3,226 research outputs found
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
Stability and examples of some approximate MCMC algorithms.
Approximate Monte Carlo algorithms are not uncommon these days, their applicability is related to the possibility of controlling the computational cost by introducing some noise or approximation in the method. We focus on the stability properties of a particular approximate MCMC algorithm, which we term noisy Metropolis-Hastings. Such properties have been studied before in tandem with the pseudo-marginal algorithm, but under fairly strong assumptions. Here, we examine the noisy Metropolis-Hastings algorithm in more detail and explore possible corrective actions for reducing the introduced bias. In this respect, a novel approximate method is presented, motivated by the class of exact algorithms with randomised acceptance. We also discuss some applications and theoretical guarantees of this new approach
Noisy Monte Carlo: Convergence of Markov chains with approximate transition kernels
Monte Carlo algorithms often aim to draw from a distribution by
simulating a Markov chain with transition kernel such that is
invariant under . However, there are many situations for which it is
impractical or impossible to draw from the transition kernel . For instance,
this is the case with massive datasets, where is it prohibitively expensive to
calculate the likelihood and is also the case for intractable likelihood models
arising from, for example, Gibbs random fields, such as those found in spatial
statistics and network analysis. A natural approach in these cases is to
replace by an approximation . Using theory from the stability of
Markov chains we explore a variety of situations where it is possible to
quantify how 'close' the chain given by the transition kernel is to
the chain given by . We apply these results to several examples from spatial
statistics and network analysis.Comment: This version: results extended to non-uniformly ergodic Markov chain
Perturbation theory for Markov chains via Wasserstein distance
Perturbation theory for Markov chains addresses the question how small
differences in the transitions of Markov chains are reflected in differences
between their distributions. We prove powerful and flexible bounds on the
distance of the th step distributions of two Markov chains when one of them
satisfies a Wasserstein ergodicity condition. Our work is motivated by the
recent interest in approximate Markov chain Monte Carlo (MCMC) methods in the
analysis of big data sets. By using an approach based on Lyapunov functions, we
provide estimates for geometrically ergodic Markov chains under weak
assumptions. In an autoregressive model, our bounds cannot be improved in
general. We illustrate our theory by showing quantitative estimates for
approximate versions of two prominent MCMC algorithms, the Metropolis-Hastings
and stochastic Langevin algorithms.Comment: 31 pages, accepted at Bernoulli Journa
Quasi-Newton particle Metropolis-Hastings
Particle Metropolis-Hastings enables Bayesian parameter inference in general
nonlinear state space models (SSMs). However, in many implementations a random
walk proposal is used and this can result in poor mixing if not tuned correctly
using tedious pilot runs. Therefore, we consider a new proposal inspired by
quasi-Newton algorithms that may achieve similar (or better) mixing with less
tuning. An advantage compared to other Hessian based proposals, is that it only
requires estimates of the gradient of the log-posterior. A possible application
is parameter inference in the challenging class of SSMs with intractable
likelihoods. We exemplify this application and the benefits of the new proposal
by modelling log-returns of future contracts on coffee by a stochastic
volatility model with -stable observations.Comment: 23 pages, 5 figures. Accepted for the 17th IFAC Symposium on System
Identification (SYSID), Beijing, China, October 201
Geometric ergodicity of the Random Walk Metropolis with position-dependent proposal covariance
We consider a Metropolis-Hastings method with proposal kernel
, where is the current state. After discussing
specific cases from the literature, we analyse the ergodicity properties of the
resulting Markov chains. In one dimension we find that suitable choice of
can change the ergodicity properties compared to the Random Walk
Metropolis case , either for the better or worse. In
higher dimensions we use a specific example to show that judicious choice of
can produce a chain which will converge at a geometric rate to its
limiting distribution when probability concentrates on an ever narrower ridge
as grows, something which is not true for the Random Walk Metropolis.Comment: 15 pages + appendices, 4 figure
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