I show how it can be beneficial to express Metropolis accept/reject decisions in terms of comparison with a uniform [0,1] value, u, and to then update u non-reversibly, as part of the Markov chain state, rather than sampling it independently each iteration. This provides a small improvement for random walk Metropolis and Langevin updates in high dimensions. It produces a larger improvement when using Langevin updates with persistent momentum, giving performance comparable to that of Hamiltonian Monte Carlo (HMC) with long trajectories. This is of significance when some variables are updated by other methods, since if HMC is used, these updates can be done only between trajectories, whereas they can be done more often with Langevin updates. I demonstrate that for a problem with some continuous variables, updated by HMC or Langevin updates, and also discrete variables, updated by Gibbs sampling between updates of the continuous variables, Langevin with persistent momentum and non-reversible updates to u samples nearly a factor of two more efficiently than HMC. Benefits are also seen for a Bayesian neural network model in which hyperparameters are updated by Gibbs sampling.
MCMC algorithms such as Metropolis-Hastings algorithms are slowed down by the computation of complex target distributions as exemplified by huge datasets. We offer in this paper a useful generalisation of the Delayed Acceptance approach, devised to reduce the computational costs of such algorithms by a simple and universal divide-and-conquer strategy. The idea behind the generic acceleration is to divide the acceptance step into several parts, aiming at a major reduction in computing time that out-ranks the corresponding reduction in acceptance probability. Each of the components can be sequen…
MCMC algorithms such as Metropolis-Hastings algorithms are slowed down by the computation of complex target distributions as exemplified by huge datasets. We offer in this paper an approach to reduce the computational costs of such algorithms by a simple and universal divide-and-conquer strategy. The idea behind the generic acceleration is to divide the acceptance step into several parts, aiming at a major reduction in computing time that outranks the corresponding reduction in acceptance probability. The division decomposes the "prior x likelihood" term into a product such that some of its co…
Practitioners of Markov chain Monte Carlo (MCMC) may hesitate to use random walk Metropolis-Hastings algorithms, especially variable-at-a-time algorithms with many parameters, because these algorithms require users to select values of tuning parameters (step sizes). These algorithms perform poorly if the step sizes are set to be too low or too high. We show in this paper that it is not difficult for an algorithm to tune these step sizes automatically to obtain a desired acceptance probability, since the logit of the acceptance probability is very nearly linear in the log of the step size, with…
The problem of optimally scaling the proposal distribution in a Markov chain Monte Carlo algorithm is critical to the quality of the generated samples. Much work has gone into obtaining such results for various Metropolis-Hastings (MH) algorithms. Recently, acceptance probabilities other than MH are being employed in problems with intractable target distributions. There is little resource available on tuning the Gaussian proposal distributions for this situation. We obtain optimal scaling results for a general class of acceptance functions, which includes Barker's and Lazy-MH. In particular, o…
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…
Recently developed adaptive Markov chain Monte Carlo (MCMC) methods have been applied successfully to many problems in Bayesian statistics. Grapham is a new open source implementation covering several such methods, with emphasis on graphical models for directed acyclic graphs. The implemented algorithms include the seminal Adaptive Metropolis algorithm adjusting the proposal covariance according to the history of the chain and a Metropolis algorithm adjusting the proposal scale based on the observed acceptance probability. Different variants of the algorithms allow one, for example, to use the…
The multiple Try Metropolis (MTM) algorithm is an advanced MCMC technique based on drawing and testing several candidates at each iteration of the algorithm. One of them is selected according to certain weights and then it is tested according to a suitable acceptance probability. Clearly, since the computational cost increases as the employed number of tries grows, one expects that the performance of an MTM scheme improves as the number of tries increases, as well. However, there are scenarios where the increase of number of tries does not produce a corresponding enhancement of the performance…
