Sampling Algorithms, from Survey Sampling to Monte Carlo Methods: Tutorial and Literature Review

Despite a few notable uses of simulation of random processes in the pre-computer era (Hammersley and Handscomb, 1964, Section 1.2; Stigler, 2002, Chapter 7), practical widespread use of simulation had to await the invention of computers. Almost as soon as computers were invented, they were used for simulation (Hammersley and Handscomb, 1964, Section 1.2). The name “Monte Carlo” started as cuteness-gambling was then (around 1950) illegal in most places, and the casino at Monte Carlo was the most famous in the world-but it soon became a colorless technical term for simulation of random processes. Markov chain Monte Carlo (MCMC) was invented soon after ordinary Monte Carlo at Los Alamos, one of the few places where computers were available at the time. Metropolis et al. (1953)∗ simulated a liquid in equilibrium with its gas phase. The obvious way to find out about the thermodynamic equilibrium is to simulate the dynamics of the system, and let it run until it reaches equilibrium. The tour …

Monte Carlo (MC) methods are widely used for Bayesian inference and optimization in statistics, signal processing and machine learning. A well-known class of MC methods are Markov Chain Monte Carlo (MCMC) algorithms. In order to foster better exploration of the state space, specially in high-dimensional applications, several schemes employing multiple parallel MCMC chains have been recently introduced. In this work, we describe a novel parallel interacting MCMC scheme, called {\it orthogonal MCMC} (O-MCMC), where a set of "vertical" parallel MCMC chains share information using some "horizontal" MCMC techniques working on the entire population of current states. More specifically, the vertical chains are led by random-walk proposals, whereas the horizontal MCMC techniques employ independent proposals, thus allowing an efficient combination of global exploration and local approximation. The interaction is contained in these horizontal iterations. Within the analysis of different implemen …

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.

We consider the task of MCMC sampling from a distribution defined on a discrete space. Building on recent insights provided in [Zan19], we devise a class of efficient continuous-time, non-reversible algorithms which make active use of the structure of the underlying space. Particular emphasis is placed on how symmetries and other group-theoretic notions can be used to improve exploration of the space. We test our algorithms on a range of examples from statistics, computational physics, machine learning, and cryptography, which show improvement on alternative algorithms. We provide practical recommendations on how to design and implement these algorithms, and close with remarks on the outlook for both discrete sampling and continuous-time Monte Carlo more broadly.

Markov chain Monte Carlo is a key computational tool in Bayesian statistics, but it can be challenging to monitor the convergence of an iterative stochastic algorithm. In this paper we show that the convergence diagnostic Rˆ of Gelman and Rubin (1992) has serious flaws. Traditional Rˆ will fail to correctly diagnose convergence failures when the chain has a heavy tail or when the variance varies across the chains. In this paper we propose an alternative rank-based diagnostic that fixes these problems. We also introduce a collection of quantile-based local efficiency measures, along with a practical approach for computing Monte Carlo error estimates for quantiles. We suggest that common trace plots should be replaced with rank plots from multiple chains. Finally, we give recommendations for how these methods should be used in practice.