Stochastic differential equations (SDEs) are an important class of time-series models, used to describe stochastic systems evolving in continuous time. Simulating paths from these processes, particularly after conditioning on noisy observations of the latent path, remains a challenge. Existing methods often introduce bias through time-discretization, require involved rejection sampling or debiasing schemes or are restricted to a narrow family of diffusions. In this work, we propose an exact Markov chain Monte Carlo (MCMC) sampling algorithm that is applicable to a broad subset of all SDEs with unit diffusion coefficient; after suitable transformation, this includes an even larger class of multivariate SDEs and most 1-d SDEs. We develop a Gibbs sampling framework that allows exact MCMC for such diffusions, without any discretization error. We demonstrate how our MCMC methodology requires only fairly straightforward simulation steps. Our framework can be extended to include parameter simulation, and allows tools from the Gaussian process literature to be easily applied. We evaluate our method on synthetic and real datasets, demonstrating superior performance to particle MCMC approaches.
Gibbs sampling is a Markov Chain Monte Carlo (MCMC) method often used in Bayesian learning. MCMC methods can be difficult to deploy on parallel and distributed systems due to their inherently sequential nature. We study asynchronous Gibbs sampling, which achieves parallelism by simply ignoring sequential requirements. This method has been shown to produce good empirical results for some hierarchical models, and is popular in the topic modeling community, but was also shown to diverge for other targets. We introduce a theoretical framework for analyzing asynchronous Gibbs sampling and other ext…
Standard Gibbs sampling applied to a multivariate normal distribution with a specified precision matrix is equivalent in fundamental ways to the Gauss-Seidel iterative solution of linear equations in the precision matrix. Specifically, the iteration operators, the conditions under which convergence occurs, and geometric convergence factors (and rates) are identical. These results hold for arbitrary matrix splittings from classical iterative methods in numerical linear algebra giving easy access to mature results in that field, including existing convergence results for antithetic-variable Gibb…
The particle Gibbs sampler is a Markov chain Monte Carlo (MCMC) algorithm to sample from the full posterior distribution of a state-space model. It does so by executing Gibbs sampling steps on an extended target distribution defined on the space of the auxiliary variables generated by an interacting particle system. This paper makes the following contributions to the theoretical study of this algorithm. Firstly, we present a coupling construction between two particle Gibbs updates from different starting points and we show that the coupling probability may be made arbitrarily close to one by i…
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…
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 article is a discussion of Zanella and Roberts' paper: Multilevel linear models, gibbs samplers and multigrid decompositions. We consider several extensions in which the multigrid decomposition would bring us interesting insights, including vector hierarchical models, linear mixed effects models and partial centering parametrizations.
We consider Particle Gibbs (PG) as a tool for Bayesian analysis of non-linear non-Gaussian state-space models. PG is a Monte Carlo (MC) approximation of the standard Gibbs procedure which uses sequential MC (SMC) importance sampling inside the Gibbs procedure to update the latent and potentially high-dimensional state trajectories. We propose to combine PG with a generic and easily implementable SMC approach known as Particle Efficient Importance Sampling (PEIS). By using SMC importance sampling densities which are approximately fully globally adapted to the targeted density of the states, PEI…
Cognitive diagnosis models (CDMs) are useful statistical tools to provide rich information relevant for intervention and learning. As a popular approach to estimate and make inference of CDMs, the Markov chain Monte Carlo (MCMC) algorithm is widely used in practice. However, when the number of attributes, $K$, is large, the existing MCMC algorithm may become time-consuming, due to the fact that $O(2^K)$ calculations are usually needed in the process of MCMC sampling to get the conditional distribution for each attribute profile. To overcome this computational issue, motivated by Culpepper and …
Gibbs sampling is a widely used Markov chain Monte Carlo (MCMC) method for numerically approximating integrals of interest in Bayesian statistics and other mathematical sciences. Many implementations of MCMC methods do not extend easily to parallel computing environments, as their inherently sequential nature incurs a large synchronization cost. In the case study illustrated by this paper, we show how to do Gibbs sampling in a fully data-parallel manner on a graphics processing unit, for a large class of exchangeable models that admit latent variable representations. Our approach takes a syste…