Scaling of proposals for Metropolis algorithms is an important practical problem in MCMC implementation. Criteria for scaling based on empirical acceptance rates of algorithms have been found to work consistently well across a broad range of problems. Essentially, proposal jump sizes are increased when acceptance rates are high and decreased when rates are low. In recent years, considerable theoretical support has been given for rules of this type which work on the basis that acceptance rates around 0.234 should be preferred. This has been based on asymptotic results that approximate high dime…
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…
Couplings play a central role in contemporary Markov chain Monte Carlo methods and in the analysis of their convergence to stationarity. In most cases, a coupling must induce relatively fast meeting between chains to ensure good performance. In this paper we fix attention on the random walk Metropolis algorithm and examine a range of coupling design choices. We introduce proposal and acceptance step couplings based on geometric, optimal transport, and maximality considerations. We consider the theoretical properties of these choices and examine their implication for the meeting time of the cha…
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）
A significant part of MCMC methods can be considered as the Metropolis-Hastings (MH) algorithm with different proposal distributions. From this point of view, the problem of constructing a sampler can be reduced to the question - how to choose a proposal for the MH algorithm? To address this question, we propose to learn an independent sampler that maximizes the acceptance rate of the MH algorithm, which, as we demonstrate, is highly related to the conventional variational inference. For Bayesian inference, the proposed method compares favorably against alternatives to sample from the posterio…
There has been considerable interest in designing Markov chain Monte Carlo algorithms by exploiting numerical methods for Langevin dynamics, which includes Hamiltonian dynamics as a deterministic case. A prominent approach is Hamiltonian Monte Carlo (HMC), where a leapfrog discretization of Hamiltonian dynamics is employed. We investigate a recently proposed class of irreversible sampling algorithms, called Hamiltonian assisted Metropolis sampling (HAMS), which uses an augmented target density similarly as in HMC, but involves a flexible proposal scheme and a carefully formulated acceptance-re…
Smoothing in state-space models amounts to computing the conditional distribution of the latent state trajectory, given observations, or expectations of functionals of the state trajectory with respect to this distributions. For models that are not linear Gaussian or possess finite state space, smoothing distributions are in general infeasible to compute as they involve intergrals over a space of dimensionality at least equal to the number of observations. Recent years have seen an increased interest in Monte Carlo-based methods for smoothing, often involving particle filters. One such method …
University of California, Irvine（加州大学尔湾分校）；University of Cambridge（剑桥大学）；University of Amsterdam（阿姆斯特丹大学）
Can we make Bayesian posterior MCMC sampling more efficient when faced with very large datasets? We argue that computing the likelihood for N datapoints in the Metropolis-Hastings (MH) test to reach a single binary decision is computationally inefficient. We introduce an approximate MH rule based on a sequential hypothesis test that allows us to accept or reject samples with high confidence using only a fraction of the data required for the exact MH rule. While this method introduces an asymptotic bias, we show that this bias can be controlled and is more than offset by a decrease in variance …
One main limitation of the existing optimal scaling results for Metropolis--Hastings algorithms is that the assumptions on the target distribution are unrealistic. In this paper, we consider optimal scaling of random-walk Metropolis algorithms on general target distributions in high dimensions arising from practical MCMC models from Bayesian statistics. For optimal scaling by maximizing expected squared jumping distance (ESJD), we show the asymptotically optimal acceptance rate $0.234$ can be obtained under general realistic sufficient conditions on the target distribution. The new sufficient …
We investigate nonlinear state-space models without a closed-form transition density, and propose reformulating such models over their latent noise variables rather than their latent state variables. In doing so the tractable noise density emerges in place of the intractable transition density. For importance sampling methods such as the auxiliary particle filter, this enables importance weights to be computed where they could not be otherwise. As case studies we take two multivariate marine biogeochemical models and perform state and parameter estimation using the particle marginal Metropolis…
