Communication costs, resulting from synchronization requirements during learning, can greatly slow down many parallel machine learning algorithms. In this paper, we present a parallel Markov chain Monte Carlo (MCMC) algorithm in which subsets of data are processed independently, with very little communication. First, we arbitrarily partition data onto multiple machines. Then, on each machine, any classical MCMC method (e.g., Gibbs sampling) may be used to draw samples from a posterior distribution given the data subset. Finally, the samples from each machine are combined to form samples from the full posterior. This embarrassingly parallel algorithm allows each machine to act independently on a subset of the data (without communication) until the final combination stage. We prove that our algorithm generates asymptotically exact samples and empirically demonstrate its ability to parallelize burn-in and sampling in several models.
Consensus Monte Carlo @cite_12 is perhaps the most relevant work to ours. In this algorithm, data is also portioned into different machines and MCMC is performed independently on each machine. Thus, it roughly has the same time complexity as our algorithm. However, the prior is not explicitly reweighted during sampling as we do in Eq , and final samples for the full posterior are generated by averaging subposterior samples. Furthermore, this algorithm has few theoretical guarantees. We find that this algorithm can be viewed as a relaxation of our nonparametric, asymptotically exact sampling procedure, where samples are generated from an evenly weighted mixture (instead of each component having weight @math ) and where each sample is set to @math instead of being drawn from @math . This algorithm is one of our experimental baselines.
@cite_12
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We present a new Markov Chain Monte Carlo (MCMC) sampling algorithm for probabilistic programs. Our approach and tool, called R2, has the unique feature of employing program analysis in order to improve the efficiency of MCMC sampling. Given an input program P, R2 propagates observations in P backwards to obtain a semantically equivalent program P′ in which every probabilistic assignment is immediately followed by an observe statement. Inference is performed by a suitably modified version of the Metropolis-Hastings algorithm that exploits the structure of the program P′. This has the overall effect of preventing rejections due to program executions that fail to satisfy observations in P. We formalize the semantics of probabilistic programs and rigorously prove the correctness of R2. We also empirically demonstrate the effectiveness of R2--in particular, we show that R2 is able to produce results of similar quality as the CHURCH and STAN probabilistic programming tools with much shorter execution time. Importance sampling methods can be iterated like MCMC algorithms, while being more robust against dependence and starting values. The population Monte Carlo principle consists of iterated generations of importance samples, with importance functions depending on the previously generated importance samples. The advantage over MCMC algorithms is that the scheme is unbiased at any iteration and can thus be stopped at any time, while iterations improve the performances of the importance function, thus leading to an adaptive importance sampling. We illustrate this method on a mixture example with multiscale importance functions. A second example reanalyzes the ion channel model using an importance sampling scheme based on a hidden Markov representation, and compares population Monte Carlo with a corresponding MCMC algorithm. Stimulus reconstruction or decoding methods provide an important tool for understanding how sensory and motor information is represented in neural activity. We discuss Bayesian decoding methods based on an encoding generalized linear model (GLM) that accurately describes how stimuli are transformed into the spike trains of a group of neurons. The form of the GLM likelihood ensures that the posterior distribution over the stimuli that caused an observed set of spike trains is log concave so long as the prior is. This allows the maximum a posteriori (MAP) stimulus estimate to be obtained using efficient optimization algorithms. Unfortunately, the MAP estimate can have a relatively large average error when the posterior is highly nongaussian. Here we compare several Markov chain Monte Carlo (MCMC) algorithms that allow for the calculation of general Bayesian estimators involving posterior expectations (conditional on model parameters). An efficient version of the hybrid Monte Carlo (HMC) algorithm was significantly superior to other MCMC methods for gaussian priors. When the prior distribution has sharp edges and corners, on the other hand, the “hit-and-run” algorithm performed better than other MCMC methods. Using these algorithms, we show that for this latter class of priors, the posterior mean estimate can have a considerably lower average error than MAP, whereas for gaussian priors, the two estimators have roughly equal efficiency. We also address the application of MCMC methods for extracting nonmarginal properties of the posterior distribution. For example, by using MCMC to calculate the mutual information between the stimulus and response, we verify the validity of a computationally efficient Laplace approximation to this quantity for gaussian priors in a wide range of model parameters; this makes direct model-based computation of the mutual information tractable even in the case of large observed neural populations, where methods based on binning the spike train fail. Finally, we consider the effect of uncertainty in the GLM parameters on the posterior estimators. Tasks such as record linkage