Modern computational advances have enabled easy parallel implementations of Markov chain Monte Carlo (MCMC). However, almost all work in estimating the variance of Monte Carlo averages, including the efficient batch means (BM) estimator, focuses on a single-chain MCMC run. We demonstrate that simply averaging covariance matrix estimators from multiple chains can yield critical underestimates in small Monte Carlo sample sizes, especially for slow-mixing Markov chains. We extend the work of \cite{arg:and:2006} and propose a multivariate replicated batch means (RBM) estimator that utilizes information from parallel chains, thereby correcting for the underestimation. Under weak conditions on the mixing rate of the process, RBM is strongly consistent and exhibits similar large-sample bias and variance to the BM estimator. We also exhibit superior theoretical properties of RBM by showing that the (negative) bias in the RBM estimator is less than the average BM estimator in the presence of positive correlation in MCMC. Consequently, in small runs, the RBM estimator can be dramatically superior and this is demonstrated through a variety of examples.
In this paper, we study the asymptotic variance of sample path averages for inhomogeneous Markov chains that evolve alternatingly according to two different $π$-reversible Markov transition kernels $P$ and $Q$. More specifically, our main result allows us to compare directly the asymptotic variances of two inhomogeneous Markov chains associated with different kernels $P_i$ and $Q_i$, $i\in\{0,1\}$, as soon as the kernels of each pair $(P_0,P_1)$ and $(Q_0,Q_1)$ can be ordered in the sense of lag-one autocovariance. As an important application, we use this result for comparing different data-au…
Approximate Bayesian computation has emerged as a standard computational tool when dealing with the increasingly common scenario of completely intractable likelihood functions in Bayesian inference. We show that many common Markov chain Monte Carlo kernels used to facilitate inference in this setting can fail to be variance bounding, and hence geometrically ergodic, which can have consequences for the reliability of estimates in practice. This phenomenon is typically independent of the choice of tolerance in the approximation. We then prove that a recently introduced Markov kernel in this sett…
This paper addresses the problem of estimating the Potts parameter B jointly with the unknown parameters of a Bayesian model within a Markov chain Monte Carlo (MCMC) algorithm. Standard MCMC methods cannot be applied to this problem because performing inference on B requires computing the intractable normalizing constant of the Potts model. In the proposed MCMC method the estimation of B is conducted using a likelihood-free Metropolis-Hastings algorithm. Experimental results obtained for synthetic data show that estimating B jointly with the other unknown parameters leads to estimation results…
Markov chain Monte Carlo (MCMC) methods generate samples that are asymptotically distributed from a target distribution of interest as the number of iterations goes to infinity. Various theoretical results provide upper bounds on the distance between the target and marginal distribution after a fixed number of iterations. These upper bounds are on a case by case basis and typically involve intractable quantities, which limits their use for practitioners. We introduce L-lag couplings to generate computable, non-asymptotic upper bound estimates for the total variation or the Wasserstein distance…
The parameters of a discrete stationary Markov model are transition probabilities between states. Traditionally, data consist in sequences of observed states for a given number of individuals over the whole observation period. In such a case, the estimation of transition probabilities is straightforwardly made by counting one-step moves from a given state to another. In many real-life problems, however, the inference is much more difficult as state sequences are not fully observed, namely the state of each individual is known only for some given values of the time variable. A review of the pro…
Interest is in evaluating, by Markov chain Monte Carlo (MCMC) simulation, the expected value of a function with respect to a, possibly unnormalized, probability distribution. A general purpose variance reduction technique for the MCMC estimator, based on the zero-variance principle introduced in the physics literature, is proposed. Conditions for asymptotic unbiasedness of the zero-variance estimator are derived. A central limit theorem is also proved under regularity conditions. The potential of the idea is illustrated with real applications to probit, logit and GARCH Bayesian models. For all…
In this paper we propose an efficient variance reduction approach for additive functionals of Markov chains relying on a novel discrete time martingale representation. Our approach is fully non-asymptotic and does not require the knowledge of the stationary distribution (and even any type of ergodicity) or specific structure of the underlying density. By rigorously analyzing the convergence properties of the proposed algorithm, we show that its cost-to-variance product is indeed smaller than one of the naive algorithm. The numerical performance of the new method is illustrated for the Langevin…
There are known expressions to calculate the moments of the first passage time in Markov chains. Nevertheless, it is commonly forgotten that in most applications the parameters of the Markov chain are constructed using estimates based upon empirical data and in those cases the data sample size should play an important role in estimating the variance. Here we provide a Monte Carlo approach to estimate the first two moments of the passage time in this situation. We illustrate this method with an example using data from the biological field.
