Monte Carlo Methods
Monte Carlo methods approximate integrals, expectations and probabilities by averaging over simulated random draws, replacing intractable analytic computation with the law of large numbers applied to a stream of samples.
Definition
Monte Carlo methods are computational techniques that estimate a deterministic quantity, typically an integral or expectation, as the average of a function evaluated at samples drawn from an appropriate probability distribution.
Scope
This area covers plain Monte Carlo estimation of integrals and expectations, importance sampling as a reweighting strategy, and Markov chain Monte Carlo for sampling from complex high-dimensional distributions, including the Gibbs sampler. It treats the statistical theory of these estimators (consistency, error rates, effective sample size) rather than physics-specific simulation models.
Sub-topics
Core questions
- How does averaging over random samples estimate an integral, and at what rate does the error decay?
- How can sampling from one distribution estimate expectations under another?
- How can a Markov chain be constructed so that its stationary distribution is the target of interest?
- How is the accuracy of a Monte Carlo estimate quantified when the draws are dependent?
Key theories
- Monte Carlo estimation
- By the law of large numbers, the sample mean of a function evaluated at independent draws converges to its expectation, and the central limit theorem gives a root-n error rate independent of dimension.
- Markov chain Monte Carlo
- Constructing a Markov chain whose invariant distribution is the target lets one sample from distributions known only up to a constant, with ergodic averages of the chain estimating expectations.
- Change of measure via importance sampling
- Drawing from a tractable proposal and reweighting by the ratio of target to proposal density yields unbiased estimates of expectations under the target, with efficiency governed by the weight variance.
Clinical relevance
Monte Carlo methods are the computational engine of modern statistics: they evaluate Bayesian posteriors, integrate out latent variables, propagate uncertainty through complex models, and estimate p-values and risk in settings where closed-form answers do not exist, with applications spanning physics, genetics, finance and epidemiology.
History
Monte Carlo methods originated in nuclear physics computations at Los Alamos in the 1940s and were named after the casino; the Metropolis algorithm followed in 1953, Hastings generalized it in 1970, and the 1990s rediscovery of Gibbs sampling by statisticians turned Markov chain Monte Carlo into the dominant tool of computational Bayesian statistics.
Key figures
- Nicholas Metropolis
- Stanislaw Ulam
- Christian P. Robert
- Andrew Gelman
Related topics
Seminal works
- robert2004
- metropolis1949
Frequently asked questions
- Why does Monte Carlo error not grow with dimension?
- The standard error of a plain Monte Carlo average shrinks like one over the square root of the number of draws regardless of the dimension of the integral. This dimension-independence is why Monte Carlo often beats grid-based quadrature for high-dimensional problems.
- What is the difference between plain Monte Carlo and Markov chain Monte Carlo?
- Plain Monte Carlo uses independent draws from the target distribution. Markov chain Monte Carlo instead simulates a dependent sequence whose long-run distribution is the target, which lets it sample distributions that cannot be drawn from directly.