In this article, we derive a novel non-reversible, continuous-time Markov chain Monte Carlo (MCMC) sampler, called Coordinate Sampler, based on a piecewise deterministic Markov process (PDMP), which can be seen as a variant of the Zigzag sampler. In addition to proving a theoretical validation for this new sampling algorithm, we show that the Markov chain it induces exhibits geometrical ergodicity convergence, for distributions whose tails decay at least as fast as an exponential distribution and at most as fast as a Gaussian distribution. Several numerical examples highlight that our coordinate sampler is more efficient than the Zigzag sampler, in terms of effective sample size.
In this paper, we introduce efficient ensemble Markov Chain Monte Carlo (MCMC) sampling methods for Bayesian computations in the univariate stochastic volatility model. We compare the performance of our ensemble MCMC methods with an improved version of a recent sampler of Kastner and Fruwirth-Schnatter (2014). We show that ensemble samplers are more efficient than this state of the art sampler by a factor of about 3.1, on a data set simulated from the stochastic volatility model. This performance gain is achieved without the ensemble MCMC sampler relying on the assumption that the latent proce…
A novel computationally efficient Markov chain Monte Carlo (MCMC) scheme for latent Gaussian models (LGMs) is proposed in this paper. The sampling scheme is a two block Gibbs sampling scheme designed to exploit the model structure of LGMs. We refer to the proposed sampling scheme as the MCMC split sampler. The principle idea behind the MCMC split sampler is to split the latent Gaussian parameters into two vectors. The former vector consists of latent parameters which appear in the data density function, while the latter vector consists of latent parameters which do not appear in it. The former…
This work introduces a class of rejection-free Markov chain Monte Carlo (MCMC) samplers, named the Bouncy Hybrid Sampler, which unifies several existing methods from the literature. Examples include the Bouncy Particle Sampler of Peters and de With (2012), Bouchard-Cote et al. (2015) and the Hamiltonian MCMC. Following the introduced general framework, we derive a new sampler called the Quadratic Bouncy Hybrid Sampler. We apply this novel sampler to the problem of sampling from a truncated Gaussian distribution.
A general methodology is presented for the construction and effective use of control variates for reversible MCMC samplers. The values of the coefficients of the optimal linear combination of the control variates are computed, and adaptive, consistent MCMC estimators are derived for these optimal coefficients. All methodological and asymptotic arguments are rigorously justified. Numerous MCMC simulation examples from Bayesian inference applications demonstrate that the resulting variance reduction can be quite dramatic.
Markov Chain Monte Carlo (MCMC) is a well-established family of algorithms which are primarily used in Bayesian statistics to sample from a target distribution when direct sampling is challenging. Single instances of MCMC methods are widely considered hard to parallelise in a problem-agnostic fashion and hence, unsuitable to meet both constraints of high accuracy and high throughput. Sequential Monte Carlo (SMC) Samplers can address the same problem, but are parallelisable: they share with Particle Filters the same key tasks and bottleneck. Although a rich literature already exists on MCMC met…
The R package MfUSampler provides Monte Carlo Markov Chain machinery for generating samples from multivariate probability distributions using univariate sampling algorithms such as Slice Sampler and Adaptive Rejection Sampler. The sampler function performs a full cycle of univariate sampling steps, one coordinate at a time. In each step, the latest sample values obtained for other coordinates are used to form the conditional distributions. The concept is an extension of Gibbs sampling where each step involves, not an independent sample from the conditional distribution, but a Markov transition…
Sequential Monte Carlo (SMC) methods are not only a popular tool in the analysis of state space models, but offer an alternative to MCMC in situations where Bayesian inference must proceed via simulation. This paper introduces a new SMC method that uses adaptive MCMC kernels for particle dynamics. The proposed algorithm features an online stochastic optimization procedure to select the best MCMC kernel and simultaneously learn optimal tuning parameters. Theoretical results are presented that justify the approach and give guidance on how it should be implemented. Empirical results, based on ana…
In this paper we propose a general framework of performing MCMC with only a mini-batch of data. We show by estimating the Metropolis-Hasting ratio with only a mini-batch of data, one is essentially sampling from the true posterior raised to a known temperature. We show by experiments that our method, Mini-batch Tempered MCMC (MINT-MCMC), can efficiently explore multiple modes of a posterior distribution. Based on the Equi-Energy sampler (Kou et al. 2006), we developed a new parallel MCMC algorithm based on the Equi-Energy sampler, which enables efficient sampling from high-dimensional multi-mo…
