Parallel tempering (PT) is a class of Markov chain Monte Carlo algorithms that constructs a path of distributions annealing between a tractable reference and an intractable target, and then interchanges states along the path to improve mixing in the target. The performance of PT depends on how quickly a sample from the reference distribution makes its way to the target, which in turn depends on the particular path of annealing distributions. However, past work on PT has used only simple paths constructed from convex combinations of the reference and target log-densities. This paper begins by demonstrating that this path performs poorly in the setting where the reference and target are nearly mutually singular. To address this issue, we expand the framework of PT to general families of paths, formulate the choice of path as an optimization problem that admits tractable gradient estimates, and propose a flexible new family of spline interpolation paths for use in practice. Theoretical and empirical results both demonstrate that our proposed methodology breaks previously-established upper performance limits for traditional paths.
Auxiliary variable methods such as the Parallel Tempering and the cluster Monte Carlo methods generate samples that follow a target distribution by using proposal and auxiliary distributions. In sampling from complex distributions, these algorithms are highly more efficient than the standard Markov chain Monte Carlo methods. However, their performance strongly depends on their parameters and determining the parameters is critical. In this paper, we proposed an algorithm for adapting the parameters during drawing samples and proved the convergence theorem of the adaptive algorithm. We applied o…
Parallel tempering is a generic Markov chain Monte Carlo sampling method which allows good mixing with multimodal target distributions, where conventional Metropolis-Hastings algorithms often fail. The mixing properties of the sampler depend strongly on the choice of tuning parameters, such as the temperature schedule and the proposal distribution used for local exploration. We propose an adaptive algorithm which tunes both the temperature schedule and the parameters of the random-walk Metropolis kernel automatically. We prove the convergence of the adaptation and a strong law of large numbers…
Developing efficient MCMC algorithms is indispensable in Bayesian inference. In parallel tempering, multiple interacting MCMC chains run to more efficiently explore the state space and improve performance. The multiple chains advance independently through local moves, and the performance enhancement steps are exchange moves, where the chains pause to exchange their current sample amongst each other. To accelerate the independent local moves, they may be performed simultaneously on multiple processors. Another problem is then encountered: depending on the MCMC implementation and inference probl…
The study of animal behavioural states inferred through hidden Markov models and similar state switching models has seen a significant increase in popularity in recent years. The ability to account for varying levels of behavioural scale has become possible through hierarchical hidden Markov models, but additional levels lead to higher complexity and increased correlation between model components. Maximum likelihood approaches to inference using the EM algorithm and direct optimisation of likelihoods are more frequently used, with Bayesian approaches being less favoured due to computational de…
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…
Parallel tempering is popular method for allowing MCMC algorithms to properly explore a $d$-dimensional multimodal target density. One problem with traditional power-based parallel tempering for multimodal targets is that the proportion of probability mass associated to the modes can change for different inverse-temperature values, sometimes dramatically so. Complementary work by the authors proposes a novel solution involving auxiliary targets that preserve regional weight upon powering up the density. This paper attempts to address the question of how to choose the temperature spacings in an…
Adaptive Importance Tempering: A flexible approach to improve computational efficiency of Metropolis Coupled Markov Chain Monte Carlo algorithms on binary spaces
This paper describes an algorithm for selecting parameter values (e.g. temperature values) at which to measure equilibrium properties with Parallel Tempering Monte Carlo simulation. Simple approaches to choosing parameter values can lead to poor equilibration of the simulation, especially for Ising spin systems that undergo $1^st$-order phase transitions. However, starting from an initial set of parameter values, the careful, iterative respacing of these values based on results with the previous set of values greatly improves equilibration. Example spin systems presented here appear in the con…
Approximate Bayesian Computational (ABC) methods (or likelihood-free methods) have appeared in the past fifteen years as useful methods to perform Bayesian analyses when the likelihood is analytically or computationally intractable. Several ABC methods have been proposed: Monte Carlo Markov Chains (MCMC) methods have been developped by Marjoramet al. (2003) and by Bortotet al. (2007) for instance, and sequential methods have been proposed among others by Sissonet al. (2007), Beaumont et al. (2009) and Del Moral et al. (2009). Until now, while ABC-MCMC methods remain the reference, sequential A…
Bouncy particle sampler with infinite exchanging parallel tempering
Center for Mathematics and Data Science, Gunma University（数学与数据科学中心，gunma大学）；Graduate School of Science and Engineering, Ibaraki University（科学与工程研究生院，ibaraki大学）
Accelerated Parallel Tempering via Neural Transports
University of Oxford（牛津大学）；University of Cambridge（剑桥大学）；Cornell University（康奈尔大学）；Xaira Therapeutics；National University of Singapore（新加坡国立大学）；University of British Columbia（不列颠哥伦比亚大学）
In this paper we develop a new general Bayesian methodology that simultaneously estimates parameters of interest and the marginal likelihood of the model. The proposed methodology builds on Simulated Tempering, which is a powerful algorithm that enables sampling from multi-modal distributions. However, Simulated Tempering comes with the practical limitation of needing to specify a prior for the temperature along a chosen discretization schedule that will allow calculation of normalizing constants at each temperature. Our proposed model defines the prior for the temperature so as to remove the …
In this paper we present extensions to the original adaptive parallel tempering algorithm. Two different approaches are presented. In the first one we introduce state-dependent strategies using current information to perform a swap step. It encompasses a wide family of potential moves including the standard one and Equi Energy type move, without any loss in tractability. In the second one, we introduce online adjustment of the number of temperatures. Numerical experiments demonstrate the effectiveness of the proposed method.
