Balancing covariates is critical for credible and efficient randomized experiments. Rerandomization addresses this by repeatedly generating treatment assignments until covariate balance meets a prespecified threshold. By shrinking this threshold, it can achieve arbitrarily strong balance, with established results guaranteeing optimal estimation and valid inference in both finite-sample and asymptotic settings across diverse complex experimental settings. Despite its rigorous theoretical foundations, practical use is limited by the extreme inefficiency of rejection sampling, which becomes prohibitively slow under small thresholds and often forces practitioners to adopt suboptimal settings, leading to degraded performance. Existing work focusing on acceleration typically fail to maintain the uniformity over the acceptable assignment space, thus losing the theoretical grounds of classical rerandomization. Building upon a Metropolis-Hastings framework, we address this challenge by introducing an additional sampling-importance resampling step, which restores uniformity and preserves statistical guarantees. Our proposed algorithm, PSRSRR, achieves speedups ranging from 10 to 10,000 times while maintaining exact and asymptotic validity, as demonstrated by simulations and two real-data applications.
We show that it is feasible to carry out exact Bayesian inference for non-Gaussian state space models using an adaptive Metropolis Hastings sampling scheme with the likelihood approximated by the particle filter. Furthermore, an adapyive independent Metropolis Hastings sampler based on a mixture of normals proposal is computationally much more efficient than an adaptive random walk proposal because the cost of constructing a good adaptive proposal is negligible compared to the cost of approximating the likelihood. Independent Metropolis Hastings proposals are also attractive because they are e…
We consider the approximation of expectations with respect to the distribution of a latent Markov process given noisy measurements. This is known as the smoothing problem and is often approached with particle and Markov chain Monte Carlo (MCMC) methods. These methods provide consistent but biased estimators when run for a finite time. We propose a simple way of coupling two MCMC chains built using Particle Independent Metropolis-Hastings (PIMH) to produce unbiased smoothing estimators. Unbiased estimators are appealing in the context of parallel computing, and facilitate the construction of co…
Markov Chain Monte Carlo methods are widely used in signal processing and communications for statistical inference and stochastic optimization. In this work, we introduce an efficient adaptive Metropolis-Hastings algorithm to draw samples from generic multi-modal and multi-dimensional target distributions. The proposal density is a mixture of Gaussian densities with all parameters (weights, mean vectors and covariance matrices) updated using all the previously generated samples applying simple recursive rules. Numerical results for the one and two-dimensional cases are provided.
Couplings play a central role in the analysis of Markov chain Monte Carlo algorithms and appear increasingly often in the algorithms themselves, e.g. in convergence diagnostics, parallelization, and variance reduction techniques. Existing couplings of the Metropolis-Hastings algorithm handle the proposal and acceptance steps separately and fall short of the upper bound on one-step meeting probabilities given by the coupling inequality. This paper introduces maximal couplings which achieve this bound while retaining the practical advantages of current methods. We consider the properties of thes…
We construct an adaptive independent Metropolis-Hastings sampler that uses a mixture of normals as a proposal distribution. To take full advantage of the potential of adaptive sampling our algorithm updates the mixture of normals frequently, starting early in the chain. The algorithm is built for speed and reliability and its sampling performance is evaluated with real and simulated examples. Our article outlines conditions for adaptive sampling to hold and gives a readily accessible proof that under these conditions the sampling scheme generates iterates that converge to the target distributi…
Casella and Robert [Biometrika 83 (1996) 81--94] presented a general Rao--Blackwellization principle for accept-reject and Metropolis--Hastings schemes that leads to significant decreases in the variance of the resulting estimators, but at a high cost in computation and storage. Adopting a completely different perspective, we introduce instead a universal scheme that guarantees variance reductions in all Metropolis--Hastings-based estimators while keeping the computation cost under control. We establish a central limit theorem for the improved estimators and illustrate their performances on to…
A Two-step Metropolis Hastings Method for Bayesian Empirical Likelihood Computation with Application to Quantile Regression and Bayesian Model Selection
University of Nebraska-Lincoln（内布拉斯加大学林肯分校）；DFS Group Ltd.（DFS集团有限公司）；Texas A& M University（德克萨斯农工大学）
Differentiating Metropolis-Hastings to Optimize Intractable Densities
Massachusetts Institute of Technology（麻省理工学院）；Chalmers University of Technology（查尔姆斯理工大学）；University of Gothenburg（哥德堡大学）；University of Oxford（牛津大学）；JuliaHub；Pumas-AI
MCMC algorithms such as Metropolis-Hastings algorithms are slowed down by the computation of complex target distributions as exemplified by huge datasets. We offer in this paper an approach to reduce the computational costs of such algorithms by a simple and universal divide-and-conquer strategy. The idea behind the generic acceleration is to divide the acceptance step into several parts, aiming at a major reduction in computing time that outranks the corresponding reduction in acceptance probability. The division decomposes the "prior x likelihood" term into a product such that some of its co…
