Hamiltonian Monte Carlo（HMC）是一种从贝叶斯模型中采样潜变量的强大算法。概率编程语言（PPL）的出现使用户无需编写推理算法，从而能够专注于建模。然而，许多模型难以由 HMC 直接求解，并且通常需要模型重参数化等技巧。我们的动机来自这样一个事实：许多此类模型可以通过边缘化得到简化。我们提出在从 PPL 提取的图模型中，将自动边缘化作为使用 HMC 采样过程的一部分，从而显著改进对真实世界分层模型的采样。
英文摘要：
Hamiltonian Monte Carlo (HMC) is a powerful algorithm to sample latent variables from Bayesian models. The advent of probabilistic programming languages (PPLs) frees users from writing inference algorithms and lets users focus on modeling. However, many models are difficult for HMC to solve directly, and often require tricks like model reparameterization. We are motivated by the fact that many of those models could be simplified by marginalization. We propose to use automatic marginalization as part of the sampling process using HMC in a graphical model extracted from a PPL, which substantially improves sampling from real-world hierarchical models.
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 sh…
In the quest for scalable Bayesian computational algorithms we need to exploit the full potential of existing methodologies. In this note we point out that message passing algorithms, which are very well developed for inference in graphical models, appear to be largely unexplored for scalable inference in Bayesian multilevel regression models. We show that nested multilevel regression models with Gaussian errors lend themselves very naturally to the combined use of belief propagation and MCMC. Specifically, the posterior distribution of the regression parameters conditionally on covariance hyp…
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…
Purdue University（普渡大学）；University of Southern California（南加州大学）
Replica exchange Monte Carlo (reMC), also known as parallel tempering, is an important technique for accelerating the convergence of the conventional Markov Chain Monte Carlo (MCMC) algorithms. However, such a method requires the evaluation of the energy function based on the full dataset and is not scalable to big data. The naïve implementation of reMC in mini-batch settings introduces large biases, which cannot be directly extended to the stochastic gradient MCMC (SGMCMC), the standard sampling method for simulating from deep neural networks (DNNs). In this paper, we propose an adaptive repl…
Single-site Markov Chain Monte Carlo (MCMC) is a variant of MCMC in which a single coordinate in the state space is modified in each step. Structured relational models are a good candidate for this style of inference. In the single-site context, second order methods become feasible because the typical cubic costs associated with these methods is now restricted to the dimension of each coordinate. Our work, which we call Newtonian Monte Carlo (NMC), is a method to improve MCMC convergence by analyzing the first and second order gradients of the target density to determine a suitable proposal de…
ParaMonte::Python (standing for Parallel Monte Carlo in Python) is a serial and MPI-parallelized library of (Markov Chain) Monte Carlo (MCMC) routines for sampling mathematical objective functions, in particular, the posterior distributions of parameters in Bayesian modeling and analysis in data science, Machine Learning, and scientific inference in general. In addition to providing access to fast high-performance serial/parallel Monte Carlo and MCMC sampling routines, the ParaMonte::Python library provides extensive post-processing and visualization tools that aim to automate and streamline t…
We propose cube thinning, a novel method for compressing the output of a MCMC (Markov chain Monte Carlo) algorithm when control variates are available. It amounts to resampling the initial MCMC sample (according to weights derived from control variates), while imposing equality constraints on averages of these control variates, using the cube method of \cite{Deville2004}. Its main advantage is that its CPU cost is linear in $N$, the original sample size, and is constant in $M$, the required size for the compressed sample. This compares favourably to Stein thinning \citep{Riabiz2020}, which has…
Sequential Monte Carlo squared (SMC$^2$) methods can be used for parameter inference of intractable likelihood state-space models. These methods replace the likelihood with an unbiased particle filter estimator, similarly to particle Markov chain Monte Carlo (MCMC). As with particle MCMC, the efficiency of SMC$^2$ greatly depends on the variance of the likelihood estimator, and therefore on the number of state particles used within the particle filter. We introduce novel methods to adaptively select the number of state particles within SMC$^2$ using the expected squared jumping distance to tri…
DeepMind（深度思维）；University College of London（伦敦大学学院）；Univ. of Econ. and Business（经济与商业大学）；The Alan Turing Institute（艾伦·图灵研究所）
We introduce a gradient-based learning method to automatically adapt Markov chain Monte Carlo (MCMC) proposal distributions to intractable targets. We define a maximum entropy regularised objective function, referred to as generalised speed measure, which can be robustly optimised over the parameters of the proposal distribution by applying stochastic gradient optimisation. An advantage of our method compared to traditional adaptive MCMC methods is that the adaptation occurs even when candidate state values are rejected. This is a highly desirable property of any adaptation strategy because th…
In geostatistics, Gaussian random fields are often used to model heterogeneities of soil or subsurface parameters. To give spatial approximations of these random fields, they are discretized. Then, different techniques of geostatistical inversion are used to condition them on measurement data. Among these techniques, Markov chain Monte Carlo (MCMC) techniques stand out, because they yield asymptotically unbiased conditional realizations. However, standard Markov Chain Monte Carlo (MCMC) methods suffer the curse of dimensionality when refining the discretization. This means that their efficienc…
