马尔可夫链蒙特卡罗（MCMC）

当直接从复杂的高维概率分布 P(x) 中采样变得困难时，MCMC 方法至关重要。MCMC 不生成独立样本，而是生成一系列相关的样本，这些样本构成马尔可夫链。马尔可夫链是一种随机过程，其中转移到下一状态的概率仅取决于当前状态，而与之前的事件序列无关。关键在于设计链的转移概率，使其平稳分布等于目标分布 P(x)。

The process starts at an arbitrary state [latex]x_0[/latex]. At each step [latex]t[/latex], a new state [latex]x_{t+1}[/latex] is generated based on the current state [latex]x_t[/latex] using a specific algorithm (like Metropolis-Hastings). After an initial “burn-in” period, during which the chain converges from its starting point to the high-probability regions of the target distribution, the subsequent states [latex]x_t, x_{t+1}, …[/latex] can be considered as (correlated) samples from [latex]P(x)[/latex]. These samples can then be used to estimate expectations of functions [latex]f(x)[/latex] with respect to [latex]P(x)[/latex] by averaging [latex]f(x_t)[/latex] over the samples. This is particularly useful in Bayesian inference, where [latex]P(x)[/latex] is a posterior distribution of model parameters, and direct calculation is often impossible due to a complex denominator (the evidence or marginal likelihood).

而且： MCMC differs from the basic Monte Carlo method in how it generates samples to estimate a desired distribution or integral. While Monte Carlo methods rely on drawing independent and identically distributed random samples directly from a target distribution or a proposal distribution, MCMC generates samples through a correlated sequence (a Markov chain) where each sample depends on the previous one. This dependency allows MCMC to efficiently explore complex, high-dimensional distributions that are difficult to sample from directly, by constructing a chain that converges to the target distribution over time. In contrast, traditional Monte Carlo methods may struggle with such problems due to inefficiencies in sampling or requiring explicit knowledge of the distribution’s form. Thus, MCMC extends Monte Carlo by harnessing dependence between samples to facilitate sampling in challenging statistical and computational settings.