A more efficient approach to parallel-tempered Markov-chain Monte Carlo for the highly structured posteriors of gravitational-wave signals

We introduce a new Markov-chain Monte Carlo (MCMC) approach designed for the efficient sampling of highly correlated and multimodal posteriors. Parallel tempering, though effective, is a costly technique for sampling such posteriors. Our approach minimizes the use of parallel tempering, only applying it for a short time to build a proposal distribution that is based upon estimation of the kernel density and tuned to the target posterior. This proposal makes subsequent use of parallel tempering unnecessary, allowing all chains to be cooled to sample the target distribution. Gains in efficiency are found to increase with increasing posterior complexity, ranging from tens of percent in the simplest cases to over a factor of 10 for the more complex cases. Our approach is particularly useful in the context of parameter estimation of gravitational-wave signals measured by ground-based detectors, which is currently done through Bayesian inference with MCMC, one of the leading sampling methods. Posteriors for these signals are typically multimodal with strong nonlinear correlations, making sampling difficult. As we enter the advanced-detector era, improved sensitivities and wider bandwidths will drastically increase the computational cost of analyses, demanding more efficient search algorithms to meet these challenges.

Figure 1

Schematic of the parallel-tempering tuned approach. Each line represents a chain with temperature increasing vertically. Swaps in chain locations show when PT is in effect. Phase I is the parallel-tempered burn-in which ends after $\sim 500$ effective samples, at which point the $T=1$ chain shares its differential evolution buffer with the other chains and the specialized proposal is tuned using its samples. During phase II the $T>1$ chains are linearly annealed to $T=1$ over the course of $\sim 10$ ACTs. Phase III produces all samples used to estimate the posterior, where chains sample independently using a jump proposal optimized to the target posterior.

Figure 2

An illustration of the clustered-KDE approach to estimating a distribution from a set of samples. (a) Samples drawn from a distribution composed of several 2D Gaussians with random sizes and orientations. The distribution was then estimated from these samples using both the simple KDE and the clustered-KDE methods. (b) Comparison of samples drawn from these estimates and the original set drawn from the true distribution.