Markov Chain Monte Carlo (MCMC) Methods

Markov Chain Monte Carlo (MCMC) methods are algorithms used to estimate complex probability distributions by generating a sequence of samples. Imagine trying to understand a complicated landscape without knowing its shape; MCMC explores this landscape by taking steps guided by probability rules, gradually "wandering" towards areas most likely to represent the true distribution. Over time, these samples help approximate difficult-to-calculate probabilities, which is useful in statistics, physics, and machine learning. Essentially, MCMC is a clever way to "sample" from complex distributions when direct calculation is challenging.