Markov Chain Monte Carlo (MCMC) requires to evaluate the full data likelihood at different parameter values iteratively and is often computationally infeasible for large data sets. This paper proposes to approximate the log-likelihood with subsamples taken according to nonuniform subsampling probabilities, and derives the most likely optimal (MLO) subsampling probabilities for better approximation. Compared with existing subsampled MCMC algorithm with equal subsampling probabilities, the MLO subsampled MCMC has a higher estimation efficiency with the same subsampling ratio. The authors also derive a formula using the asymptotic distribution of the subsampled log-likelihood to determine the required subsample size in each MCMC iteration for a given level of precision. This formula is used to develop an adaptive version of the MLO subsampled MCMC algorithm. Numerical experiments demonstrate that the proposed method outperforms the uniform subsampled MCMC.

马尔可夫链蒙特卡洛（MCMC）要求迭代评估不同参数值下的完整数据似然性，并且对于大型数据集通常在计算上不可行。本文提出根据非均匀子采样概率对子样本进行对数似然估计，并推导最可能的最优（MLO）子采样概率，以实现更好的近似。与具有相同子采样概率的现有子采样MCMC算法相比，在相同子采样率的情况下，MLO子采样MCMC具有更高的估计效率。作者还使用子采样对数似然的渐近分布导出了一个公式，以确定给定精度水平下每次MCMC迭代中所需的子采样大小。此公式用于开发MLO二次采样MCMC算法的自适应版本。数值实验表明，该方法优于均匀采样MCMC。