Since its inception the Metropolis-Hastings kernel has been applied in sophisticated ways to address ever more challenging and diverse sampling problems. Its success stems from the flexibility brought by the fact that its verification and sampling implementation rests on a local ``detailed balance'' condition, as opposed to a global condition in the form of a typically intractable integral equation. While checking the local condition is routine in the simplest scenarios, this proves much more difficult for complicated applications involving auxiliary structures and variables. Our aim is to dev…
We consider the problem of inference for nonlinear, multivariate diffusion processes, satisfying Itô stochastic differential equations (SDEs), using data at discrete times that may be incomplete and subject to measurement error. Our starting point is a state-of-the-art correlated pseudo-marginal Metropolis-Hastings algorithm, that uses correlated particle filters to induce strong and positive correlation between successive likelihood estimates. However, unless the measurement error or the dimension of the SDE is small, correlation can be eroded by the resampling steps in the particle filter. W…
We introduce a general framework that constructs estimators with reduced variance for random walk Metropolis and Metropolis-adjusted Langevin algorithms. The resulting estimators require negligible computational cost and are derived in a post-process manner utilising all proposal values of the Metropolis algorithms. Variance reduction is achieved by producing control variates through the approximate solution of the Poisson equation associated with the target density of the Markov chain. The proposed method is based on approximating the target density with a Gaussian and then utilising accurate…
We present an adaptive method for the automatic scaling of Random-Walk Metropolis-Hastings algorithms, which quickly and robustly identifies the scaling factor that yields a specified overall sampler acceptance probability. Our method relies on the use of the Robbins-Monro search process, whose performance is determined by an unknown steplength constant. We give a very simple estimator of this constant for proposal distributions that are univariate or multivariate normal, together with a sampling algorithm for automating the method. The effectiveness of the algorithm is demonstrated with both …
This paper discusses the challenges presented by tall data problems associated with Bayesian classification (specifically binary classification) and the existing methods to handle them. Current methods include parallelizing the likelihood, subsampling, and consensus Monte Carlo. A new method based on the two-stage Metropolis-Hastings algorithm is also proposed. The purpose of this algorithm is to reduce the exact likelihood computational cost in the tall data situation. In the first stage, a new proposal is tested by the approximate likelihood based model. The full likelihood based posterior c…
The missing data issue often complicates the task of estimating generalized linear models (GLMs). We describe why the pseudo-marginal Metropolis-Hastings algorithm, used in this setting, is an effective strategy for parameter estimation. This approach requires fewer assumptions, it provides joint inferences on the parameters in the likelihood, the covariate model, and the parameters of the missingness-mechanism, and there is no logical inconsistency of assuming that there are multiple posterior distributions. Moreover, this approach is asymptotically exact, just like most other Markov chain Mo…
This paper defines an approximation scheme for a solution of the Poisson equation of a geometrically ergodic Metropolis-Hastings chain $Φ$. The approximations give rise to a natural sequence of control variates for the ergodic average $S_k(F)=(1/k)\sum_{i=1}^{k} F(Φ_i)$, where $F$ is the force function in the Poisson equation. The main result of the paper shows that the sequence of the asymptotic variances (in the CLTs for the control-variate estimators) converges to zero and gives a rate of this convergence. Numerical examples in the case of a double-well potential are discussed.
The ability to generate samples of the random effects from their conditional distributions is fundamental for inference in mixed effects models. Random walk Metropolis is widely used to conduct such sampling, but such a method can converge slowly for medium dimension problems, or when the joint structure of the distributions to sample is complex. We propose a Metropolis Hastings (MH) algorithm based on a multidimensional Gaussian proposal that takes into account the joint conditional distribution of the random effects and does not require any tuning, in contrast with more sophisticated sampler…
High quality risk adjustment in health insurance markets weakens insurer incentives to engage in inefficient behavior to attract lower-cost enrollees. We propose a novel methodology based on Markov Chain Monte Carlo methods to improve risk adjustment by clustering diagnostic codes into risk groups optimal for health expenditure prediction. We test the performance of our methodology against common alternatives using panel data from 500 thousand enrollees of the Colombian Healthcare System. Results show that our methodology outperforms common alternatives and suggest that it has potential to imp…
We consider a pseudo-marginal Metropolis--Hastings kernel $P_m$ that is constructed using an average of $m$ exchangeable random variables, as well as an analogous kernel $P_s$ that averages $s<m$ of these same random variables. Using an embedding technique to facilitate comparisons, we show that the asymptotic variances of ergodic averages associated with $P_m$ are lower bounded in terms of those associated with $P_s$. We show that the bound provided is tight and disprove a conjecture that when the random variables to be averaged are independent, the asymptotic variance under $P_m$ is never le…
We present MH-MGT, a multivariate technique for sampling from twice-differentiable, log-concave probability density functions. MH-MGT is Metropolis-Hastings sampling using asymmetric, multivariate Gaussian proposal functions constructed from Taylor-series expansion of the log-density function. The mean of the Gaussian proposal function represents the full Newton step, and thus MH-MGT is the stochastic counterpart to Newton optimization. Convergence analysis shows that MH-MGT is well suited for sampling from computationally-expensive log-densities with contributions from many independent observ…