This file is the Sweave documentation for the examples provided in Flegal, J. M. and Jones, G. L. (2010), "Implementing Markov chain Monte Carlo: Estimating with confidence", in Handbook of Markov Chain Monte Carlo, edited by Brooks, S., Gelman, A., Jones, G., and Meng, X. published by Chapman & Hall/CRC Press.
Markov chain Monte Carlo (MCMC) is a simulation method commonly used for estimating expectations with respect to a given distribution. We consider estimating the covariance matrix of the asymptotic multivariate normal distribution of a vector of sample means. Geyer (1992) developed a Monte Carlo error estimation method for estimating a univariate mean. We propose a novel multivariate version of Geyer's method that provides an asymptotically valid estimator for the covariance matrix and results in stable Monte Carlo estimates. The finite sample properties of the proposed method are investigated…
We study the asymptotic behavior of kernel estimators of asymptotic variances (or long-run variances) for a class of adaptive Markov chains. The convergence is studied both in $L^p$ and almost surely. The results also apply to Markov chains and improve on the existing literature by imposing weaker conditions. We illustrate the results with applications to the $\operatorname {GARCH}(1,1)$ Markov model and to an adaptive MCMC algorithm for Bayesian logistic regression.
In this paper we propose a novel variance reduction approach for additive functionals of Markov chains based on minimization of an estimate for the asymptotic variance of these functionals over suitable classes of control variates. A distinctive feature of the proposed approach is its ability to significantly reduce the overall finite sample variance. This feature is theoretically demonstrated by means of a deep non asymptotic analysis of a variance reduced functional as well as by a thorough simulation study. In particular we apply our method to various MCMC Bayesian estimation problems where…
The naive importance sampling estimator, based on samples from a single importance density, can be numerically unstable. Instead, we consider generalized importance sampling estimators where samples from more than one probability distribution are combined. We study this problem in the Markov chain Monte Carlo context, where independent samples are replaced with Markov chain samples. If the chains converge to their respective target distributions at a polynomial rate, then under two finite moment conditions, we show a central limit theorem holds for the generalized estimators. Further, we devel…
It is commonly admitted that non-reversible Markov chain Monte Carlo (MCMC) algorithms usually yield more accurate MCMC estimators than their reversible counterparts. In this note, we show that in addition to their variance reduction effect, some non-reversible MCMC algorithms have also the undesirable property to slow down the convergence of the Markov chain. This point, which has been overlooked by the literature, has obvious practical implications. We illustrate this phenomenon for different non-reversible versions of the Metropolis-Hastings algorithm on several discrete state space example…
Estimating Transition Rates in Two-State Non-Homogeneous Markov Jump Processes with Intermittent Observations: A Pseudo-Marginal McMC Approach via Honest Times
The utility of a Markov chain Monte Carlo algorithm is, in large part, determined by the size of the spectral gap of the corresponding Markov operator. However, calculating (and even approximating) the spectral gaps of practical Monte Carlo Markov chains in statistics has proven to be an extremely difficult and often insurmountable task, especially when these chains move on continuous state spaces. In this paper, a method for accurate estimation of the spectral gap is developed for general state space Markov chains whose operators are non-negative and trace-class. The method is based on the fa…
We consider batch size selection for a general class of multivariate batch means variance estimators, which are computationally viable for high-dimensional Markov chain Monte Carlo simulations. We derive the asymptotic mean squared error for this class of estimators. Further, we propose a parametric technique for estimating optimal batch sizes and discuss practical issues regarding the estimating process. Vector auto-regressive, Bayesian logistic regression, and Bayesian dynamic space-time examples illustrate the quality of the estimation procedure where the proposed optimal batch sizes outper…
In this paper, a new modification of ranked set sampling (RSS) is suggested, namely; unified ranked set sampling (URSS) for estimating the population mean and variance. The performance of the empirical mean and variance estimators based on URSS are compared with their counterparts in ranked set sampling and simple random sampling (SRS) via Monte Carlo simulation. Simulation results indicate that the URSS estimators perform better than their counterparts using RSS and SRS designs when the ranking is perfect. When the ranking is imperfect, the URSS estimators still are superior than their counte…
The Eigenvector Method for Umbrella Sampling (EMUS) belongs to a popular class of methods in statistical mechanics which adapt the principle of stratified survey sampling to the computation of free energies. We develop a detailed theoretical analysis of EMUS. Based on this analysis, we show that EMUS is an efficient general method for computing averages over arbitrary target distributions. In particular, we show that EMUS can be dramatically more efficient than direct MCMC when the target distribution is multimodal or when the goal is to compute tail probabilities. To illustrate these theoreti…