and multi-target tracking, which involve reconstructing the set of objects that underlie some observed data, are particularly challenging for probabilistic inference. Recent work has achieved efficient and accurate inference on such problems using Markov chain Monte Carlo (MCMC) techniques with customized proposal distributions. Currently, implementing such a system requires coding MCMC state representations and acceptance probability calculations that are specific to a particular application. An alternative approach, which we pursue in this paper, is to use a general-purpose probabilistic modeling language (such as BLOG) and a generic Metropolis-Hastings MCMC algorithm that supports user-supplied proposal distributions. Our algorithm gains flexibility by using MCMC states that are only partial descriptions of possible worlds; we provide conditions under which MCMC over partial worlds yields correct answers to queries. We also show how to use a context-specific Bayes net to identify the factors in the acceptance probability that need to be computed for a given proposed move. Experimental results on a citation matching task show that our general-purpose MCMC engine compares favorably with an application-specific system. In human-robot collaborative tasks, incorporating qualitative information provided by humans can greatly enhance the robustness and efficacy of robot state estimation. We introduce an algorithmic framework to model qualitative information as quantitative constraints on and between states. Our approach, named Sequentially Constrained Hamiltonian Monte Carlo, integrates Hamiltonian dynamics into Sequentially Constrained Monte Carlo sampling. We are able to generate samples that satisfy arbitrarily complex, non-smooth and discontinuous constraints, which in turn allows us to support a wide range of qualitative information. We evaluate our approach for constrained sampling qualitatively and quantitatively with several classes of constraints. SCHMC significantly outperforms the Metropolis-Hastings algorithm (a standard Markov Chain Monte Carlo (MCMC) method) and the Hamiltonian Monte Carlo (HMC) method, in terms of both the accuracy of the sampling (for satisfying constraints) and the quality of approximation. Compared to Sequentially Constrained Monte Carlo (SCMC), which supports similar kinds of constraints, our SCHMC approach has faster convergence rates and lower parameter sensitivity. ABSTRACTWe propose subsampling Markov chain Monte Carlo (MCMC), an MCMC framework where the likelihood function for n observations is estimated from a random subset of m observations. We introduce a highly efficient unbiased estimator of the log-likelihood based on control variates, such that the computing cost is much smaller than that of the full log-likelihood in standard MCMC. The likelihood estimate is bias-corrected and used in two dependent pseudo-marginal algorithms to sample from a perturbed posterior, for which we derive the asymptotic error with respect to n and m, respectively. We propose a practical estimator of the error and show that the error is negligible even for a very small m in our applications. We demonstrate that subsampling MCMC is substantially more efficient than standard MCMC in terms of sampling efficiency for a given computational budget, and that it outperforms other subsampling methods for MCMC proposed in the literature. Supplementary materials for this article are availabl... Recently, Stochastic Gradient Markov Chain Monte Carlo (SG-MCMC) methods have been proposed for scaling up Monte Carlo computations to large data problems. Whilst these approaches have proven useful in many applications, vanilla SG-MCMC might suffer from poor mixing rates when random variables exhibit strong couplings under the target densities or big scale differences. In this study, we propose a novel SG-MCMC method that takes the local geometry into account by using ideas from Quasi-Newton optimization methods. These second order methods directly approximate the inverse Hessian by using a limited history of samples and their gradients. Our method uses dense approximations of the inverse Hessian while keeping the time and memory complexities linear with the dimension of the problem. We provide a formal theoretical analysis where we show that the proposed method is asymptotically unbiased and consistent with the posterior expectations. We illustrate the effectiveness of the approach on both synthetic and real datasets. Our experiments on two challenging applications show that our method achieves fast convergence rates similar to Riemannian approaches while at the same time having low computational requirements similar to diagonal preconditioning approaches. Existing Markov Chain Monte Carlo (MCMC) methods are either based on general-purpose and domain-agnostic schemes which can lead to slow convergence, or hand-crafting of problem-specific proposals by an expert. We propose A-NICE-MC, a novel method to train flexible parametric Markov chain kernels to produce samples with desired properties. First, we propose an efficient likelihood-free adversarial training method to train a Markov chain and mimic a given data distribution. Then, we leverage flexible volume preserving flows to obtain parametric kernels for MCMC. Using a bootstrap approach, we show how to train efficient Markov chains to sample from a prescribed posterior distribution by iteratively improving the quality of both the model and the samples. A-NICE-MC provides the first framework to automatically design efficient domain-specific MCMC proposals. Empirical results demonstrate that A-NICE-MC combines the strong guarantees of MCMC with the expressiveness of deep neural networks, and is able to significantly outperform competing methods such as Hamiltonian Monte Carlo. We