We analyse computational efficiency of Metropolis-Hastings algorithms with stochastic AR(1) process proposals. These proposals include, as a subclass, discretized Langevin diffusion (e.g. MALA) and discretized Hamiltonian dynamics (e.g. HMC). We derive expressions for the expected acceptance rate and expected jump size for MCMC methods with general stochastic AR(1) process proposals for the case where the target distribution is absolutely continuous with respect to a Gaussian and the covariance of the Gaussian is allowed to have off-diagonal terms. This allows us to extend what is known about …
We study the mixing time of the Metropolis-adjusted Langevin algorithm (MALA) for sampling from a log-smooth and strongly log-concave distribution. We establish its optimal minimax mixing time under a warm start. Our main contribution is two-fold. First, for a $d$-dimensional log-concave density with condition number $κ$, we show that MALA with a warm start mixes in $\tilde O(κ\sqrt{d})$ iterations up to logarithmic factors. This improves upon the previous work on the dependency of either the condition number $κ$ or the dimension $d$. Our proof relies on comparing the leapfrog integrator with …
Proposals for Metropolis-Hastings MCMC derived by discretizing Langevin diffusion or Hamiltonian dynamics are examples of stochastic autoregressive proposals that form a natural wider class of proposals with equivalent computability. We analyze Metropolis-Hastings MCMC with stochastic autoregressive proposals applied to target distributions that are absolutely continuous with respect to some Gaussian distribution to derive expressions for expected acceptance probability and expected jump size, as well as measures of computational cost, in the limit of high dimension. Thus, we are able to unify…
We present an adaptive method for the automatic scaling of Random-Walk Metropolis-Hastings algorithms, which quickly and robustly identifies the scaling factor that yields a specified overall sampler acceptance probability. Our method relies on the use of the Robbins-Monro search process, whose performance is determined by an unknown steplength constant. We give a very simple estimator of this constant for proposal distributions that are univariate or multivariate normal, together with a sampling algorithm for automating the method. The effectiveness of the algorithm is demonstrated with both …
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…
Stanford University（斯坦福大学）；University of Sydney（悉尼大学）
Traditional MCMC algorithms are computationally intensive and do not scale well to large data. In particular, the Metropolis-Hastings (MH) algorithm requires passing over the entire dataset to evaluate the likelihood ratio in each iteration. We propose a general framework for performing MH-MCMC using mini-batches of the whole dataset and show that this gives rise to approximately a tempered stationary distribution. We prove that the algorithm preserves the modes of the original target distribution and derive an error bound on the approximation with mild assumptions on the likelihood. To furthe…
The Metropolis-Hastings (MH) algorithm is the prototype for a class of Markov chain Monte Carlo methods that propose transitions between states and then accept or reject the proposal. These methods generate a correlated sequence of random samples that convey information about the desired probability distribution. Deciding how this information gets recorded is an important step in the practical design of MH-class algorithm implementations. Many implementations discard most of this information in order to reduce demands on storage capacity and disk writing throughput. Here, we describe how recor…
Hybrid Monte Carlo (HMC) generates samples from a prescribed probability distribution in a configuration space by simulating Hamiltonian dynamics, followed by the Metropolis (-Hastings) acceptance/rejection step. Compressible HMC (CHMC) generalizes HMC to a situation in which the dynamics is reversible but not necessarily Hamiltonian. This article presents a framework to further extend the algorithm. Within the existing framework, each trajectory of the dynamics must be integrated for the same amount of (random) time to generate a valid Metropolis proposal. Our generalized acceptance/rejection…
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…
Reversible jump MCMC (RJ-MCMC) sampling techniques, which allow to jointly tackle model selection and parameter estimation problems in a coherent Bayesian framework, have become increasingly popular in the signal processing literature since the seminal paper of Andrieu and Doucet (IEEE Trans. Signal Process., 47(10), 1999). Crucial to the implementation of any RJ-MCMC sampler is the computation of the so-called Metropolis-Hastings-Green (MHG) ratio, which determines the acceptance probability for the proposed moves. It turns out that the expression of the MHG ratio that was given in the paper …
