Bayesian Computation and MCMC
Bayesian computation makes inference practical by drawing samples from posterior distributions that cannot be evaluated in closed form, chiefly through Markov chain Monte Carlo.
Definition
Bayesian computation is the set of numerical methods for approximating posterior distributions and the expectations taken over them; Markov chain Monte Carlo constructs a Markov chain whose stationary distribution is the target posterior, so that its samples can be used for inference.
Scope
This area covers the algorithms that power modern Bayesian analysis: the Metropolis-Hastings framework, Gibbs sampling, gradient-based Hamiltonian Monte Carlo, and deterministic variational approximations, together with the convergence diagnostics and Monte Carlo error assessment that make their output trustworthy.
Sub-topics
Core questions
- How can samples be drawn from a posterior known only up to a normalizing constant?
- How do Metropolis-Hastings and Gibbs sampling construct chains with the correct stationary distribution?
- How does gradient information let Hamiltonian Monte Carlo explore high-dimensional posteriors efficiently?
- When are deterministic approximations such as variational inference preferable to sampling?
- How is convergence of an MCMC sampler diagnosed and Monte Carlo error quantified?
Key concepts
- Markov chain Monte Carlo
- stationary distribution
- detailed balance
- burn-in
- mixing
- effective sample size
- convergence diagnostics
- Monte Carlo standard error
Key theories
- Markov chain Monte Carlo
- By constructing a Markov chain whose invariant distribution is the posterior, MCMC turns intractable integration into the problem of simulating and averaging over a chain.
- Detailed balance
- Reversibility with respect to the target distribution is the standard sufficient condition guaranteeing that a sampler leaves the posterior invariant, underpinning Metropolis-Hastings and Gibbs.
- Convergence diagnostics
- Practical inference relies on diagnostics such as the potential scale reduction factor and effective sample size to judge whether chains have reached and mixed across the stationary distribution.
Clinical relevance
MCMC and related computation enable fitting realistic hierarchical and nonlinear models throughout science, from population pharmacokinetics and genetics to cosmology and ecology, where posteriors have no analytic form.
History
The Metropolis algorithm (1953) and Hastings' generalization (1970) originated in physics; Geman and Geman's Gibbs sampler (1984) and Gelfand and Smith's 1990 paper brought these methods into mainstream statistics, triggering the Bayesian computational revolution that continues with Hamiltonian Monte Carlo and variational methods.
Debates
- Sampling versus deterministic approximation
- MCMC offers asymptotically exact samples at high computational cost, while variational inference is fast but approximate; the trade-off between accuracy and scalability remains an active concern.
Key figures
- Nicholas Metropolis
- W. Keith Hastings
- Stuart Geman
- Donald Geman
- Radford Neal
Related topics
Seminal works
- robert2004
- brooks2011
- gelman2013
Frequently asked questions
- Why is MCMC needed at all?
- For most realistic models the posterior has no closed form and its normalizing constant is an intractable high-dimensional integral; MCMC sidesteps this by producing samples from the posterior using only its unnormalized density.
- How do I know an MCMC run has converged?
- Convergence is assessed with diagnostics such as the potential scale reduction factor across multiple chains, trace plots, and effective sample size, though these can detect failure to converge but never prove convergence with certainty.