Calculating a Monte Carlo standard error (MCSE) is an important step in the statistical analysis of the simulation output obtained from a Markov chain Monte Carlo experiment. An MCSE is usually based on an estimate of the variance of the asymptotic normal distribution. We consider spectral and batch means methods for estimating this variance. In particular, we establish conditions which guarantee that these estimators are strongly consistent as the simulation effort increases. In addition, for the batch means and overlapping batch means methods we establish conditions ensuring consistency in t…
This paper proposes a family of weighted batch means variance estimators, which are computationally efficient and can be conveniently applied in practice. The focus is on Markov chain Monte Carlo simulations and estimation of the asymptotic covariance matrix in the Markov chain central limit theorem, where conditions ensuring strong consistency are provided. Finite sample performance is evaluated through auto-regressive, Bayesian spatial-temporal, and Bayesian logistic regression examples, where the new estimators show significant computational gains with a minor sacrifice in variance compared…
An important problem in the implementation of Markov Chain Monte Carlo algorithms is to determine the convergence time, or the number of iterations before the chain is close to stationarity. For many Markov chains used in practice this time is not known. Even in cases where the convergence time is known to be polynomial, the theoretical bounds are often too crude to be practical. Thus, practitioners like to carry out some form of statistical analysis in order to assess convergence. This has led to the development of a number of methods known as convergence diagnostics which attempt to diagnose…
Markov chain Monte Carlo (MCMC) algorithms are used to estimate features of interest of a distribution. The Monte Carlo error in estimation has an asymptotic normal distribution whose multivariate nature has so far been ignored in the MCMC community. We present a class of multivariate spectral variance estimators for the asymptotic covariance matrix in the Markov chain central limit theorem and provide conditions for strong consistency. We examine the finite sample properties of the multivariate spectral variance estimators and its eigenvalues in the context of a vector autoregressive process …
This paper investigates methods for estimating the optimal stochastic control policy for a Markov Decision Process with unknown transition dynamics and an unknown reward function. This form of model-free reinforcement learning comprises many real world systems such as playing video games, simulated control tasks, and real robot locomotion. Existing methods for estimating the optimal stochastic control policy rely on high variance estimates of the policy descent. However, these methods are not guaranteed to find the optimal stochastic policy, and the high variance gradient estimates make conver…
Markov cohort state-transition models have been the standard approach for simulating the prognosis of patients or, more generally, the life trajectories of individuals over a time period. Current approaches for estimating the variance of a Markov model using a Monte Carlo sampling or a master equation representation are computationally expensive and analytically difficult to express and solve. We introduce an alternative representation of a Markov model in the form of a multinomial distribution. We derive this representation from principles and then verify its veracity in a simulation exercise…
The estimation of normalizing constants is a fundamental step in probabilistic model comparison. Sequential Monte Carlo methods may be used for this task and have the advantage of being inherently parallelizable. However, the standard choice of using a fixed number of particles at each iteration is suboptimal because some steps will contribute disproportionately to the variance of the estimate. We introduce an adaptive version of the Resample-Move algorithm, in which the particle set is adaptively expanded whenever a better approximation of an intermediate distribution is needed. The algorithm…
Sogang University（西江大学）；University of Pennsylvania（宾夕法尼亚大学）；Princeton University（普林斯顿大学）
Minimizing the inclusive Kullback-Leibler (KL) divergence with stochastic gradient descent (SGD) is challenging since its gradient is defined as an integral over the posterior. Recently, multiple methods have been proposed to run SGD with biased gradient estimates obtained from a Markov chain. This paper provides the first non-asymptotic convergence analysis of these methods by establishing their mixing rate and gradient variance. To do this, we demonstrate that these methods-which we collectively refer to as Markov chain score ascent (MCSA) methods-can be cast as special cases of the Markov c…
By making use of martingale representations, we derive the asymptotic normality of particle filters in hidden Markov models and a relatively simple formula for their asymptotic variances. Although repeated resamplings result in complicated dependence among the sample paths, the asymptotic variance formula and martingale representations lead to consistent estimates of the standard errors of the particle filter estimates of the hidden states.