We show how the random coefficient logistic demand (BLP) model can be phrased as an automatically differentiable moment function, including the incorporation of numerical safeguards proposed in the literature. This allows gradient-based frequentist and quasi-Bayesian estimation using the Continuously Updating Estimator (CUE). Drawing from the machine learning literature, we outline hitherto under-utilized best practices in both frequentist and Bayesian estimation techniques. Our Monte Carlo experiments compare the performance of CUE, 2S-GMM, and LTE estimation. Preliminary findings indicate th…
While the Matrix Generalized Inverse Gaussian ($\mathcal{MGIG}$) distribution arises naturally in some settings as a distribution over symmetric positive semi-definite matrices, certain key properties of the distribution and effective ways of sampling from the distribution have not been carefully studied. In this paper, we show that the $\mathcal{MGIG}$ is unimodal, and the mode can be obtained by solving an Algebraic Riccati Equation (ARE) equation [7]. Based on the property, we propose an importance sampling method for the $\mathcal{MGIG}$ where the mode of the proposal distribution matches …
We explore a general framework in Markov chain Monte Carlo (MCMC) sampling where sequential proposals are tried as a candidate for the next state of the Markov chain. This sequential-proposal framework can be applied to various existing MCMC methods, including Metropolis-Hastings algorithms using random proposals and methods that use deterministic proposals such as Hamiltonian Monte Carlo (HMC) or the bouncy particle sampler. Sequential-proposal MCMC methods construct the same Markov chains as those constructed by the delayed rejection method under certain circumstances. In the context of HMC,…
Markov Chain Monte Carlo (MCMC) algorithms are widely used for stochastic optimization, sampling, and integration of mathematical objective functions, in particular, in the context of Bayesian inverse problems and parameter estimation. For decades, the algorithm of choice in MCMC simulations has been the Metropolis-Hastings (MH) algorithm. An advancement over the traditional MH-MCMC sampler is the Delayed-Rejection Adaptive Metropolis (DRAM). In this paper, we present MatDRAM, a stochastic optimization, sampling, and Monte Carlo integration toolbox in MATLAB which implements a variant of the D…
We introduce a dynamic mechanism for the solution of analytically-tractable substructure in probabilistic programs, using conjugate priors and affine transformations to reduce variance in Monte Carlo estimators. For inference with Sequential Monte Carlo, this automatically yields improvements such as locally-optimal proposals and Rao-Blackwellization. The mechanism maintains a directed graph alongside the running program that evolves dynamically as operations are triggered upon it. Nodes of the graph represent random variables, edges the analytically-tractable relationships between them. Rando…
University of Helsinki（赫尔辛基大学）；Universidad Carlos III de Madrid（马德里卡洛斯三世大学）；Universidad Politécnica de Madrid（马德里理工大学）
Monte Carlo methods represent the "de facto" standard for approximating complicated integrals involving multidimensional target distributions. In order to generate random realizations from the target distribution, Monte Carlo techniques use simpler proposal probability densities to draw candidate samples. The performance of any such method is strictly related to the specification of the proposal distribution, such that unfortunate choices easily wreak havoc on the resulting estimators. In this work, we introduce a layered (i.e., hierarchical) procedure to generate samples employed within a Mon…
Probabilistic inference procedures are usually coded painstakingly from scratch, for each target model and each inference algorithm. We reduce this effort by generating inference procedures from models automatically. We make this code generation modular by decomposing inference algorithms into reusable program-to-program transformations. These transformations perform exact inference as well as generate probabilistic programs that compute expectations, densities, and MCMC samples. The resulting inference procedures are about as accurate and fast as other probabilistic programming systems on rea…
Variational inference is a scalable technique for approximate Bayesian inference. Deriving variational inference algorithms requires tedious model-specific calculations; this makes it difficult to automate. We propose an automatic variational inference algorithm, automatic differentiation variational inference (ADVI). The user only provides a Bayesian model and a dataset; nothing else. We make no conjugacy assumptions and support a broad class of models. The algorithm automatically determines an appropriate variational family and optimizes the variational objective. We implement ADVI in Stan (…
Sequential Monte Carlo (SMC) samplers form an attractive alternative to MCMC for Bayesian computation. However, their performance depends strongly on the Markov kernels used to rejuvenate particles. We discuss how to calibrate automatically (using the current particles) Hamiltonian Monte Carlo kernels within SMC. To do so, we build upon the adaptive SMC approach of Fearnhead and Taylor (2013), and we also suggest alternative methods. We illustrate the advantages of using HMC kernels within an SMC sampler via an extensive numerical study.
