마르코프 체인 몬테카를로(MCMC)

MCMC(모노코핑) 방법은 복잡하고 고차원적인 확률 분포 [latex]P(x)[/latex]에서 직접 샘플링하는 것이 불가능할 때 필수적입니다. MCMC는 독립적인 샘플을 생성하는 대신, 마르코프 체인을 형성하는 상관된 샘플 시퀀스를 생성합니다. 마르코프 체인은 다음 상태로의 전이 확률이 이전 사건의 순서가 아닌 현재 상태에만 의존하는 확률 과정입니다. 핵심은 체인의 정상 상태 분포가 목표 분포 [latex]P(x)[/latex]가 되도록 전이 확률을 설계하는 것입니다.

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).

게다가: 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.