Herding is a technique to sequentially generate deterministic samples from a probability distribution. In this work, we propose a continuous herded Gibbs sampler that combines kernel herding on continuous densities with the Gibbs sampling idea. Our algorithm allows for deterministically sampling from high-dimensional multivariate probability densities, without directly sampling from the joint density. Experiments with Gaussian mixture densities indicate that the L2 error decreases similarly to kernel herding, while the computation time is significantly lower, i.e., linear in the number of dime…
Harvard University（哈佛大学）；UC Irvine（加州大学欧文分校）；UBC（不列颠哥伦比亚大学）；Facebook（脸书）；University of Amsterdam（阿姆斯特丹大学）
The Gibbs sampler is one of the most popular algorithms for inference in statistical models. In this paper, we introduce a herding variant of this algorithm, called herded Gibbs, that is entirely deterministic. We prove that herded Gibbs has an $O(1/T)$ convergence rate for models with independent variables and for fully connected probabilistic graphical models. Herded Gibbs is shown to outperform Gibbs in the tasks of image denoising with MRFs and named entity recognition with CRFs. However, the convergence for herded Gibbs for sparsely connected probabilistic graphical models is still an ope…
University of Amsterdam（阿姆斯特丹大学）；Qualcomm AI Research, Qualcomm Technologies Netherlands B.V.（高通技术荷兰有限公司高通人工智能研究院）
Deterministic dynamics is an essential part of many MCMC algorithms, e.g. Hybrid Monte Carlo or samplers utilizing normalizing flows. This paper presents a general construction of deterministic measure-preserving dynamics using autonomous ODEs and tools from differential geometry. We show how Hybrid Monte Carlo and other deterministic samplers follow as special cases of our theory. We then demonstrate the utility of our approach by constructing a continuous non-sequential version of Gibbs sampling in terms of an ODE flow and extending it to discrete state spaces. We find that our deterministic…
A central task in many applications is reasoning about processes that change over continuous time. Continuous-Time Bayesian Networks is a general compact representation language for multi-component continuous-time processes. However, exact inference in such processes is exponential in the number of components, and thus infeasible for most models of interest. Here we develop a novel Gibbs sampling procedure for multi-component processes. This procedure iteratively samples a trajectory for one of the components given the remaining ones. We show how to perform exact sampling that adapts to the na…
Bayesian inference in state-space models is challenging due to high-dimensional state trajectories. A viable approach is particle Markov chain Monte Carlo, combining MCMC and sequential Monte Carlo to form "exact approximations" to otherwise intractable MCMC methods. The performance of the approximation is limited to that of the exact method. We focus on particle Gibbs and particle Gibbs with ancestor sampling, improving their performance beyond that of the underlying Gibbs sampler (which they approximate) by marginalizing out one or more parameters. This is possible when the parameter prior i…
Gibbs sampling is a Markov chain Monte Carlo method that is often used for learning and inference on graphical models. Minibatching, in which a small random subset of the graph is used at each iteration, can help make Gibbs sampling scale to large graphical models by reducing its computational cost. In this paper, we propose a new auxiliary-variable minibatched Gibbs sampling method, {\it Poisson-minibatching Gibbs}, which both produces unbiased samples and has a theoretical guarantee on its convergence rate. In comparison to previous minibatched Gibbs algorithms, Poisson-minibatching Gibbs su…
This paper proposes an efficient implementation of the generalized labeled multi-Bernoulli (GLMB) filter by combining the prediction and update into a single step. In contrast to the original approach which involves separate truncations in the prediction and update steps, the proposed implementation requires only one single truncation for each iteration, which can be performed using a standard ranked optimal assignment algorithm. Furthermore, we propose a new truncation technique based on Markov Chain Monte Carlo methods such as Gibbs sampling, which drastically reduces the complexity of the f…
We develop a framework for approximating collapsed Gibbs sampling in generative latent variable cluster models. Collapsed Gibbs is a popular MCMC method, which integrates out variables in the posterior to improve mixing. Unfortunately for many complex models, integrating out these variables is either analytically or computationally intractable. We efficiently approximate the necessary collapsed Gibbs integrals by borrowing ideas from expectation propagation. We present two case studies where exact collapsed Gibbs sampling is intractable: mixtures of Student-t's and time series clustering. Our …