As social network analysis (SNA) has drawn much attention in recent years, one bottleneck of SNA is these network data are too massive to handle. Furthermore, some network data are not accessible due to privacy problems. Therefore, we have to develop sampling methods to draw representative sample graphs from the population graph. In this paper, Metropolis-Hastings Random Walk (MHRW) and Random Walk with Jumps (RWwJ) sampling strategies are introduced, including the procedure of collecting nodes, the underlying mathematical theory, and corresponding estimators. We compared our methods and exist…
Markov chain Monte Carlo (MCMC) methods to sample from a probability distribution $π$ defined on a space $(Θ,\mathcal{T})$ consist of the simulation of realisations of Markov chains $\{θ_{n},n\geq1\}$ of invariant distribution $π$ and such that the distribution of $θ_{i}$ converges to $π$ as $i\rightarrow\infty$. In practice one is typically interested in the computation of expectations of functions, say $f$, with respect to $π$ and it is also required that averages $M^{-1}\sum_{n=1}^{M}f(θ_{n})$ converge to the expectation of interest. The iterative nature of MCMC makes it difficult to develo…
This work develops a powerful and versatile framework for determining acceptance ratios in Metropolis-Hastings type Markov kernels widely used in statistical sampling problems. Our approach allows us to derive new classes of kernels which unify random walk or diffusion-type sampling methods with more complicated "extended phase space" algorithms based around ideas from Hamiltonian dynamics. Our starting point is an abstract result developed in the generality of measurable state spaces that addresses proposal kernels that possess a certain involution structure. Note that, while this underlying …
Graph sampling via crawling has been actively considered as a generic and important tool for collecting uniform node samples so as to consistently estimate and uncover various characteristics of complex networks. The so-called simple random walk with re-weighting (SRW-rw) and Metropolis-Hastings (MH) algorithm have been popular in the literature for such unbiased graph sampling. However, an unavoidable downside of their core random walks -- slow diffusion over the space, can cause poor estimation accuracy. In this paper, we propose non-backtracking random walk with re-weighting (NBRW-rw) and M…
The Metropolis-Hastings (MH) algorithm is the prototype for a class of Markov chain Monte Carlo methods that propose transitions between states and then accept or reject the proposal. These methods generate a correlated sequence of random samples that convey information about the desired probability distribution. Deciding how this information gets recorded is an important step in the practical design of MH-class algorithm implementations. Many implementations discard most of this information in order to reduce demands on storage capacity and disk writing throughput. Here, we describe how recor…
This article develops a general-purpose adaptive sampler that approximates the target density by a mixture of multivariate t densities. The adaptive sampler is based on reversible proposal distributions each of which has the mixture of multivariate t densities as its invariant density. The reversible proposals consist of a combination of independent and correlated steps that allow the sampler to traverse the parameter space efficiently as well as allowing the sampler to keep moving and locally exploring the parameter space. We employ a two-chain approach, in which a trial chain is used to adap…
When an unbiased estimator of the likelihood is used within a Metropolis--Hastings chain, it is necessary to trade off the number of Monte Carlo samples used to construct this estimator against the asymptotic variances of averages computed under this chain. Many Monte Carlo samples will typically result in Metropolis--Hastings averages with lower asymptotic variances than the corresponding Metropolis--Hastings averages using fewer samples. However, the computing time required to construct the likelihood estimator increases with the number of Monte Carlo samples. Under the assumption that the d…
The Multiple Try Metropolis (MTM) method is a generalization of the classical Metropolis-Hastings algorithm in which the next state of the chain is chosen among a set of samples, according to normalized weights. In the literature, several extensions have been proposed. In this work, we show and remark upon the flexibility of the design of MTM-type methods, fulfilling the detailed balance condition. We discuss several possibilities and show different numerical results.
We consider the Random Walk Metropolis algorithm on $\mathbb{R}^n$ with Gaussian proposals, and when the target probability measure is the $n$-fold product of a one-dimensional law. It is well known (see Roberts et al. (Ann. Appl. Probab. 7 (1997) 110-120)) that, in the limit $n\to\infty$, starting at equilibrium and for an appropriate scaling of the variance and of the timescale as a function of the dimension $n$, a diffusive limit is obtained for each component of the Markov chain. In Jourdain et al. (Optimal scaling for the transient phase of the random walk Metropolis algorithm: The mean-f…
Our article deals with Bayesian inference for a general state space model with the simulated likelihood computed by the particle filter. We show empirically that the partially or fully adapted particle filters can be much more efficient than the standard particle, especially when the signal to noise ratio is high. This is especially important because using the particle filter within MCMC sampling is O(T^2), where T is the sample size. We also show that an adaptive independent proposal for the unknown parameters based on a mixture of normals can be much more efficient than the usual optimal ran…