A classical approach for approximating expectations of functions w.r.t. partially known distributions is to compute the average of function values along a trajectory of a Metropolis-Hastings (MH) Markov chain. A key part in the MH algorithm is a suitable acceptance/rejection of a proposed state, which ensures the correct stationary distribution of the resulting Markov chain. However, the rejection of proposals causes highly correlated samples. In particular, when a state is rejected it is not taken any further into account. In contrast to that we consider a MH importance sampling estimator whi…
Latent position network models are a versatile tool in network science; applications include clustering entities, controlling for causal confounders, and defining priors over unobserved graphs. Estimating each node's latent position is typically framed as a Bayesian inference problem, with Metropolis within Gibbs being the most popular tool for approximating the posterior distribution. However, it is well-known that Metropolis within Gibbs is inefficient for large networks; the acceptance ratios are expensive to compute, and the resultant posterior draws are highly correlated. In this article,…
The independence sampler is one of the most commonly used MCMC algorithms usually as a component of a Metropolis-within-Gibbs algorithm. The common focus for the independence sampler is on the choice of proposal distribution to obtain an as high as possible acceptance rate. In this paper we have a somewhat different focus concentrating on the use of the independence sampler for updating augmented data in a Bayesian framework where a natural proposal distribution for the independence sampler exists. Thus we concentrate on the proportion of the augmented data to update to optimise the independen…
Markov Chain Monte Carlo (MCMC) algorithms are routinely used to draw samples from distributions with intractable normalization constants. However, standard MCMC algorithms do not apply to doubly-intractable distributions in which there are additional parameter-dependent normalization terms; for example, the posterior over parameters of an undirected graphical model. An ingenious auxiliary-variable scheme (Moeller et al., 2004) offers a solution: exact sampling (Propp and Wilson, 1996) is used to sample from a Metropolis-Hastings proposal for which the acceptance probability is tractable. Unfo…
University of Cambridge（剑桥大学）；ETH Zürich（苏黎世联邦理工学院）
Stochastic gradient Markov Chain Monte Carlo algorithms are popular samplers for approximate inference, but they are generally biased. We show that many recent versions of these methods (e.g. Chen et al. (2014)) cannot be corrected using Metropolis-Hastings rejection sampling, because their acceptance probability is always zero. We can fix this by employing a sampler with realizable backwards trajectories, such as Gradient-Guided Monte Carlo (Horowitz, 1991), which generalizes stochastic gradient Langevin dynamics (Welling and Teh, 2011) and Hamiltonian Monte Carlo. We show that this sampler c…
Sequential state estimation in non-linear and non-Gaussian state spaces has a wide range of applications in statistics and signal processing. One of the most effective non-linear filtering approaches, particle filtering, suffers from weight degeneracy in high-dimensional filtering scenarios. Several avenues have been pursued to address high-dimensionality. Among these, particle flow particle filters construct effective proposal distributions by using invertible flow to migrate particles continuously from the prior distribution to the posterior, and sequential Markov chain Monte Carlo (SMCMC) m…
Very recently, Transformation based Markov Chain Monte Carlo (TMCMC) was proposed by Dutta and Bhattcharya (2013) as a much efficient alternative to the Metropolis-Hastings algorithm, Random Walk Metropolis (RWM) algorithm, especially in high dimensions. The main advantage of this algorithm is that it simultaneously updates all components of a high dimensional parameter by some appropriate deterministic transformation of a single random variable, thereby reducing time complexity and enhancing the acceptance rate. The optimal scaling of the additive TMCMC approach has already been studied for t…
We consider the recently introduced Transformation-based Markov Chain Monte Carlo (TMCMC) (Dutta and Bhattacharya (2014)), a methodology that is designed to update all the parameters simultaneously using some simple deterministic transformation of a onedimensional random variable drawn from some arbitrary distribution on a relevant support. The additive transformation based TMCMC is similar in spirit to random walk Metropolis, except the fact that unlike the latter, additive TMCMC uses a single draw from a onedimensional proposal distribution to update the high-dimensional parameter. In this p…