We present a novel method in the family of particle MCMC methods that we refer to as particle Gibbs with ancestor sampling (PG-AS). Similarly to the existing PG with backward simulation (PG-BS) procedure, we use backward sampling to (considerably) improve the mixing of the PG kernel. Instead of using separate forward and backward sweeps as in PG-BS, however, we achieve the same effect in a single forward sweep. We apply the PG-AS framework to the challenging class of non-Markovian state-space models. We develop a truncation strategy of these models that is applicable in principle to any backwa…
This paper considers how to obtain MCMC quantitative convergence bounds which can be translated into tight complexity bounds in high-dimensional {settings}. We propose a modified drift-and-minorization approach, which establishes generalized drift conditions defined in subsets of the state space. The subsets are called the "large sets", and are chosen to rule out some "bad" states which have poor drift property when the dimension of the state space gets large. Using the "large sets" together with a "fitted family of drift functions", a quantitative bound can be obtained which can be translated…
Gibbs sampling is the de facto Markov chain Monte Carlo method used for inference and learning on large scale graphical models. For complicated factor graphs with lots of factors, the performance of Gibbs sampling can be limited by the computational cost of executing a single update step of the Markov chain. This cost is proportional to the degree of the graph, the number of factors adjacent to each variable. In this paper, we show how this cost can be reduced by using minibatching: subsampling the factors to form an estimate of their sum. We introduce several minibatched variants of Gibbs, sh…
University of California Berkeley（加州大学伯克利分校）；Tsinghua University（清华大学）
Streaming variational Bayes (SVB) is successful in learning LDA models in an online manner. However previous attempts toward developing online Monte-Carlo methods for LDA have little success, often by having much worse perplexity than their batch counterparts. We present a streaming Gibbs sampling (SGS) method, an online extension of the collapsed Gibbs sampling (CGS). Our empirical study shows that SGS can reach similar perplexity as CGS, much better than SVB. Our distributed version of SGS, DSGS, is much more scalable than SVB mainly because the updates' communication complexity is small.
Particle Markov chain Monte Carlo (PMCMC) is a systematic way of combining the two main tools used for Monte Carlo statistical inference: sequential Monte Carlo (SMC) and Markov chain Monte Carlo (MCMC). We present a novel PMCMC algorithm that we refer to as particle Gibbs with ancestor sampling (PGAS). PGAS provides the data analyst with an off-the-shelf class of Markov kernels that can be used to simulate the typically high-dimensional and highly autocorrelated state trajectory in a state-space model. The ancestor sampling procedure enables fast mixing of the PGAS kernel even when using seem…
University of British Columbia（英属哥伦比亚大学）；BC Cancer Agency（不列颠哥伦比亚癌症机构）
Bayesian feature allocation models are a popular tool for modelling data with a combinatorial latent structure. Exact inference in these models is generally intractable and so practitioners typically apply Markov Chain Monte Carlo (MCMC) methods for posterior inference. The most widely used MCMC strategies rely on an element wise Gibbs update of the feature allocation matrix. These element wise updates can be inefficient as features are typically strongly correlated. To overcome this problem we have developed a Gibbs sampler that can update an entire row of the feature allocation matrix in a s…
Markov Chain Monte Carlo (MCMC) methods are a popular technique in Bayesian statistical modeling. They have long been used to obtain samples from posterior distributions, but recent research has focused on the scalability of these techniques for large problems. We do not develop new sampling methods but instead describe a blocked Gibbs sampler which is sufficiently scalable to accomodate many interesting problems. The sampler we describe applies to a restricted subset of the Generalized Linear Mixed-effects Models (GLMM's); this subset includes Poisson and Gaussian regression models. The block…
The objective of this paper is to study the Gibbs sampling for computing the mean of observable in very high dimension - a powerful Markov chain Monte Carlo method. Under the Dobrushin's uniqueness condition, we establish some explicit and sharp estimate of the exponential convergence rate and prove some Gaussian concentration inequalities for the empirical mean.
