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/.

VLT/SPHERE robust astrometry of the HR8799 planets at milliarcsecond-level accuracy

“…-The integrated autocorrelation time τ (see e.g. ForemanMackey et al 2013;Christen & Fox 2010;Goodman & Weare 2010), also called inefficiency factor, aims to give an estimate of the number of posterior PDF evaluations required to draw an independent sample. The smaller τ, the better.…”

A Robust Mass Estimator for Dark Matter Subhalo Perturbations in Strong Gravitational Lenses

“…In this paper, since we are working with simulated data, we skip this step entirely; from a practical standpoint, this is possible thanks to an adaptive Markov Chain Monte Carlo (MCMC) algorithm called T-Walk (Christen & Fox 2010), which is uniquely suited to efficient exploration of multimodal distributions. The T-Walk algorithm uses a Metropolis-Hastings step, but at each step, certain chains perform a "jump" over the current point in another, randomly chosen chain in an attempt to find new modes in the posterior distribution.…”

Comparison of statistical sampling methods with ScannerBit, the GAMBIT scanning module

“…In the serial version of the T-Walk algorithm [41], chains are advanced one at a time, with the proposal density based on the current parameter points of all chains not chosen for advancement, and the chain to be advanced chosen randomly at each iteration. In the parallel version, each MPI process randomly selects a chain for advancement at each iteration, and the proposal distribution used for advancing all chains is based only on the state of the remaining chains not chosen for advancement by any process in that iteration.…”

“…The version of the T-Walk algorithm described above, and implemented in ScannerBit, differs slightly from the original algorithm [41] in two ways. The first is the use of the full concurrent covariance matrix for the Gaussian jumps in the hop and blow moves, making them similar to the "walk" move of Ref.…”