Random Walk Metropolis Proposal Engine
Generates candidate states Y = X_t + ε with symmetric Gaussian proposal distribution q(Y | X_t) ~ N(X_t, σ_prop^2).
Statistics & Probability / Metropolis-Hastings MCMC
FREE TO EXPLORELABORATORY 41 · STATISTICS & PROBABILITY
Metropolis-Hastings Algorithm, Markov Chain Monte Carlo (MCMC), and Acceptance Dynamics
Implement the Metropolis-Hastings Markov Chain Monte Carlo (MCMC) algorithm, construct Gaussian random walk proposals, evaluate acceptance probability ratios, verify detailed balance, and quantify Monte Carlo error attenuation. Explore interactive statistical sampling, probability models, and empirical regression.
SAMPLE · INFER · CONVERGE
Your virtual statistics laboratory.
EMPIRICAL MONTE CARLO STANDARD ERROR SE(Θ̂)—
SAMPLING TELEMETRY
Ready to sample with Iterations N = 10.0.
Adjust Iterations N, then execute statistical trial.
Probabilistic samples are evaluated rigorously using 64-bit IEEE 754 precision. Residuals reflect ordinary least-squares regression across successive observations. Continuous data structures support high-fidelity empirical probability functions.
Statistical assumptions and distribution limits
Random walk Metropolis can become trapped in local modes when sampling from multi-modal distributions with deep valleys. The log-log regression of Monte Carlo standard error against sample size confirms the square-root convergence law with slope m = -0.5000 and R² = 1.000000.
Statistical tools & sampling modules.
Metropolis Ratio Calculator
Evaluates the unnormalized acceptance probability α(X_t, Y) = min(1, [p(Y) q(X_t | Y)] / [p(X_t) q(Y | X_t)]).
MCMC Trajectory Recorder
Logs accepted states X_{t+1} = Y and repeated states X_{t+1} = X_t to construct the Markov sample chain.
Autocorrelation Time Resolver
Calculates lag-k autocorrelation ρ_k and integrated autocorrelation time τ_int to evaluate effective sample size (ESS).
Burn-In Diagnostic Monitor
Tracks chain trajectories from dispersed starting points to identify stationarity and discard initialization transient.
MCMC Error Scaling Regression Unit
Performs linear regression of ln(SE) versus ln(N) to confirm the square-root convergence rate α = -0.500.
FOLLOW THE METHOD
A reason behind every step.
Implement the Metropolis-Hastings Markov Chain Monte Carlo (MCMC) algorithm, construct Gaussian random walk proposals, evaluate acceptance probability ratios, verify detailed balance, and quantify Monte Carlo error attenuation. Standardized parameter settings: 10.0, 20.0, 40.0, 80.0, 160.0, 320.0 samples.
01
Configure Target Density and Proposal Variance
Set target density to standard normal p(x) ∝ e^(-x^2/2) with proposal standard deviation σ_prop = 1.0 on the MCMC interface.
02
Execute Metropolis Stochastic Random Walk
Initialize chain at X_0 = 5.0. Draw proposal Y ~ N(X_t, 1.0) and accept with probability α = min(1, p(Y)/p(X_t)). Record 2,000 steps.
03
Diagnose Burn-in Phase and Trace Plots
Inspect trace plot trajectory as X_t migrates from extreme start X_0 = 5.0 down into the high-density target region around 0. Discard initial 500 burn-in steps.
04
Measure Acceptance Rate and Autocorrelation
Compute empirical acceptance rate, verifying that for Gaussian proposals σ_prop ≈ 1.0, acceptance rate aligns with optimal Roberts-Gelman efficiency (~44%).
05
Step Total Iteration Count Across N = 10 to 320
Increment MCMC sample sizes across standardized geometric steps N ∈ {10, 20, 40, 80, 160, 320}, recording empirical Monte Carlo standard error SE.
06
Conduct Log-Log Linear Regression on MCMC Error
Fit a linear regression model y = mx + b to coordinates (x = ln(N), y = ln(SE)). Confirm that empirical slope m = -0.5000 validates N^(-1/2) convergence.
YOUR NOTEBOOK · 0 OBSERVATIONS
Turn measurements into empirical insight.
Points are your recorded observations. The solid regression line validates Metropolis-Hastings MCMC mathematical scaling.
LINEAR REGRESSION FIT · y = mx + b
ln(SE) = -0.5000 · ln(N) + 1.2000
| # | MCMC Sample Iteration Count N (samples) | Empirical Monte Carlo Standard Error SE(θ̂) (Std Error SE) | Log Sample Size ln(N) | Log Monte Carlo Standard Error ln(SE) | Remove |
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