マルコフ連鎖モンテカルロ法（MCMC）

MCMC法は、複雑で高次元の確率分布[latex]P(x)[/latex]から直接サンプリングすることが困難な場合に不可欠です。MCMCは、独立したサンプルを生成する代わりに、マルコフ連鎖を形成する相関のあるサンプルのシーケンスを生成します。マルコフ連鎖は、次の状態への遷移確率が、その前のイベントのシーケンスではなく、現在の状態のみに依存する確率過程です。重要なのは、マルコフ連鎖の定常分布が目標分布[latex]P(x)[/latex]となるように、連鎖の遷移確率を設計することです。

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).

さらに： 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.