Likelihood-free methods such as approximate Bayesian computation (ABC) have extended the reach of statistical inference to problems with computationally intractable likelihoods. Such approaches perform well for small-to-moderate dimensional problems, but suffer a curse of dimensionality in the number of model parameters. We introduce a likelihood-free approximate Gibbs sampler that naturally circumvents the dimensionality issue by focusing on lower-dimensional conditional distributions. These distributions are estimated by flexible regression models either before the sampler is run, or adaptiv…
In this paper, we present a new idea for Transfer Learning (TL) based on Gibbs Sampling. Gibbs sampling is an algorithm in which instances are likely to transfer to a new state with a higher possibility with respect to a probability distribution. We find that such an algorithm can be employed to transfer instances between domains. Restricted Boltzmann Machine (RBM) is an energy based model that is very feasible for being trained to represent a data distribution and also for performing Gibbs sampling. We used RBM to capture data distribution of the source domain and use it in order to cast targ…
We present a new notion of probabilistic duality for random variables involving mixture distributions. Using this notion, we show how to implement a highly-parallelizable Gibbs sampler for weakly coupled discrete pairwise graphical models with strictly positive factors that requires almost no preprocessing and is easy to implement. Moreover, we show how our method can be combined with blocking to improve mixing. Even though our method leads to inferior mixing times compared to a sequential Gibbs sampler, we argue that our method is still very useful for large dynamic networks, where factors ar…
Departement of Computer Science, KU Leuven（计算机科学系，鲁汶大学）；Department of Electrical Engineering, ESAT - STADIUS, KU Leuven（电气工程系，ESAT - STADIUS，鲁汶大学）
Bayesian regression remains a simple but effective tool based on Bayesian inference techniques. For large-scale applications, with complicated posterior distributions, Markov Chain Monte Carlo methods are applied. To improve the well-known computational burden of Markov Chain Monte Carlo approach for Bayesian regression, we developed a multilevel Gibbs sampler for Bayesian regression of linear mixed models. The level hierarchy of data matrices is created by clustering the features and/or samples of data matrices. Additionally, the use of correlated samples is investigated for variance reductio…
We study parameter inference in large-scale latent variable models. We first propose an unified treatment of online inference for latent variable models from a non-canonical exponential family, and draw explicit links between several previously proposed frequentist or Bayesian methods. We then propose a novel inference method for the frequentist estimation of parameters, that adapts MCMC methods to online inference of latent variable models with the proper use of local Gibbs sampling. Then, for latent Dirich-let allocation,we provide an extensive set of experiments and comparisons with existin…
On skew-symmetric distributions and their use in Monte Carlo sampling algorithms: coordinate-free, Gibbs-style and manifold versions of the Barker proposal
University of Oxford（牛津大学）；University College London（伦敦大学学院）；MRC Biostatistics Unit, University of Cambridge（剑桥大学医学研究理事会生物统计学单元）
The Griddy Gibbs sampling was proposed by Ritter and Tanner (1992) as a computationally efficient approximation of the well-known Gibbs sampling method. The algorithm is simple and effective and has been used successfully to address problems in various fields of applied science. However, the approximate nature of the algorithm has prevented it from being widely used: the Markov chains generated by the Griddy Gibbs sampling method are not reversible in general, so the existence and uniqueness of its invariant measure is not guaranteed. Even when such an invariant measure uniquely exists, there …
The pairwise influence matrix of Dobrushin has long been used as an analytical tool to bound the rate of convergence of Gibbs sampling. In this work, we use Dobrushin influence as the basis of a practical tool to certify and efficiently improve the quality of a discrete Gibbs sampler. Our Dobrushin-optimized Gibbs samplers (DoGS) offer customized variable selection orders for a given sampling budget and variable subset of interest, explicit bounds on total variation distance to stationarity, and certifiable improvements over the standard systematic and uniform random scan Gibbs samplers. In ou…