Restricted Boltzmann Machines (RBMs) are one of the fundamental building blocks of deep learning. Approximate maximum likelihood training of RBMs typically necessitates sampling from these models. In many training scenarios, computationally efficient Gibbs sampling procedures are crippled by poor mixing. In this work we propose a novel method of sampling from Boltzmann machines that demonstrates a computationally efficient way to promote mixing. Our approach leverages an under-appreciated property of deep generative models such as the Deep Belief Network (DBN), where Gibbs sampling from deeper…
We propose a new sampler that integrates the protocol of parallel tempering with the Nosé-Hoover (NH) dynamics. The proposed method can efficiently draw representative samples from complex posterior distributions with multiple isolated modes in the presence of noise arising from stochastic gradient. It potentially facilitates deep Bayesian learning on large datasets where complex multimodal posteriors and mini-batch gradient are encountered.
Department of Computer Science, Aalto University, Finland（阿姆斯特丹大学计算机科学系，芬兰）；Department of Engineering, University of Cambridge, United Kingdom（剑桥大学工程系，英国）；Department of Empirical Inference, Max Planck Institute for Intelligent Systems, Tübingen, Germany（智能系统研究所经验推断系，德国）
Sampling from complex target distributions is a challenging task fundamental to Bayesian inference. Parallel tempering (PT) addresses this problem by constructing a Markov chain on the expanded state space of a sequence of distributions interpolating between the posterior distribution and a fixed reference distribution, which is typically chosen to be the prior. However, in the typical case where the prior and posterior are nearly mutually singular, PT methods are computationally prohibitive. In this work we address this challenge by constructing a generalized annealing path connecting the pos…
In the current work we present two generalizations of the Parallel Tempering algorithm, inspired by the so-called continuous-time Infinite Swapping algorithm. Such a method, found its origins in the molecular dynamics community, and can be understood as the limit case of the continuous-time Parallel Tempering algorithm, where the (random) time between swaps of states between two parallel chains goes to zero. Thus, swapping states between chains occurs continuously. In the current work, we extend this idea to the context of time-discrete Markov chains and present two Markov chain Monte Carlo al…
Purdue University（普渡大学）；Morgan Stanley（摩根士丹利）；University of Michigan（密歇根大学）
Parallel tempering (PT), also known as replica exchange, is the go-to workhorse for simulations of multi-modal distributions. The key to the success of PT is to adopt efficient swap schemes. The popular deterministic even-odd (DEO) scheme exploits the non-reversibility property and has successfully reduced the communication cost from $O(P^2)$ to $O(P)$ given sufficiently many $P$ chains. However, such an innovation largely disappears in big data due to the limited chains and few bias-corrected swaps. To handle this issue, we generalize the DEO scheme to promote non-reversibility and propose a …
This paper introduces the parallel hierarchical sampler (PHS), a Markov chain Monte Carlo algorithm using several chains simultaneously. The connections between PHS and the parallel tempering (PT) algorithm are illustrated, convergence of PHS joint transition kernel is proved and and its practical advantages are emphasized. We illustrate the inferences obtained using PHS, parallel tempering and the Metropolis-Hastings algorithm for three Bayesian model selection problems, namely Gaussian clustering, the selection of covariates for a linear regression model and the selection of the structure of…