formulate gradient-based Markov chain Monte Carlo (MCMC) sampling as optimization on the space of probability measures, with Kullback-Leibler (KL) divergence as the objective function. We show that an underdamped form of the Langevin algorithm perform accelerated gradient descent in this metric. To characterize the convergence of the algorithm, we construct a Lyapunov functional and exploit hypocoercivity of the underdamped Langevin algorithm. As an application, we show that accelerated rates can be obtained for a class of nonconvex functions with the Langevin algorithm. One of the main shortcomings of Markov chain Monte Carlo samplers is their inability to mix between modes of the target distribution. In this paper we show that advance knowledge of the location of these modes can be incorporated into the MCMC sampler by introducing mode-hopping moves that satisfy detailed balance. The proposed sampling algorithm explores local mode structure through local MCMC moves (e.g. diffusion or Hybrid Monte Carlo) but in addition also represents the relative strengths of the different modes correctly using a set of global moves. This “mode-hopping” MCMC sampler can be viewed as a generalization of the darting method [1]. This paper presents a new algorithm for mobile robot localization, called Monte Carlo Localization (MCL). MCL is a version of Markov localization, a family of probabilistic approaches that have recently been applied with great practical success. However, previous approaches were either computationally cumbersome (such as grid-based approaches that represent the state space by high-resolution 3D grids), or had to resort to extremely coarse-grained resolutions. Our approach is computationally efficient while retaining the ability to represent (almost) arbitrary distributions. MCL applies sampling-based methods for approximating probability distributions, in a way that places computation "where needed." The number of samples is adapted on-line, thereby invoking large sample sets only when necessary. Empirical results illustrate that MCL yields improved accuracy while requiring an order of magnitude less computation when compared to previous approaches. It is also much easier to implement. We present a new sampling approach to Bayesian learning of the Bayesian network structure. Like some earlier sampling methods, we sample linear orders on nodes rather than directed acyclic graphs (DAGs). The key difference is that we replace the usual Markov chain Monte Carlo (MCMC) method by the method of annealed importance sampling (AIS). We show that AIS is not only competitive to MCMC in exploring the posterior, but also superior to MCMC in two ways: it enables easy and efficient parallelization, due to the independence of the samples, and lower-bounding of the marginal likelihood of the model with good probabilistic guarantees. We also provide a principled way to correct the bias due to order-based sampling, by implementing a fast algorithm for counting the linear extensions of a given partial order. Stochastic gradient Markov chain Monte Carlo (SG-MCMC) methods are Bayesian analogs to popular stochastic optimization methods; however, this connection is not well studied. We explore this relationship by applying simulated annealing to an SGMCMC algorithm. Furthermore, we extend recent SG-MCMC methods with two key components: i) adaptive preconditioners (as in ADAgrad or RMSprop), and ii) adaptive element-wise momentum weights. The zerotemperature limit gives a novel stochastic optimization method with adaptive elementwise momentum weights, while conventional optimization methods only have a shared, static momentum weight. Under certain assumptions, our theoretical analysis suggests the proposed simulated annealing approach converges close to the global optima. Experiments on several deep neural network models show state-of-the-art results compared to related stochastic optimization algorithms. This paper presents Markov chain Monte Carlo data association (MCMCDA) for solving data association problems arising in multitarget tracking in a cluttered environment. When the number of targets is fixed, the single-scan version of MCMCDA approximates joint probabilistic data association (JPDA). Although the exact computation of association probabilities in JPDA is NP-hard, we prove that the single-scan MCMCDA algorithm provides a fully polynomial randomized approximation scheme for JPDA. For general multitarget tracking problems, in which unknown numbers of targets appear and disappear at random times, we present a multi-scan MCMCDA algorithm that approximates the optimal Bayesian filter. We also present extensive simulation studies supporting theoretical results in this paper. Our simulation results also show that MCMCDA outperforms multiple hypothesis tracking (MHT) by a significant margin in terms of accuracy and efficiency under extreme conditions, such as a large number of targets in a dense environment, low detection probabilities, and high false alarm rates. We describe a Markov chain Monte Carlo based particle filter that effectively deals with interacting targets, i.e., targets that are influenced by the proximity and or behavior of other targets. Such interactions cause problems for traditional approaches to the data association problem. In response, we developed a joint tracker that includes a more sophisticated motion model to maintain the identity of targets throughout an interaction, drastically reducing tracker failures. The paper presents two main contributions: (1) we show how a Markov random field (MRF) motion prior, built on the fly at each time step, can substantially improve tracking when targets interact, and (2) we show how this can be done efficiently using Markov chain Monte Carlo (MCMC) sampling. We prove that