Walter R. Gilks is a British statistician whose contributions to computational Bayesian statistics helped make MCMC methods accessible to practitioners across medicine, ecology, and the social sciences. His development of adaptive rejection sampling provided an efficient, automatic method for generating samples from log-concave distributions, while his work on the BUGS (Bayesian inference Using Gibbs Sampling) project created the software infrastructure that democratized Bayesian computation in the 1990s and 2000s.
Life and Career
Trained in statistics in the United Kingdom, developing expertise in biostatistics and computational methods at the Medical Research Council (MRC) Biostatistics Unit in Cambridge.
Publishes "Derivative-free Adaptive Rejection Sampling for Gibbs Sampling" with P. Wild, introducing the ARS algorithm that becomes a standard component in MCMC software.
Contributes to the development of BUGS alongside David Spiegelhalter and colleagues at the MRC Biostatistics Unit, transforming Bayesian computation from an expert-only pursuit into a tool for applied researchers.
Co-edits Markov Chain Monte Carlo in Practice with S. Richardson and D. Spiegelhalter, one of the most influential applied MCMC textbooks.
Continues research on adaptive MCMC methods and their applications in health sciences, contributing to the development of WinBUGS and OpenBUGS.
Adaptive Rejection Sampling
The problem that adaptive rejection sampling (ARS) solves is fundamental to Gibbs sampling: how to draw samples efficiently from the full conditional distributions that arise at each step. When these conditionals are log-concave but not of a standard form, conventional rejection sampling requires a hand-crafted envelope function. ARS constructs this envelope automatically by building a piecewise linear upper bound on the log-density, refining it with each rejected point. The result is an algorithm that adapts to the shape of the target distribution on the fly, requiring no user intervention.
Sample from exp(h(x)) by inverse CDF of piecewise exponential
Accept with probability f(x) / exp(h(x))
On rejection, add the evaluation point to refine h(x)
The elegance of ARS lies in its self-improving nature: every rejected sample provides information that tightens the envelope, increasing the acceptance rate over time. For log-concave targets, the method is exact (not approximate) and increasingly efficient. Gilks later extended the method to handle non-log-concave distributions through the adaptive rejection Metropolis sampling (ARMS) variant, which adds a Metropolis-Hastings accept/reject step.
A distribution is log-concave if the logarithm of its density function is concave. Many common distributions are log-concave, including the normal, exponential, gamma (with shape parameter at least 1), and beta (with both parameters at least 1). Log-concavity guarantees that tangent lines to the log-density provide valid upper bounds, which is the mathematical foundation of ARS. This property makes the algorithm both theoretically sound and practically useful for a wide range of statistical models.
The BUGS Project
The BUGS software project, initiated by David Spiegelhalter with contributions from Gilks and others at the MRC Biostatistics Unit, was arguably the most important software development in the history of Bayesian statistics. Before BUGS, fitting a Bayesian model required writing custom MCMC samplers from scratch, a task that demanded deep expertise in both the statistical model and the computational algorithm. BUGS allowed users to specify models in a declarative language and automatically constructed and ran the appropriate Gibbs sampler.
Gilks contributed both to the computational engine, particularly the use of ARS within the Gibbs sampling framework, and to the development of practical methods for monitoring convergence and diagnosing problems with MCMC output. His co-edited volume Markov Chain Monte Carlo in Practice provided the theoretical and practical grounding that enabled a generation of applied researchers to adopt Bayesian methods.
Impact on Applied Bayesian Statistics
Gilks's work exemplifies the principle that theoretical advances in statistics achieve their full impact only when accompanied by accessible software and clear exposition. The combination of ARS as a building block, BUGS as a platform, and MCMC in Practice as a guide created the ecosystem that made Bayesian statistics practical. Thousands of published analyses in epidemiology, ecology, pharmacology, and other fields relied on the tools he helped build.
"MCMC methods have transformed Bayesian statistics from a theoretical curiosity to a practical tool for data analysis. The challenge now is not whether we can fit complex models, but whether we can do so reliably." — Walter Gilks, Markov Chain Monte Carlo in Practice (1996)