IMT Lille Douai（IMT里尔-杜埃大学）；CRISTAL (UMR 9189)（CRISTAL实验室（UMR 9189））；Universidad Carlos III de Madrid（马德里卡洛斯三世大学）；Image Processing Laboratory, Universitat de València（瓦伦西亚大学图像处理实验室）
Bayesian methods and their implementations by means of sophisticated Monte Carlo techniques have become very popular in signal processing over the last years. Importance Sampling (IS) is a well-known Monte Carlo technique that approximates integrals involving a posterior distribution by means of weighted samples. In this work, we study the assignation of a single weighted sample which compresses the information contained in a population of weighted samples. Part of the theory that we present as Group Importance Sampling (GIS) has been employed implicitly in different works in the literature. T…
We propose a new sampling algorithm combining two quite powerful ideas in the Markov chain Monte Carlo literature -- adaptive Metropolis sampler and two-stage Metropolis-Hastings sampler. The proposed sampling method will be particularly very useful for high-dimensional posterior sampling in Bayesian models with expensive likelihoods. In the first stage of the proposed algorithm, an adaptive proposal is used based on the previously sampled states and the corresponding acceptance probability is computed based on an approximated inexpensive target density. The true expensive target density is ev…
Delocalization of bias in unadjusted Hamiltonian Monte Carlo and underdamped Langevin
Department of Mathematics, University of California, Los Angeles, CA 90095, USA（加州大学洛杉矶分校数学系）；Courant Institute, New York University, NY 10012, USA（纽约大学Courant研究所）
Random Walk Metropolis Hastings (RWMH) algorithm, is quite inefficient in high dimensions because of its abysmally slow acceptance rate. The slow acceptance rate results from the fact that RWMH separately updates each coordinate of the chain at every step. Dutta and Bhattacharya (2013) proposed a new technique called Transformation based Markov Chain Monte Carlo (TMCMC) aimed at overcoming these problems. This method updates all co-ordinates at a time- ensuring stable acceptance in all dimensions. We have shown here that geometric ergodicity is achieved for sub-exponential targets for two vers…
Markov chain Monte Carlo (MCMC) algorithms have long been the main workhorses of Bayesian inference. Among them, Hamiltonian Monte Carlo (HMC) has recently become very popular due to its efficiency resulting from effective use of the gradients of the target distribution. In privacy-preserving machine learning, differential privacy (DP) has become the gold standard in ensuring that the privacy of data subjects is not violated. Existing DP MCMC algorithms either use random-walk proposals, or do not use the Metropolis--Hastings (MH) acceptance test to ensure convergence without decreasing their s…
One of the most widely used samplers in practice is the component-wise Metropolis-Hastings (CMH) sampler that updates in turn the components of a vector valued Markov chain using accept-reject moves generated from a proposal distribution. When the target distribution of a Markov chain is irregularly shaped, a `good' proposal distribution for one part of the state space might be a `poor' one for another part of the state space. We consider a component-wise multiple-try Metropolis (CMTM) algorithm that can automatically choose from a set of candidate moves sampled from different distributions. T…
Department of Statistics, University of Oxford（1 统计系，牛津大学）；Inria, CNRS, École Normale Supérieure, PSL Research University（2 Inria，法国国家科学研究中心，巴黎高等师范学院，PSL研究大学）；Department of Mathematical Sciences, University of Copenhagen（3 数学科学系，哥本哈根大学）
We consider the problem of sampling from a strongly log-concave density in $\mathbb{R}^d$, and prove a non-asymptotic upper bound on the mixing time of the Metropolis-adjusted Langevin algorithm (MALA). The method draws samples by simulating a Markov chain obtained from the discretization of an appropriate Langevin diffusion, combined with an accept-reject step. Relative to known guarantees for the unadjusted Langevin algorithm (ULA), our bounds show that the use of an accept-reject step in MALA leads to an exponentially improved dependence on the error-tolerance. Concretely, in order to obtai…
Metropolis Hastings nested sampling evolves a Markov chain, accepting new points along the chain according to a version of the Metropolis Hastings acceptance ratio, which has been modified to satisfy the nested sampling likelihood constraint. The geometric nested sampling algorithm I present here is based on the Metropolis Hastings method, but treats parameters as though they represent points on certain geometric objects, namely circles, tori and spheres. For parameters which represent points on a circle or torus, the trial distribution is "wrapped" around the domain of the posterior distribut…
