Solving the Poisson equation using coupled Markov chains

Douc, Randal; Jacob, Pierre; Lee, Anthony; Vats, Dootika

doi:10.48550/arxiv.2206.05691

Cited by 3 publications

(3 citation statements)

References 41 publications

(69 reference statements)

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“…where h is referred to as the solution to the Poisson equation for f or as the "fishy function" because the French word Poisson translates to fish (Douc et al, 2022). The fishy function is typically intractable so Section 1.3.2 describes approaches to approximating it.…”

Section: Control Variates Using the Poisson Equationmentioning

confidence: 99%

Regularized Zero-Variance Control Variates

et al. 2023

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Zero-variance control variates (ZV-CV) are a post-processing method to reduce the variance of Monte Carlo estimators of expectations using the derivatives of the log target. Once the derivatives are available, the only additional computational effort lies in solving a linear regression problem. Significant variance reductions have been achieved with this method in low dimensional examples, but the number of covariates in the regression rapidly increases with the dimension of the target. In this paper, we present compelling empirical evidence that the use of penalized regression techniques in the selection of high-dimensional control variates provides performance gains over the classical least squares method. Another type of regularization based on using subsets of derivatives, or a priori regularization as we refer to it in this paper, is also proposed to reduce computational and storage requirements. Several examples showing the utility and limitations of regularized ZV-CV for Bayesian inference are given. The methods proposed in this paper are accessible through the R package ZVCV.

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Section: Control Variates Using the Poisson Equationmentioning

confidence: 99%

Regularized Zero-Variance Control Variates

et al. 2023

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“…Specifically, it is proved in Hu and Zhang (2020) that for each z ∈ , the within-stratum variance, var(n −1∕2 S n (z)), converges to a stratum specific limit 𝜎 2 z . With similar techniques and the Cramér-Wold Device, one may show that the covariance matrix of n −1∕2 S n indeed converges to 𝚺 ∞ , whose computation, however, requires solving Poisson equations associated with an induced Markov chain, and is challenging even when the dynamics of the Markov chain are given; see, for example, Douc et al (2022).…”

Section: Introductionmentioning

confidence: 99%

Consistent covariances estimation for stratum imbalances under minimization method for covariate‐adaptive randomization

Zhao,

Song,

Jiang

et al. 2024

Scandinavian J Statistics

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Pocock and Simon's minimization method is a popular approach for covariate‐adaptive randomization in clinical trials. Valid statistical inference with data collected under the minimization method requires the knowledge of the limiting covariance matrix of within‐stratum imbalances, whose existence is only recently established. In this work, we propose a bootstrap‐based estimator for this limit and establish its consistency, in particular, by Le Cam's third lemma. As an application, we consider in simulation studies adjustments to existing robust tests for treatment effects with survival data by the proposed estimator. It shows that the adjusted tests achieve a size close to the nominal level, and unlike other designs, the robust tests without adjustment may have an asymptotic size inflation issue under the minimization method.This article is protected by copyright. All rights reserved.

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“…Specifically, it is proved in [8] that for each z ∈ Z, the within-stratum variance, var(n −1/2 S n (z)), converges to a stratum specific limit σ 2 z . By similar techniques and the Cramér-Wold Device, one may show that the covariance matrix of n −1/2 S n indeed converges to Σ ∞ , whose computation, however, requires solving Poisson equations associated with an induced Markov chain, and is challenging even when the number of strata is small and the dynamics of the Markov chain are given; see, e.g., [5].…”

Section: Introductionmentioning

confidence: 99%

Consistent covariances estimation for stratum imbalances under marginal design for covariate adaptive randomization

Zhao¹,

Song²,

Jiang³

et al. 2022

Preprint

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Marginal design, also called as the minimization method, is a popular approach used for covariate adaptive randomization when the number of strata is large in a clinical trial. It aims to achieve a balanced allocation of treatment groups at the marginal levels of some prespecified and discrete stratification factors. Valid statistical inference with data collected under covariate adaptive randomization requires the knowledge of the limiting covariance matrix of within-stratum imbalances. The existence of the limit under the marginal design is recently established, which can be estimated by Monte Carlo simulations when the distribution of the stratification factors is known. This assumption may not hold, however, in practice. In this work, we propose to replace the usually unknown distribution with an estimator, such as the empirical distribution, in the Monte Carlo approach and establish its consistency, in particular, by Le Cam's third lemma. As an application, we consider in simulation studies adjustments to existing robust tests for treatment effects with survival data by the proposed covariances estimator. It shows that the adjusted tests achieve a size close to the nominal level, and unlike other designs, the robust tests without adjustment may have an asymptotic size inflation issue under the marginal design.

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Solving the Poisson equation using coupled Markov chains

Cited by 3 publications

References 41 publications

Regularized Zero-Variance Control Variates

Regularized Zero-Variance Control Variates

Consistent covariances estimation for stratum imbalances under minimization method for covariate‐adaptive randomization

Consistent covariances estimation for stratum imbalances under marginal design for covariate adaptive randomization

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