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摘要翻译：
在本文中，我们研究了统计大样本理论对使用大都市随机游动进行贝叶斯估计和准贝叶斯估计的计算复杂性的影响。我们的分析是基于Laplace-Bernstein-Von Mises中心极限定理，该定理指出，在大样本中，后验或准后验接近正常密度。利用中心极限定理成立的条件，建立了大样本下一般Metropolis随机游动方法计算复杂度的多项式界。我们的分析涵盖了潜在的对数似然函数或极值准则函数可能是非凹的、不连续的和参数维数增加的情况。然而，中心极限定理以一种非常具体的方式限制了对数似然或极值准则函数对连续性和对数凹性的偏离。在使中心极限定理在参数维数增加时成立所需的最小假设下，我们证明了Metropolis算法在理论上是有效的，即使对正则高斯行走也是有效的。具体地，我们证明了该算法在大样本中的运行时间在参数维$d$上的概率有界于一个多项式，特别是在老化期后的领先情况下，它的运行时间具有随机阶$d^2$。然后给出了指数族、曲线指数族和增维Z-估计的应用。
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英文标题：
《On the Computational Complexity of MCMC-based Estimators in Large
Samples》
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作者：
Alexandre Belloni and Victor Chernozhukov
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最新提交年份：
2012
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分类信息：
一级分类：Mathematics 数学
二级分类：Statistics Theory 统计理论
分类描述：Applied, computational and theoretical statistics: e.g. statistical inference, regression, time series, multivariate analysis, data analysis, Markov chain Monte Carlo, design of experiments, case studies
应用统计、计算统计和理论统计：例如统计推断、回归、时间序列、多元分析、数据分析、马尔可夫链蒙特卡罗、实验设计、案例研究
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一级分类：Mathematics 数学
二级分类：Probability 概率
分类描述：Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory
概率论与随机过程的理论与应用：例如中心极限定理，大偏差，随机微分方程，统计力学模型，排队论
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一级分类：Statistics 统计学
二级分类：Computation 计算
分类描述：Algorithms, Simulation, Visualization
算法、模拟、可视化
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一级分类：Statistics 统计学
二级分类：Statistics Theory 统计理论
分类描述：stat.TH is an alias for math.ST. Asymptotics, Bayesian Inference, Decision Theory, Estimation, Foundations, Inference, Testing.
Stat.Th是Math.St的别名。渐近，贝叶斯推论，决策理论，估计，基础，推论，检验。
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英文摘要：
In this paper we examine the implications of the statistical large sample theory for the computational complexity of Bayesian and quasi-Bayesian estimation carried out using Metropolis random walks. Our analysis is motivated by the Laplace-Bernstein-Von Mises central limit theorem, which states that in large samples the posterior or quasi-posterior approaches a normal density. Using the conditions required for the central limit theorem to hold, we establish polynomial bounds on the computational complexity of general Metropolis random walks methods in large samples. Our analysis covers cases where the underlying log-likelihood or extremum criterion function is possibly non-concave, discontinuous, and with increasing parameter dimension. However, the central limit theorem restricts the deviations from continuity and log-concavity of the log-likelihood or extremum criterion function in a very specific manner. Under minimal assumptions required for the central limit theorem to hold under the increasing parameter dimension, we show that the Metropolis algorithm is theoretically efficient even for the canonical Gaussian walk which is studied in detail. Specifically, we show that the running time of the algorithm in large samples is bounded in probability by a polynomial in the parameter dimension $d$, and, in particular, is of stochastic order $d^2$ in the leading cases after the burn-in period. We then give applications to exponential families, curved exponential families, and Z-estimation of increasing dimension.
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PDF链接：
https://arxiv.org/pdf/704.2167
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