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.
Markov Chain Monte Carlo (MCMC) methods are a powerful tool for computation with complex probability distributions. However the performance of such methods is critically dependant on properly tuned parameters, most of which are difficult if not impossible to know a priori for a given target distribution. Adaptive MCMC methods aim to address this by allowing the parameters to be updated during sampling based on previous samples from the chain at the expense of requiring a new theoretical analysis to ensure convergence. In this work we extend the convergence theory of adaptive MCMC methods to a …
Couplings play a central role in the analysis of Markov chain Monte Carlo algorithms and appear increasingly often in the algorithms themselves, e.g. in convergence diagnostics, parallelization, and variance reduction techniques. Existing couplings of the Metropolis-Hastings algorithm handle the proposal and acceptance steps separately and fall short of the upper bound on one-step meeting probabilities given by the coupling inequality. This paper introduces maximal couplings which achieve this bound while retaining the practical advantages of current methods. We consider the properties of thes…
Bayesian modelling and computational inference by Markov chain Monte Carlo (MCMC) is a principled framework for large-scale uncertainty quantification, though is limited in practice by computational cost when implemented in the simplest form that requires simulating an accurate computer model at each iteration of the MCMC. The delayed acceptance Metropolis--Hastings MCMC leverages a reduced model for the forward map to lower the compute cost per iteration, though necessarily reduces statistical efficiency that can, without care, lead to no reduction in the computational cost of computing estim…
Note on the computation of the Metropolis-Hastings ratio for Birth-or-Death moves in trans-dimensional MCMC algorithms for signal decomposition problems
可逆跳跃MCMC（RJ-MCMC）采样技术允许在相干贝叶斯框架中联合处理模型选择和参数估计问题，自从Andrieu和Doucet的开创性论文（IEEE Trans. Signal Process.，47（10），1999）。任何RJ-MCMC采样器的实现的关键是计算所谓的大都会-黑斯廷斯-格林（MHG）比率，它决定了建议移动的接受概率。事实证明，在Andrieu和Doucet的论文中给出的MHG比的表达式对于“要么死要么死”移动-最简单的一种跨维移动，几乎在RJ-MCMC的所有应用中都使用到信号分解问题-是错误的。不幸的是，这个错误已经被复制在许多后续的论文处理的RJ-MCMC采样在信号处理文献。本文讨论了MHG比的计算，重点讨论了建议核可以分解为简单核的混合的情况，MHG比很容易计算。我们提供了充分的条件下，MHG比的混合物可以推导出MHG比的基本内核，它是由。作为应用，我们考虑了非死即死的移动情形，并给出了Andrieu和Doucet论文中错误率的修正表达式。
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 dev…
Couplings play a central role in the analysis of Markov chain convergence and in the construction of novel Markov chain Monte Carlo estimators, diagnostics, and variance reduction techniques. The set of possible couplings is often intractable, frustrating the search for tight bounds and efficient estimators. To address this challenge for algorithms in the Metropolis-Hastings (MH) family, we establish a simple characterization of the set of MH transition kernel couplings. We then extend this result to describe the set of maximal couplings of the MH kernel, resolving an open question of O'Leary …
We consider the problem of inference for nonlinear, multivariate diffusion processes, satisfying Itô stochastic differential equations (SDEs), using data at discrete times that may be incomplete and subject to measurement error. Our starting point is a state-of-the-art correlated pseudo-marginal Metropolis-Hastings algorithm, that uses correlated particle filters to induce strong and positive correlation between successive likelihood estimates. However, unless the measurement error or the dimension of the SDE is small, correlation can be eroded by the resampling steps in the particle filter. W…
We introduce a general framework that constructs estimators with reduced variance for random walk Metropolis and Metropolis-adjusted Langevin algorithms. The resulting estimators require negligible computational cost and are derived in a post-process manner utilising all proposal values of the Metropolis algorithms. Variance reduction is achieved by producing control variates through the approximate solution of the Poisson equation associated with the target density of the Markov chain. The proposed method is based on approximating the target density with a Gaussian and then utilising accurate…
Markov chain Monte Carlo (MCMC) methods to sample from a probability distribution $π$ defined on a space $(Θ,\mathcal{T})$ consist of the simulation of realisations of Markov chains $\{θ_{n},n\geq1\}$ of invariant distribution $π$ and such that the distribution of $θ_{i}$ converges to $π$ as $i\rightarrow\infty$. In practice one is typically interested in the computation of expectations of functions, say $f$, with respect to $π$ and it is also required that averages $M^{-1}\sum_{n=1}^{M}f(θ_{n})$ converge to the expectation of interest. The iterative nature of MCMC makes it difficult to develo…
Hamiltonian Monte Carlo (HMC) is a powerful Markov Chain Monte Carlo (MCMC) method for sampling from complex high-dimensional continuous distributions. However, in many situations it is necessary or desirable to combine HMC with other Metropolis-Hastings (MH) samplers. The common HMC-within-Gibbs strategy implies a trade-off between long HMC trajectories and more frequent other MH updates. Addressing this trade-off has been the focus of several recent works. In this paper we propose Metropolis Augmented Hamiltonian Monte Carlo (MAHMC), an HMC variant that allows MH updates within HMC and elimi…
In order to construct accurate proposers for Metropolis-Hastings Markov Chain Monte Carlo, we integrate ideas from probabilistic graphical models and neural networks in an open-source framework we call Lightweight Inference Compilation (LIC). LIC implements amortized inference within an open-universe declarative probabilistic programming language (PPL). Graph neural networks are used to parameterize proposal distributions as functions of Markov blankets, which during "compilation" are optimized to approximate single-site Gibbs sampling distributions. Unlike prior work in inference compilation …
Metropolis-Hastings (MH) is a commonly-used MCMC algorithm, but it can be intractable on large datasets due to requiring computations over the whole dataset. In this paper, we study minibatch MH methods, which instead use subsamples to enable scaling. We observe that most existing minibatch MH methods are inexact (i.e. they may change the target distribution), and show that this inexactness can cause arbitrarily large errors in inference. We propose a new exact minibatch MH method, TunaMH, which exposes a tunable trade-off between its batch size and its theoretically guaranteed convergence rat…
Markov Chain Monte Carlo (MCMC) algorithms are widely used for stochastic optimization, sampling, and integration of mathematical objective functions, in particular, in the context of Bayesian inverse problems and parameter estimation. For decades, the algorithm of choice in MCMC simulations has been the Metropolis-Hastings (MH) algorithm. An advancement over the traditional MH-MCMC sampler is the Delayed-Rejection Adaptive Metropolis (DRAM). In this paper, we present MatDRAM, a stochastic optimization, sampling, and Monte Carlo integration toolbox in MATLAB which implements a variant of the D…
Metropolis-Hastings view on variational inference and adversarial training
National Research University Higher School of Economics（国家研究型高等经济学院）；Skolkovo Institute of Science and Technology（斯科尔科沃科学技术研究院）；Samsung AI Center Moscow（三星莫斯科人工智能中心）；HSE（高等经济学院）；Skoltech（斯科尔科沃科学技术研究院）
Component-wise MCMC algorithms, including Gibbs and conditional Metropolis-Hastings samplers, are commonly used for sampling from multivariate probability distributions. A long-standing question regarding Gibbs algorithms is whether a deterministic-scan (systematic-scan) sampler converges faster than its random-scan counterpart. We answer this question when the samplers involve two components by establishing an exact quantitative relationship between the $L^2$ convergence rates of the two samplers. The relationship shows that the deterministic-scan sampler converges faster. We also establish q…