Markov chain Monte Carlo (MCMC) methods are ubiquitous tools for simulation-based inference in many fields but designing and identifying good MCMC samplers is still an open question. This paper introduces a novel MCMC algorithm, namely, Auto Adapt MCMC. For sampling variables or blocks of variables, we use two levels of adaptation where the inner adaptation optimizes the MCMC performance within each sampler, while the outer adaptation explores the valid space of kernels to find the optimal samplers. We provide a theoretical foundation for our approach. To show the generality and usefulness of …
It is commonly admitted that non-reversible Markov chain Monte Carlo (MCMC) algorithms usually yield more accurate MCMC estimators than their reversible counterparts. In this note, we show that in addition to their variance reduction effect, some non-reversible MCMC algorithms have also the undesirable property to slow down the convergence of the Markov chain. This point, which has been overlooked by the literature, has obvious practical implications. We illustrate this phenomenon for different non-reversible versions of the Metropolis-Hastings algorithm on several discrete state space example…
We introduce a new Markov chain Monte Carlo (MCMC) sampler called the Markov Interacting Importance Sampler (MIIS). The MIIS sampler uses conditional importance sampling (IS) approximations to jointly sample the current state of the Markov Chain and estimate conditional expectations, possibly by incorporating a full range of variance reduction techniques. We compute Rao-Blackwellized estimates based on the conditional expectations to construct control variates for estimating expectations under the target distribution. The control variates are particularly efficient when there are substantial c…
With the rapidly growing scales of statistical problems, subset based communication-free parallel MCMC methods are a promising future for large scale Bayesian analysis. In this article, we propose a new Weierstrass sampler for parallel MCMC based on independent subsets. The new sampler approximates the full data posterior samples via combining the posterior draws from independent subset MCMC chains, and thus enjoys a higher computational efficiency. We show that the approximation error for the Weierstrass sampler is bounded by some tuning parameters and provide suggestions for choice of the va…
In this work we present a non-reversible, tuning- and rejection-free Markov chain Monte Carlo which naturally fits in the framework of hit-and-run. The sampler only requires access to the gradient of the log-density function, hence the normalizing constant is not needed. We prove the proposed Markov chain is invariant for the target distribution and illustrate its applicability through a wide range of examples. We show that the sampler introduced in the present paper is intimately related to the continuous sampler of Peters and de With (2012), Bouchard-Cote et al. (2017). In particular, the co…
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 …
We consider various versions of adaptive Gibbs and Metropolis-within-Gibbs samplers, which update their selection probabilities (and perhaps also their proposal distributions) on the fly during a run by learning as they go in an attempt to optimize the algorithm. We present a cautionary example of how even a simple-seeming adaptive Gibbs sampler may fail to converge. We then present various positive results guaranteeing convergence of adaptive Gibbs samplers under certain conditions.
Markov jump processes (or continuous-time Markov chains) are a simple and important class of continuous-time dynamical systems. In this paper, we tackle the problem of simulating from the posterior distribution over paths in these models, given partial and noisy observations. Our approach is an auxiliary variable Gibbs sampler, and is based on the idea of uniformization. This sets up a Markov chain over paths by alternately sampling a finite set of virtual jump times given the current path and then sampling a new path given the set of extant and virtual jump times using a standard hidden Marko…
Parallel tempering (PT) methods are a popular class of Markov chain Monte Carlo schemes used to sample complex high-dimensional probability distributions. They rely on a collection of $N$ interacting auxiliary chains targeting tempered versions of the target distribution to improve the exploration of the state-space. We provide here a new perspective on these highly parallel algorithms and their tuning by identifying and formalizing a sharp divide in the behaviour and performance of reversible versus non-reversible PT schemes. We show theoretically and empirically that a class of non-reversibl…
In the context of nonparametric Bayesian estimation a Markov chain Monte Carlo algorithm is devised and implemented to sample from the posterior distribution of the drift function of a continuously or discretely observed one-dimensional diffusion. The drift is modeled by a scaled linear combination of basis functions with a Gaussian prior on the coefficients. The scaling parameter is equipped with a partially conjugate prior. The number of basis function in the drift is equipped with a prior distribution as well. For continuous data, a reversible jump Markov chain algorithm enables the explora…
The multinomial probit (MNP) model is a useful tool for describing discrete-choice data and there are a variety of methods for fitting the model. Among them, the algorithms provided by Imai and van Dyk (2005a), based on Marginal Data Augmentation, are widely used, because they are efficient in terms of convergence and allow the possibly improper prior distribution to be specified directly on identifiable parameters. Burgette and Nordheim (2012) modify a model and algorithm of Imai and van Dyk (2005a) to avoid an arbitrary choice that is often made to establish identifiability. There is an erro…
