A general purpose sampling algorithm for continuous distributions (the t-walk)

Abstract

We develop a new general purpose MCMC sampler for arbitrary continuous distributions that requires no tuning. We call this MCMC the t-walk. The t-walk maintains two independent points in the sample space, and all moves are based on proposals that are then accepted with a standard Metropolis-Hastings acceptance probability on the product space. Hence the t-walk is provably convergent under the usual mild requirements. We restrict proposal distributions, or `moves', to those that produce an algorithm that is invariant to scale, and approximately invariant to affine transformations of the state space. Hence scaling of proposals, and effectively also coordinate transformations, that might be used to increase efficiency of the sampler, are not needed since the t-walk's operation is identical on any scaled version of the target distribution. Four moves are given that result in an effective sampling algorithm.

We use the simple device of updating only a random subset of coordinates at each step to allow application of the t-walk to high-dimensional problems. In a series of test problems across dimensions we find that the t-walk is only a small factor less efficient than optimally tuned algorithms, but significantly outperforms general random-walk M-H samplers that are not tuned for specific problems. Further, the t-walk remains effective for target distributions for which no optimal affine transformation exists such as those where correlation structure is very different in differing regions of state space.

Several examples are presented showing good mixing and convergence characteristics, varying in dimensions from 1 to 200 and with radically different scale and correlation structure, using exactly the same sampler. The t-walk is available for R, Python, MatLab and C++ at http://www.cimat.mx/~jac/twalk/