Motivated by Bayesian inference with highly informative data we analyze the performance of random walk-like Metropolis-Hastings algorithms for approximate sampling of increasingly concentrating target distributions. We focus on Gaussian proposals which use a Hessian-based approximation of the target covariance. By means of pushforward transition kernels we show that for Gaussian target measures the spectral gap of the corresponding Metropolis-Hastings algorithm is independent of the concentration of the posterior, i.e., the noise level in the observational data that is used for Bayesian inference. Moreover, by exploiting the convergence of the concentrating posteriors to their Laplace approximation we extend the analysis to non-Gaussian target measures which either concentrate around a single point or along a linear manifold. In particular, in that setting we show that the average acceptance rate as well as the expected squared jump distance of suitable Metropolis-Hastings Markov chains do not deteriorate as the target concentrates.
翻译:根据贝叶斯人的推论,我们分析了随机行走式大都会-哈斯廷算法的性能,以对日益集中的目标分布进行近似抽样。我们侧重于高斯人的提案,这些提案使用了以黑森为基础的目标共变近似值。我们通过推向向过渡内核的方式表明,对于高斯人来说,相应的大都会-哈斯廷算法的光谱差距与后方的集中程度无关,即用于巴伊西亚推断的观测数据中的噪音水平。此外,我们利用集中后方的后方数据与拉普尔近距离的趋同,将分析扩大到非加萨人的目标措施,要么集中在一个单一点左右,要么沿着一条线性线性线性线性线性列。特别是,在当时我们表明,适当的马科托斯-哈斯海夫人链的平均接受率以及预期的平方跳距离不会随着目标浓缩而恶化。