incorporating an MRF to model interactions is equivalent to adding an additional interaction factor to the importance weights in a joint particle filter. Since a joint particle filter suffers from exponential complexity in the number of tracked targets, we replace the traditional importance sampling step in the particle filter with an MCMC sampling step. The resulting filter deals efficiently and effectively with complicated interactions when targets approach each other. We present both qualitative and quantitative results to substantiate the claims made in the paper, including a large scale experiment on a video-sequence of over 10,000 frames in length. We present an iterative Markov chain Monte Carlo algorithm for computing reference priors and minimax risk for general parametric families. Our approach uses MCMC techniques based on the Blahut-Arimoto algorithm for computing channel capacity in information theory. We give a statistical analysis of the algorithm, bounding the number of samples required for the stochastic algorithm to closely approximate the deterministic algorithm in each iteration. Simulations are presented for several examples from exponential families. Although we focus on applications to reference priors and minimax risk, the methods and analysis we develop are applicable to a much broader class of optimization problems and iterative algorithms. Mobile robot localization is the problem of determining a robot’s pose from sensor data. This article presents a family of probabilistic localization algorithms known as Monte Carlo Localization (MCL). MCL algorithms represent a robot’s belief by a set of weighted hypotheses (samples), which approximate the posterior under a common Bayesian formulation of the localization problem. Building on the basic MCL algorithm, this article develops a more robust algorithm called MixtureMCL, which integrates two complimentary ways of generating samples in the estimation. To apply this algorithm to mobile robots equipped with range finders, a kernel density tree is learned that permits fast sampling. Systematic empirical results illustrate the robustness and computational efficiency of the approach.  2001 Published by Elsevier Science B.V. We develop a Bayesian nonparametric approach to a general family of latent class problems in which individuals can belong simultaneously to multiple classes and where each class can be exhibited multiple times by an individual. We introduce a combinatorial stochastic process known as the negative binomial process ( @math ) as an infinite-dimensional prior appropriate for such problems. We show that the @math is conjugate to the beta process, and we characterize the posterior distribution under the beta-negative binomial process ( @math ) and hierarchical models based on the @math (the @math ). We study the asymptotic properties of the @math and develop a three-parameter extension of the @math that exhibits power-law behavior. We derive MCMC algorithms for posterior inference under the @math , and we present experiments using these algorithms in the domains of image segmentation, object recognition, and document analysis. Despite having various attractive qualities such as high prediction accuracy and the ability to quantify uncertainty and avoid over-fitting, Bayesian Matrix Factorization has not been widely adopted because of the prohibitive cost of inference. In this paper, we propose a scalable distributed Bayesian matrix factorization algorithm using stochastic gradient MCMC. Our algorithm, based on Distributed Stochastic Gradient Langevin Dynamics, can not only match the prediction accuracy of standard MCMC methods like Gibbs sampling, but at the same time is as fast and simple as stochastic gradient descent. In our experiments, we show that our algorithm can achieve the same level of prediction accuracy as Gibbs sampling an order of magnitude faster. We also show that our method reduces the prediction error as fast as distributed stochastic gradient descent, achieving a 4.1 improvement in RMSE for the Netflix dataset and an 1.8 for the Yahoo music dataset. S U M M A R Y Non-linear inverse problems in the geosciences often involve probabilistic sampling of multimodal density functions or global optimization and sometimes both. Efficient algorithmic tools for carrying out sampling or optimization in challenging cases are of major interest. Here results are presented of some numerical experiments with a technique, known as Parallel Tempering, which originated in the field of computational statistics but is finding increasing numbers of applications in fields ranging from Chemical Physics to Astronomy. To date, experience in use of Parallel Tempering within earth sciences problems is very limited. In this paper, we describe Parallel Tempering and compare it to related methods of Simulated Annealing and Simulated Tempering for optimization and sampling, respectively. A key feature of Parallel Tempering is that it satisfies the detailed balance condition required for convergence of Markov chain Monte Carlo (McMC) algorithms while improving the efficiency of probabilistic sampling. Numerical results are presented on use of Parallel Tempering for trans-dimensional inversion of synthetic seismic receiver functions and also the simultaneous fitting of multiple receiver functions using global optimization. These suggest that its use can significantly accelerate sampling algorithms and improve exploration of parameter space in optimization. Parallel Tempering is a meta-algorithm which may be used together with many existing McMC sampling and direct search optimization techniques. It’s generality and demonstrated performance suggests that there is significant potential for applications to both sampling and optimization problems in the geosciences.