The objective of this article is to study the asymptotic behavior of a new particle filtering approach in the context of hidden Markov models (HMMs). In particular, we develop an algorithm where the latent-state sequence is segmented into multiple shorter portions, with an estimation technique based upon a separate particle filter in each portion. The partitioning facilitates the use of parallel processing. Based upon this approach, we introduce new estimators of the latent states and likelihood which have similar or better variance properties compared to estimators derived from standard parti…
We present several Monte Carlo strategies for simulating discrete-time Markov chains with continuous multi-dimensional state space; we focus on stratified techniques. We first analyze the variance of the calculation of the measure of a domain included in the unit hypercube, when stratified samples are used. We then show that each step of the simulation of a Markov chain can be reduced to the numerical integration of the indicator function of a subdomain of the unit hypercube. Our approach for Markov chains simulates N copies of the chain in parallel using stratified sampling and the copies are…
In this paper, I develop a formula for estimating Bayes factors directly from minimal summary statistics produced in repeated measures analysis of variance designs. The formula, which requires knowing only the $F$-statistic, the number of subjects, and the number of repeated measurements per subject, is based on the BIC approximation of the Bayes factor, a common default method for Bayesian computation with linear models. In addition to providing computational examples, I report a simulation study in which I demonstrate that the formula compares favorably to a recently developed, more complex …
This paper provides a framework for estimating the mean and variance of a high-dimensional normal density. The main setting considered is a fixed number of vector following a high-dimensional normal distribution with unknown mean and diagonal covariance matrix. The diagonal covariance matrix can be known or unknown. If the covariance matrix is unknown, the sample size can be as small as $2$. The proposed estimator is based on the idea that the unobserved pairs of mean and variance for each dimension are drawn from an unknown bivariate distribution, which we model as a mixture of normal-inverse…
This paper introduces a general framework for estimating variance components in the linear mixed models via general unbiased estimating equations, which include some well-used estimators such as the restricted maximum likelihood estimator. We derive the asymptotic covariance matrices and second-order biases under general estimating equations without assuming the normality of the underlying distributions and identify a class of second-order unbiased estimators of variance components. It is also shown that the asymptotic covariance matrices and second-order biases do not depend on whether the re…
Missing data is a common challenge in biomedical research. This fact, along with growing dataset volumes of the modern era, make the issue of computationally-efficient analysis with missing data of crucial practical importance. A general computationally-efficient estimation framework for dealing with missing data is the pseudo-expected estimating equations (PEEE) approach. The method is applicable with any parametric model for which estimation involves the solution of a set of estimating equations, such as likelihood score equations. A key limitation of the PEEE approach is that there is curre…
Estimation of a treatment effect by a regression discontinuity design faces a severe challenge when the running variable contains measurement errors since the errors smoothen the discontinuity on which the identification depends. The existing studies show that the variance of the measurement errors plays a vital role in both bias correction and identification under such situations. However, the methodologies to estimate the variance from data are relatively undeveloped. This paper proposes two estimators for the variance of measurement errors of running variables of sharp regression continuity…
The lasso has been studied extensively as a tool for estimating the coefficient vector in the high-dimensional linear model; however, considerably less is known about estimating the error variance in this context. In this paper, we propose the natural lasso estimator for the error variance, which maximizes a penalized likelihood objective. A key aspect of the natural lasso is that the likelihood is expressed in terms of the natural parameterization of the multiparameter exponential family of a Gaussian with unknown mean and variance. The result is a remarkably simple estimator of the error var…
This paper provides a design-based framework for variance (bound) estimation in experimental analysis. Results are applicable to virtually any combination of experimental design, linear estimator (e.g., difference-in-means, OLS, WLS) and variance bound, allowing for unified treatment and a basis for systematic study and comparison of designs using matrix spectral analysis. A proposed variance estimator reproduces Eicker-Huber-White (aka. "robust", "heteroskedastic consistent", "sandwich", "White", "Huber-White", "HC", etc.) standard errors and "cluster-robust" standard errors as special cases.…
We propose a two-stage estimation method of variance components in time series models known as FDSLRMs, whose observations can be described by a linear mixed model (LMM). We based estimating variances, fundamental quantities in a time series forecasting approach called kriging, on the empirical (plug-in) best linear unbiased predictions of unobservable random components in FDSLRM. The method, providing invariant non-negative quadratic estimators, can be used for any absolutely continuous probability distribution of time series data. As a result of applying the convex optimization and the LMM m…
