MCMCteknikker

MCMCteknikker, or Markov Chain Monte Carlo techniques, are a class of computational algorithms that use random sampling to estimate properties of complex probability distributions. By constructing a Markov chain whose equilibrium distribution matches the target distribution, these methods generate dependent samples that can be used to approximate integrals, compute expectations, or sample from posterior distributions in Bayesian inference.

The foundational method, the Metropolis algorithm, proposes a new state from a symmetric proposal distribution and accepts it with probability equal to the ratio of target densities. Metropolis–Hastings generalises this by allowing asymmetric proposals, making it applicable to a wider range of problems. Gibbs sampling is a special case of Metropolis–Hastings where each variable is updated conditionally on all others; it is particularly useful when full conditional distributions are tractable. Slice sampling introduces auxiliary variables to sample from the joint distribution without requiring a proposal distribution, often improving convergence in multimodal settings.

More advanced methods such as Hamiltonian Monte Carlo (HMC) and its variants, including the No‑U‑Turn Sampler (NUTS), exploit gradient information to propose distant moves that maintain high acceptance rates, leading to efficient exploration of high‑dimensional spaces. Reversible‑jump MCMC extends the framework to trans‑dimensional problems, allowing model comparison and structure learning.

MCMC techniques underpin many areas of science, including physics, economics, machine learning, and phylogenetics. Practical use requires careful tuning of step sizes, proposal distributions, and assessment of convergence using diagnostics such as trace plots, autocorrelation, effective sample size, and Gelman–Rubin statistics. Open‑source software packages like PyMC, Stan, and JAGS provide user‑friendly implementations of these algorithms, facilitating widespread application across disciplines.