The Sampling Gallery
Interactive Monte Carlo, MCMC, and related sampling algorithms.
Build intuition by watching samplers work
Markov chain Monte Carlo is a powerful tool for simulating from probability distributions — in particular posterior distributions for Bayesian inference. MCMC algorithms have good theoretical properties and are easily utilised in modern software packages. However, on paper each MCMC algorithm is only a few lines of mathematics; its behaviour is not always obvious. This collection of visualisations across a range of target geometries will let you watch a sampler explore a landscape in real time — where it proposes, what it rejects, how gradients, trajectories and adaptation shape the search. Pick a target, choose an algorithm, tune its parameters and watch it run.
Independent Monte Carlo
Independent draws — exact, weighted or low-discrepancyDraws from a simple proposal under an envelope and keeps the lucky ones — exact samples, and a lesson in why MCMC is needed.
Keeps every draw, weighted by p/q; weight degeneracy and the effective sample size are the whole story. SIR resampling and Pareto smoothing (PSIS) as toggles.
Low-discrepancy point sets instead of random draws — evenly stratified space-filling, side by side with plain Monte Carlo.
Markov chain Monte Carlo
From random walks to gradient-guided chains
Blind Gaussian hops, accepted by the density ratio — the simplest sampler there is.
Learns the proposal covariance from its own history as it runs.
Rolls a frictionless particle along gradients to take long, informed leaps; step-size adaptation optional.
HMC that picks its own path length and step size — the self-tuning sampler behind Stan.
Nudges proposals uphill along the gradient, then corrects to stay exact.
Samples uniformly under the density curve by stepping out an interval and shrinking it — no tuning-sensitive accept/reject step.
Slice-samples the angle around an ellipse through the current state and a Gaussian prior draw — rejection-free with no step size to tune.
Updates one coordinate at a time from its exact conditional.
Uses curvature as well as gradient to shape anisotropic proposals.
Gradient plus noise, never rejecting — fast but slightly biased.
Langevin steps from noisy mini-batch gradients; toggle control variates to tame the noise at the mode.
Sets its own step size by optimising over the space of measures.
Ensemble and multimodal methods
Populations of interacting chainsWalkers propose from each other’s positions — stretch moves are affine-invariant, so correlated ridges come for free. Differential evolution as a second mode.
Replicas at rising temperatures swap states — hot chains cross between modes and hand discoveries to the cold one.
Sequential and annealed methods
Through a ladder of easier distributionsA weighted particle cloud anneals from an easy reference to the target: reweight, resample, rejuvenate. Annealed importance sampling as a no-resampling mode.
Peels the distribution in shells of increasing likelihood, drawing replacements from a constrained region.
Non-reversible samplers
Piecewise-deterministic, rejection-freeDeterministic and variational particle methods
Optimisation in the space of measures
References
[1] Haario, Saksman & Tamminen. An adaptive Metropolis algorithm. Bernoulli (2001).
[2] Hoffman & Gelman. The No-U-Turn Sampler. JMLR (2011).
[3] Roberts & Tweedie. Exponential convergence of Langevin distributions. Bernoulli (1996).
[4] Li et al. Anisotropic Gaussian mutations via Hessian-Hamiltonian dynamics. ACM TOG (2015).
[5] Liu et al. Stein Variational Gradient Descent. NeurIPS (2016).
[6] Buchner. A statistical test for Nested Sampling algorithms. Stat. & Comput. (2014).
[7] ter Braak & Vrugt. Differential Evolution Markov Chain with snooker updater. Stat. & Comput. (2008).
[8] Welling & Teh. Bayesian Learning via Stochastic Gradient Langevin Dynamics. ICML (2011).
[9] Baker et al. Control variates for stochastic gradient MCMC. Stat. & Comput. (2019).
[10] Sharrock & Nemeth. Tuning-Free Sampling via Optimization on the Space of Measures. (2025).
[11] Bierkens, Fearnhead & Roberts. The Zig-Zag process. Ann. Statist. (2019).
[12] Bouchard-Côté, Vollmer & Doucet. The Bouncy Particle Sampler. JASA (2018).
[13] Neal. Slice sampling. Ann. Statist. (2003).
[14] von Neumann. Various techniques used in connection with random digits. (1951).
[15] Kahn & Marshall. Methods of reducing sample size in Monte Carlo computations. (1953).
[16] Geyer. Markov chain Monte Carlo maximum likelihood. (1991).
[17] Del Moral, Doucet & Jasra. Sequential Monte Carlo samplers. JRSS-B (2006).
[18] Goodman & Weare. Ensemble samplers with affine invariance. CAMCoS (2010).
[19] Neal. Annealed importance sampling. Stat. & Comput. (2001).
[20] Niederreiter. Random Number Generation and Quasi-Monte Carlo Methods. SIAM (1992).
[21] Vehtari, Simpson, Gelman, Yao & Gabry. Pareto smoothed importance sampling. JMLR (2024).
[22] Murray, Adams & MacKay. Elliptical slice sampling. AISTATS (2010).