We often estimate a parameter of interest psi when the identifying conditions involve a nuisance parameter theta. Examples from causal inference are Inverse Probability Weighting, Marginal Structural Models and Structural Nested Models. To estimate treatment effects from observational data, these methods posit a (pooled) logistic regression model for the treatment and/or censoring probabilities and estimate these first. These methods are all based on unbiased estimating equations. First, we provide a general formula for the variance of the parameter of interest psi when the nuisance parameter …
In the near future, millions of load curves measuring the electricity consumption of French households in small time grids (probably half hours) will be available. All these collected load curves represent a huge amount of information which could be exploited using survey sampling techniques. In particular, the total consumption of a specific cus- tomer group (for example all the customers of an electricity supplier) could be estimated using unequal probability random sampling methods. Unfortunately, data collection may undergo technical problems resulting in missing values. In this paper we s…
In this paper we develop a statistical estimation technique to recover the transition kernel $P$ of a Markov chain $X=(X_m)_{m \in \mathbb N}$ in presence of censored data. We consider the situation where only a sub-sequence of $X$ is available and the time gaps between the observations are iid random variables. Under the assumption that neither the time gaps nor their distribution are known, we provide an estimation method which applies when some transitions in the initial Markov chain $X$ are known to be unfeasible. A consistent estimator of $P$ is derived in closed form as a solution of a m…
The development of a complex disease is an intricate interplay of genetic and environmental factors. "Heritability" is defined as the proportion of total trait variance due to genetic factors within a given population. Studies with monozygotic (MZ) and dizygotic (DZ) twins allow us to estimate heritability by fitting an "ACE" model which estimates the proportion of trait variance explained by additive genetic (A), common shared environment (C), and unique non-shared environmental (E) latent effects, thus helping us better understand disease risk and etiology. In this paper, we develop a flexib…
In this article, we develop methods for sample size and power calculations in four-level intervention studies when intervention assignment is carried out at any level, with a particular focus on cluster randomized trials (CRTs). CRTs involving four levels are becoming popular in health care research, where the effects are measured, for example, from evaluations (level 1) within participants (level 2) in divisions (level 3) that are nested in clusters (level 4). In such multi-level CRTs, we consider three types of intraclass correlations between different evaluations to account for such cluster…
The Aalen-Johansen estimator generalizes the Kaplan-Meier estimator for independently left-truncated and right-censored survival data to estimating the transition probability matrix of a time-inhomogeneous Markov model with finite state space. Such multi-state models have a wide range of applications for modelling complex courses of a disease over the course of time, but the Markov assumption may often be in doubt. If censoring is entirely unrelated to the multi-state data, it has been noted that the Aalen-Johansen estimator, standardized by the initial empirical distribution of the multi-stat…
Random-effects meta-analysis requires an estimate of the between-study variance, $τ^2$. We study methods of estimation of $τ^2$ and its confidence interval in meta-analysis of odds ratio, and also the performance of related estimators of the overall effect. We provide results of extensive simulations on five point estimators of $τ^2$ (the popular methods of DerSimonian-Laird, restricted maximum likelihood, and Mandel and Paule; the less-familiar method of Jackson; and the new method (KD) based on the improved approximation to the distribution of the Q statistic by Kulinskaya and Dollinger (201…
Methods for random-effects meta-analysis require an estimate of the between-study variance, $τ^2$. The performance of estimators of $τ^2$ (measured by bias and coverage) affects their usefulness in assessing heterogeneity of study-level effects, and also the performance of related estimators of the overall effect. For the effect measure standardized mean difference (SMD), we provide the results from extensive simulations on five point estimators of $τ^2$ (the popular methods of DerSimonian-Laird, restricted maximum likelihood, and Mandel and Paule (MP); the less-familiar method of Jackson; the…
Methods for random-effects meta-analysis require an estimate of the between-study variance, $τ^2$. The performance of estimators of $τ^2$ (measured by bias and coverage) affects their usefulness in assessing heterogeneity of study-level effects, and also the performance of related estimators of the overall effect. For the effect measure mean difference (MD), we review five point estimators of $τ^2$ (the popular methods of DerSimonian-Laird, restricted maximum likelihood, and Mandel and Paule (MP); the less-familiar method of Jackson; and a new method (WT) based on the improved approximation to…
We investigate the accuracy of the two most common estimators for the maximum expected value of a general set of random variables: a generalization of the maximum sample average, and cross validation. No unbiased estimator exists and we show that it is non-trivial to select a good estimator without knowledge about the distributions of the random variables. We investigate and bound the bias and variance of the aforementioned estimators and prove consistency. The variance of cross validation can be significantly reduced, but not without risking a large bias. The bias and variance of different va…