By facilitating the generation of samples from arbitrary probability distributions, Markov Chain Monte Carlo (MCMC) is, arguably, \emph{the} tool for the evaluation of Bayesian inference problems that yield non-standard posterior distributions. In recent years, however, it has become apparent that Sequential Monte Carlo (SMC) samplers have the potential to outperform MCMC in a number of ways. SMC samplers are better suited to highly parallel computing architectures and also feature various tuning parameters that are not available to MCMC. One such parameter - the `L-kernel' - is a user-defined…
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…
A general methodology is introduced for the construction and effective application of control variates to estimation problems involving data from reversible MCMC samplers. We propose the use of a specific class of functions as control variates, and we introduce a new, consistent estimator for the values of the coefficients of the optimal linear combination of these functions. The form and proposed construction of the control variates is derived from our solution of the Poisson equation associated with a specific MCMC scenario. The new estimator, which can be applied to the same MCMC sample, is…
We introduce a new version of particle filter in which the number of "children" of a particle at a given time has a Poisson distribution. As a result, the number of particles is random and varies with time. An advantage of this scheme is that descendants of different particles can evolve independently. It makes easy to parallelize computations. Moreover, particle filter with Poisson resampling is readily adapted to the case when a hidden process is a continuous time, piecewise deterministic semi-Markov process. We show that the basic techniques of particle MCMC, namely particle independent Met…
Redwood Center for theoretical Neuroscience（雷德伍德理论神经科学中心）；Department of Physics（物理系）；University of California Berkeley（加州大学伯克利分校）；Berkeley AI Research（伯克利人工智能研究中心）；Helen Wills Neuroscience Institute（海伦·威尔斯神经科学研究所）
Markov Chain Monte Carlo (MCMC) methods sample from unnormalized probability distributions and offer guarantees of exact sampling. However, in the continuous case, unfavorable geometry of the target distribution can greatly limit the efficiency of MCMC methods. Augmenting samplers with neural networks can potentially improve their efficiency. Previous neural network based samplers were trained with objectives that either did not explicitly encourage exploration, or used a L2 jump objective which could only be applied to well structured distributions. Thus it seems promising to instead maximize…
In this paper we present an extension of population-based Markov chain Monte Carlo (MCMC) to the trans-dimensional case. One of the main challenges in MCMC-based inference is that of simulating from high and trans-dimensional target measures. In such cases, MCMC methods may not adequately traverse the support of the target; the simulation results will be unreliable. We develop population methods to deal with such problems, and give a result proving the uniform ergodicity of these population algorithms, under mild assumptions. This result is used to demonstrate the superiority, in terms of conv…
We establish an ordering criterion for the asymptotic variances of two consistent Markov chain Monte Carlo (MCMC) estimators: an importance sampling (IS) estimator, based on an approximate reversible chain and subsequent IS weighting, and a standard MCMC estimator, based on an exact reversible chain. Essentially, we relax the criterion of the Peskun type covariance ordering by considering two different invariant probabilities, and obtain, in place of a strict ordering of asymptotic variances, a bound of the asymptotic variance of IS by that of the direct MCMC. Simple examples show that IS can …
We consider the efficient use of an approximation within Markov chain Monte Carlo (MCMC), with subsequent importance sampling (IS) correction of the Markov chain inexact output, leading to asymptotically exact inference. We detail convergence and central limit theorems for the resulting MCMC-IS estimators. We also consider the case where the approximate Markov chain is pseudo-marginal, requiring unbiased estimators for its approximate marginal target. Convergence results with asymptotic variance formulae are shown for this case, and for the case where the IS weights based on unbiased estimator…
In large-scale genomic applications vast numbers of molecular features are scanned in order to find a small number of candidates which are linked to a particular disease or phenotype. This is a variable selection problem in the "large p, small n" paradigm where many more variables than samples are available. Additionally, a complex dependence structure is often observed among the markers/genes due to their joint involvement in biological processes and pathways. Bayesian variable selection methods that introduce sparseness through additional priors on the model size are well suited to the probl…
This supplementary material includes three parts: some preliminary results, four examples, an experiment, three new algorithms, and all proofs of the results in the paper "Reversible MCMC on Markov equivalence classes of sparse directed acyclic graphs".
