MonteCarloSampling

MonteCarloSampling is a class of computational methods that use random sampling to estimate numerical quantities. It is often employed when analytical solutions are difficult or impossible.

At its core, Monte Carlo sampling estimates an expectation or integral with respect to a probability distribution. One draws independent samples, evaluates a function on each sample, and averages the results. By the law of large numbers the estimator converges as sample size grows, and the central limit theorem provides error estimates.

Variants include Monte Carlo integration for high-dimensional integrals; importance sampling to reduce variance; stratified and rejection sampling to improve efficiency; and Markov Chain Monte Carlo methods such as Metropolis–Hastings and Gibbs sampling for complex distributions.

Applications span numerical integration, Bayesian statistics, physics simulations, finance (for example option pricing), engineering, and risk assessment. MCMC is especially common for sampling from posterior distributions in Bayesian inference.

Key practical considerations include choosing an appropriate sampling distribution, applying variance reduction techniques, assessing convergence, and managing computational cost. Uncertainty is typically reported as standard errors or confidence intervals derived from the estimator.