebra exercises. It is found that in terms of asymptotic variance, the two-component random-scan Gibbs sampler is never much worse, and could be considerably better than its deterministic-scan counterpart, provided that the selection probability is appropriately chosen. This is especially the case when there is a large discrepancy in computation cost between the two components. The result contrasts with the known fact that the deterministic-scan version has a faster convergence rate. A modified version of the deterministic-scan sampler that accounts for computation cost behaves similarly to the random-scan version. As a side product, some general formulas for characterizing the convergence rate of a possibly non-reversible or time-inhomogeneous Markov chain in an operator theoretic framework are developed