Assume that we would like to estimate the expected value of a function $F$
with respect to a density $\pi$. We prove that if $\pi$ is close enough under
KL divergence to another density $q$, an independent Metropolis sampler
estimator that obtains samples from $\pi$ with proposal density $q$, enriched
with a variance reduction computational strategy based on control variates,
achieves smaller asymptotic variance than that of the crude Monte Carlo
estimator. The control variates construction requires no extra computational
effort but assumes that the expected value of $F$ under $q$ is analytically
available. We illustrate this result by calculating the marginal likelihood in
a linear regression model with prior-likelihood conflict and a non-conjugate
prior. Furthermore, we propose an adaptive independent Metropolis algorithm
that adapts the proposal density such that its KL divergence with the target is
being reduced. We demonstrate its applicability in a Bayesian logistic and
Gaussian process regression problems and we rigorously justify our asymptotic
arguments under easily verifiable and essentially minimal conditions.

基于控制变量的独立 Metropolis 采样器使用提议密度进行方差缩减计算策略，可在 KL 散度条件下对密度进行估计，以较粗糙蒙特卡洛估计器具有更小的渐近方差，同时不需要额外的计算代价，但假设提议密度下的函数期望值在分析上是可用的。研究通过计算带有先验 - 似然冲突和非共轭先验的线性回归模型的边际似然，验证了该结果。此外，我们提出了一种自适应独立 Metropolis 算法，通过调整提议密度以减小其与目标之间的 KL 散度，可在贝叶斯逻辑回归和高斯过程回归问题中应用，并且严格对易验证和基本的最小条件进行了理论说明。

独立 Metropolis 是否能胜过朴素蒙特卡罗？

Can independent Metropolis beat crude Monte Carlo?

Adaptive and interacting Markov chain Monte Carlo algorithms (MCMC) have been
recently introduced in the literature. These novel simulation algorithms are
designed to increase the simulation efficiency to sample complex distributions.
Motivated by some recently introduced algorithms (such as the adaptive
Metropolis algorithm and the interacting tempering algorithm), we develop a
general methodological and theoretical framework to establish both the
convergence of the marginal distribution and a strong law of large numbers.
This framework weakens the conditions introduced in the pioneering paper by
Roberts and Rosenthal [J. Appl. Probab. 44 (2007) 458--475]. It also covers the
case when the target distribution $\pi$ is sampled by using Markov transition
kernels with a stationary distribution that differs from $\pi$.

本文介绍一种基于 Markov chain Monte Carlo 算法的新型模拟算法，具有高效采样复杂分布的性能，并介绍一种方法学和理论框架，证明了边际分布的收敛性和大数定律成立的条件， 并且具有一定的泛化性，适用于目标分布为非平稳分布的情况。