mcmc

Visualizes Metropolis–Hastings MCMC on a 1D bimodal target density pi(x). The animation shows repeated propose-and-correct steps (a candidate x' is proposed, then accepted/rejected with probability alpha to keep pi invariant). The bottom panel accumulates the empirical histogram and a running time-average g(x)=x, illustrating ergodicity: long-run averages along a single chain converge to expectations under pi.

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t=0s

practical uses

01.Bayesian inference when the posterior is not analytically sampleable (e.g., hierarchical models)
02.Sampling high-dimensional distributions for uncertainty quantification in ML/statistics
03.Estimating expectations/normalizing constants via Monte Carlo when only an unnormalized density is available

technical notes

Uses a persistent closure to simulate a single MH random-walk chain. Target pi(x) is an unnormalized bimodal mixture; symmetric Gaussian proposals make the MH ratio alpha = min(1, pi(x')/pi(x)). Animation interpolates the proposal motion with ease(), while the histogram/mean update only on step completion for clarity. Rendering uses a snapped grid for a blocky green-on-black aesthetic and scales with Math.min(w,h)/baseSize.

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