Bayesian inference for factorial hidden Markov models is challenging due to the exponentially sized latent variable space. Standard Monte Carlo samplers can have difficulties effectively exploring the posterior landscape and are often restricted to exploration around localised regions that depend on initialisation. We introduce a general purpose ensemble Markov Chain Monte Carlo (MCMC) technique to improve on existing poorly mixing samplers. This is achieved by combining parallel tempering and an auxiliary variable scheme to exchange information between the chains in an efficient way. The latt…
In this paper, we introduce a reversible version of a genetically modified mode jumping Markov chain Monte Carlo algorithm (GMJMCMC) for inference on posterior model probabilities in complex model spaces, where the number of explanatory variables is prohibitively large for classical Markov Chain Monte Carlo methods. Unlike the earlier proposed GMJMCMC algorithm, the introduced algorithm is a proper MCMC and its limiting distribution corresponds to the posterior marginal model probabilities in the explored model space under reasonable regularity conditions.
We present a new data-driven benchmark system to evaluate the performance of new MCMC samplers. Taking inspiration from the COCO benchmark in optimization, we view this task as having critical importance to machine learning and statistics given the rate at which new samplers are proposed. The common hand-crafted examples to test new samplers are unsatisfactory; we take a meta-learning-like approach to generate benchmark examples from a large corpus of data sets and models. Surrogates of posteriors found in real problems are created using highly flexible density models including modern neural n…
In this paper we address the widely-experienced difficulty in tuning Hamiltonian-based Monte Carlo samplers. We develop an algorithm that allows for the adaptation of Hamiltonian and Riemann manifold Hamiltonian Monte Carlo samplers using Bayesian optimization that allows for infinite adaptation of the parameters of these samplers. We show that the resulting sampling algorithms are ergodic, and that the use of our adaptive algorithms makes it easy to obtain more efficient samplers, in some cases precluding the need for more complex solutions. Hamiltonian-based Monte Carlo samplers are widely k…
Motivated by the physics of strings and branes, we develop a class of Markov chain Monte Carlo (MCMC) algorithms involving extended objects. Starting from a collection of parallel Metropolis-Hastings (MH) samplers, we place them on an auxiliary grid, and couple them together via nearest neighbor interactions. This leads to a class of "suburban samplers" (i.e., spread out Metropolis). Coupling the samplers in this way modifies the mixing rate and speed of convergence for the Markov chain, and can in many cases allow a sampler to more easily overcome free energy barriers in a target distribution…
We propose approaches for testing implementations of Markov Chain Monte Carlo methods as well as of general Monte Carlo methods. Based on statistical hypothesis tests, these approaches can be used in a unit testing framework to, for example, check if individual steps in a Gibbs sampler or a reversible jump MCMC have the desired invariant distribution. Two exact tests for assessing whether a given Markov chain has a specified invariant distribution are discussed. These and other tests of Monte Carlo methods can be embedded into a sequential method that allows low expected effort if the simulati…
We propose a very fast approximate Markov Chain Monte Carlo (MCMC) sampling framework that is applicable to a large class of sparse Bayesian inference problems, where the computational cost per iteration in several models is of order $O(ns)$, where $n$ is the sample size, and $s$ the underlying sparsity of the model. This cost can be further reduced by data sub-sampling when stochastic gradient Langevin dynamics are employed. The algorithm is an extension of the asynchronous Gibbs sampler of Johnson et al. (2013), but can be viewed from a statistical perspective as a form of Bayesian iterated …
Universidad Carlos III de Madrid（马德里卡洛斯三世大学）；Universidad Rey Juan Carlos（胡安卡洛斯国王大学）；University of Edinburgh（爱丁堡大学）
Monte Carlo sampling methods are the standard procedure for approximating complicated integrals of multidimensional posterior distributions in Bayesian inference. In this work, we focus on the class of Layered Adaptive Importance Sampling (LAIS) scheme, which is a family of adaptive importance samplers where Markov chain Monte Carlo algorithms are employed to drive an underlying multiple importance sampling scheme. The modular nature of LAIS allows for different possible implementations, yielding a variety of different performance and computational costs. In this work, we propose different enh…