Gibbs sampling is a Markov Chain Monte Carlo sampling technique that iteratively samples variables from their conditional distributions. There are two common scan orders for the variables: random scan and systematic scan. Due to the benefits of locality in hardware, systematic scan is commonly used, even though most statistical guarantees are only for random scan. While it has been conjectured that the mixing times of random scan and systematic scan do not differ by more than a logarithmic factor, we show by counterexample that this is not the case, and we prove that that the mixing times do n…
Sequential Monte Carlo (SMC) algorithms were originally designed for estimating intractable conditional expectations within state-space models, but are now routinely used to generate approximate samples in the context of general-purpose Bayesian inference. In particular, SMC algorithms are often used as subroutines within larger Monte Carlo schemes, and in this context, the demands placed on SMC are different: control of mean-squared error is insufficient---one needs to control the divergence from the target distribution directly. Towards this goal, we introduce the conditional adaptive resamp…
We develop efficient simulation techniques for Bayesian inference on switching GARCH models. Our contribution to existing literature is manifold. First, we discuss different multi-move sampling techniques for Markov Switching (MS) state space models with particular attention to MS-GARCH models. Our multi-move sampling strategy is based on the Forward Filtering Backward Sampling (FFBS) applied to an approximation of MS-GARCH. Another important contribution is the use of multi-point samplers, such as the Multiple-Try Metropolis (MTM) and the Multiple trial Metropolize Independent Sampler, in com…
Monash University（莫纳什大学）；Allstate Research and Planning Center（好事达研究与规划中心）
The K-Mean and EM algorithms are popular in clustering and mixture modeling, due to their simplicity and ease of implementation. However, they have several significant limitations. Both coverage to a local optimum of their respective objective functions (ignoring the uncertainty in the model space), require the apriori specification of the number of classes/clsuters, and are inconsistent. In this work we overcome these limitations by using the Minimum Message Length (MML) principle and a variation to the K-Means/EM observation assignment and parameter calculation scheme. We maintain the simpli…
Columbia University（哥伦比亚大学）；University of Oxford（牛津大学）
Particle Markov chain Monte Carlo techniques rank among current state-of-the-art methods for probabilistic program inference. A drawback of these techniques is that they rely on importance resampling, which results in degenerate particle trajectories and a low effective sample size for variables sampled early in a program. We here develop a formalism to adapt ancestor resampling, a technique that mitigates particle degeneracy, to the probabilistic programming setting. We present empirical results that demonstrate nontrivial performance gains.
Motivated by de Finetti's representation theorem for almost exchangeable arrays, we want to sample $\mathbf p \in [0,1]^d$ from a distribution with density proportional to $\exp(-A^2\sum_{i<j}c_{ij}(p_i-p_j)^2)$, where $A$ is large and $c_{ij}$'s are non-negative weights. We analyze the rate of convergence of a coordinate Gibbs sampler used to simulate from these measures. We show that for every non-zero fixed matrix $C=(c_{ij})$, and large enough $A$, mixing happens in $Θ(A^2)$ steps in a suitable Wasserstein distance. The upper and lower bounds are explicit and depend on the matrix $C$ throu…
TNO（荷兰应用科学研究组织）；Utrecht University（乌得勒支大学）；University of Amsterdam（阿姆斯特丹大学）
We introduce and characterise the performance of the Markov chain Monte Carlo (MCMC) inference method Prune Sampling for discrete and deterministic Bayesian networks (BNs). We developed a procedure to obtain the performance of a MCMC sampling method in the limit of infinite simulation time, extrapolated from relatively short simulations. This approach was used to conduct a study to compare the accuracy, rate of convergence and the time consumption of Prune Sampling with two conventional MCMC sampling methods: Gibbs- and Metropolis sampling. We show that Markov chains created by Prune Sampling …
Regularization by denoising: Bayesian model and Langevin-within-split Gibbs sampling
Institut Denis Poisson, University of Orléans（德尼·泊松研究所，奥尔良大学）；University of Toulouse, IRIT/INP-ENSEEIHT, CNRS（图卢兹大学，法国国家信息与自动化研究所/国立高等电子、电子技术、计算机与机械学院，法国国家科学研究中心）
本文提出一种基于去噪正则化（RED）的图像反演贝叶斯框架，并设计了内含Langevin Monte Carlo步骤的分裂Gibbs采样算法，通过去模糊等任务验证了其有效性。