This paper introduces a new specialized algorithm for equilibrium Monte Carlo sampling of binary-valued systems, which allows for large moves in the state space. This is achieved by constructing self-avoiding walks (SAWs) in the state space. As a consequence, many bits are flipped in a single MCMC step. We name the algorithm SARDONICS, an acronym for Self-Avoiding Random Dynamics on Integer Complex Systems. The algorithm has several free parameters, but we show that Bayesian optimization can be used to automatically tune them. SARDONICS performs remarkably well in a broad number of sampling ta…
Hamiltonian Monte Carlo (HMC) is a Markov chain Monte Carlo (MCMC) algorithm that avoids the random walk behavior and sensitivity to correlated parameters that plague many MCMC methods by taking a series of steps informed by first-order gradient information. These features allow it to converge to high-dimensional target distributions much more quickly than simpler methods such as random walk Metropolis or Gibbs sampling. However, HMC's performance is highly sensitive to two user-specified parameters: a step size ε and a desired number of steps L. In particular, if L is too small then the algor…
We introduce a new family of MCMC samplers that combine auxiliary variables, Gibbs sampling and Taylor expansions of the target density. Our approach permits the marginalisation over the auxiliary variables yielding marginal samplers, or the augmentation of the auxiliary variables, yielding auxiliary samplers. The well-known Metropolis-adjusted Langevin algorithm (MALA) and preconditioned Crank-Nicolson Langevin (pCNL) algorithm are shown to be special cases. We prove that marginal samplers are superior in terms of asymptotic variance and demonstrate cases where they are slower in computing ti…
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…
Mixtures of linear mixed models (MLMMs) are useful for clustering grouped data and can be estimated by likelihood maximization through the EM algorithm. The conventional approach to determining a suitable number of components is to compare different mixture models using penalized log-likelihood criteria such as BIC.We propose fitting MLMMs with variational methods which can perform parameter estimation and model selection simultaneously. A variational approximation is described where the variational lower bound and parameter updates are in closed form, allowing fast evaluation. A new variation…
For modeling multivariate financial time series we propose a single factor copula model together with stochastic volatility margins. This model generalizes single factor models relying on the multivariate normal distribution and allows for symmetric and asymmetric tail dependence. We develop joint Bayesian inference using Hamiltonian Monte Carlo (HMC) within Gibbs sampling. Thus we avoid information loss caused by the two-step approach for margins and dependence in copula models as followed by Schamberger et al(2017). Further, the Bayesian approach allows for high dimensional parameter spaces …
We propose an exact slice sampler for Hierarchical Dirichlet process (HDP) and its associated mixture models (Teh et al., 2006). Although there are existing MCMC algorithms for sampling from the HDP, a slice sampler has been missing from the literature. Slice sampling is well-known for its desirable properties including its fast mixing and its natural potential for parallelization. On the other hand, the hierarchical nature of HDPs poses challenges to adopting a full-fledged slice sampler that automatically truncates all the infinite measures involved without ad-hoc modifications. In this work…
