MATLAB Implementation of Metropolis-Hastings Sampler

The Metropolis-Hastings (MH) sampler is a core algorithm in Markov Chain Monte Carlo (MCMC) methods, widely used in Bayesian inference and statistical modeling. It constructs a Markov chain whose stationary distribution converges to the target distribution, enabling sampling from complex probability distributions.

Algorithm Core Concept The key components of MH sampling are the Proposal Distribution and Acceptance Probability. The algorithmic steps are: 1. Initialization: Select an initial state as the starting point for sampling. 2. Candidate Generation: Generate a new candidate point from a proposal distribution (e.g., normal distribution). 3. Acceptance Probability Calculation: Determine whether to accept the candidate based on the ratio of target and proposal distributions. If accepted, update the current state; otherwise retain the original state. 4. Iteration: Repeat steps 2-3 until the required sample size is achieved.

MATLAB Implementation Key Points When implementing MH sampling in MATLAB, consider: - Target Distribution: Define as the logarithmic form of the target function to prevent numerical overflow. Use log probabilities in acceptance ratio calculations. - Proposal Distribution: Typically choose symmetric distributions (e.g., Gaussian) to simplify acceptance probability computation. Implement using MATLAB's normrnd() function for Gaussian proposals. - Convergence Diagnostics: Assess chain convergence by monitoring sampling paths or calculating Gelman-Rubin statistics. Use trace plots and autocorrelation functions for visual inspection.

Application Scenarios MH sampling is particularly suitable for high-dimensional spaces or distributions with complex analytical forms, such as posterior distribution sampling in Bayesian models. Its flexibility makes it a universal tool for probabilistic modeling.

Optimization Directions - Tune the width of proposal distribution to balance exploration efficiency and acceptance rate - Incorporate adaptive methods to dynamically optimize proposal distribution parameters - Implement thinning and burn-in periods to improve sample quality