The combinatorial sequential Monte Carlo (CSMC) has been demonstrated to be an efficient complementary method to the standard Markov chain Monte Carlo (MCMC) for Bayesian phylogenetic tree inference using biological sequences. It is appealing to combine the CSMC and MCMC in the framework of the particle Gibbs (PG) sampler to jointly estimate the phylogenetic trees and evolutionary parameters. However, the Markov chain of the particle Gibbs may mix poorly if the underlying SMC suffers from the path degeneracy issue. Some remedies, including the particle Gibbs with ancestor sampling and the inte…
Sampling from posterior distributions using Markov chain Monte Carlo (MCMC) methods can require an exhaustive number of iterations, particularly when the posterior is multi-modal as the MCMC sampler can become trapped in a local mode for a large number of iterations. In this paper, we introduce the pseudo-extended MCMC method as a simple approach for improving the mixing of the MCMC sampler for multi-modal posterior distributions. The pseudo-extended method augments the state-space of the posterior using pseudo-samples as auxiliary variables. On the extended space, the modes of the posterior a…
I show how any reversible Markov chain on a finite state space that is irreducible, and hence suitable for estimating expectations with respect to its invariant distribution, can be used to construct a non-reversible Markov chain on a related state space that can also be used to estimate these expectations, with asymptotic variance at least as small as that using the reversible chain (typically smaller). The non-reversible chain achieves this improvement by avoiding (to the extent possible) transitions that backtrack to the state from which the chain just came. The proof that this modification…
MCMC using $\textit{bouncy}$ Hamiltonian dynamics: A unifying framework for Hamiltonian Monte Carlo and piecewise deterministic Markov process samplers
Many recent Markov chain Monte Carlo (MCMC) samplers leverage continuous dynamics to define a transition kernel that efficiently explores a target distribution. In tandem, a focus has been on devising scalable variants that subsample the data and use stochastic gradients in place of full-data gradients in the dynamic simulations. However, such stochastic gradient MCMC samplers have lagged behind their full-data counterparts in terms of the complexity of dynamics considered since proving convergence in the presence of the stochastic gradient noise is non-trivial. Even with simple dynamics, sign…
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…
In Monte-Carlo methods the Markov processes used to sample a given target distribution usually satisfy detailed balance, i.e. they are time-reversible. However, relatively recent results have demonstrated that appropriate reversible and irreversible perturbations can accelerate convergence to equilibrium. In this paper we present some general design principles which apply to general Markov processes. Working with the generator of Markov processes, we prove that for some of the most commonly used performance criteria, i.e., spectral gap, asymptotic variance and large deviation functionals, samp…
An intriguing new class of piecewise deterministic Markov processes (PDMPs) has recently been proposed as an alternative to Markov chain Monte Carlo (MCMC). In order to facilitate the application to a larger class of problems, we propose a new class of PDMPs termed Gibbs zig-zag samplers, which allow parameters to be updated in blocks with a zig-zag sampler applied to certain parameters and traditional MCMC-style updates to others. We demonstrate the flexibility of this framework on posterior sampling for logistic models with shrinkage priors for high-dimensional regression and random effects …
Motivated by the problem of exploring discrete but very complex state spaces in Bayesian models, we propose a novel Markov Chain Monte Carlo search algorithm: the taxicab sampler. We describe the construction of this sampler and discuss how its interpretation and usage differs from that of standard Metropolis-Hastings as well as the related Hamming ball sampler. The proposed sampling algorithm is then shown to demonstrate substantial improvement in computation time without any loss of efficiency relative to a naïve Metropolis-Hastings search in a motivating Bayesian regression tree count model…
Stochastic gradient Markov chain Monte Carlo (SG-MCMC) has become increasingly popular for simulating posterior samples in large-scale Bayesian modeling. However, existing SG-MCMC schemes are not tailored to any specific probabilistic model, even a simple modification of the underlying dynamical system requires significant physical intuition. This paper presents the first meta-learning algorithm that allows automated design for the underlying continuous dynamics of an SG-MCMC sampler. The learned sampler generalizes Hamiltonian dynamics with state-dependent drift and diffusion, enabling fast t…
