Gibbs Sampling for Gaussian Distributions in MCMC (Markov Chain Monte Carlo)

In MCMC (Markov Chain Monte Carlo) methods, the Gibbs sampling algorithm serves as an efficient sampling technique for Gaussian distributions. This algorithm operates by sampling from full conditional probability distributions, meaning each variable's sampling step depends on the current values of all other variables. This sequential updating mechanism enhances both sampling efficiency and accuracy by breaking down complex multivariate sampling into manageable univariate steps. For Gaussian distributions specifically, Gibbs sampling leverages the conditional Gaussian properties where each variable's conditional distribution remains Gaussian given the others. The implementation typically involves: - Initializing all variables with starting values - Iteratively sampling each variable from its conditional distribution: x_i ~ P(x_i | x_{-i}) - Using the updated values immediately for subsequent conditional distributions A practical Python implementation would utilize NumPy for handling Gaussian distributions: import numpy as np def gibbs_gaussian(initial_params, iterations): samples = [] current = initial_params for _ in range(iterations): for i in range(len(current)): # Sample from conditional distribution P(x_i | x_{-i}) # For Gaussians, this involves computing conditional mean and variance conditional_mean = compute_conditional_mean(current, i) conditional_var = compute_conditional_variance(current, i) current[i] = np.random.normal(conditional_mean, np.sqrt(conditional_var)) samples.append(current.copy()) return np.array(samples) In computer vision applications, Gibbs sampling proves particularly valuable for tasks like image segmentation and image denoising. For segmentation, it can model pixel dependencies through Gaussian Markov Random Fields, while for denoising, it helps estimate clean image values by sampling from conditional distributions of noisy observations. The algorithm's sequential nature allows it to capture complex spatial relationships while maintaining computational tractability.