Abstract
We present a novel implementation of the adaptively annealed thermodynamic integration technique using Hamiltonian Monte Carlo (HMC). Thermodynamic integration with importance sampling and adaptive annealing is an especially useful method for estimating model evidence for problems that use physics-based mathematical models. Because it is based on importance sampling, this method requires an efficient way to refresh the ensemble of samples. Existing successful implementations use binary slice sampling on the Hilbert curve to accomplish this task. This implementation works well if the model has few parameters or if it can be broken into separate parts with identical parameter priors that can be refreshed separately. However, for models that are not separable and have many parameters, a different method for refreshing the samples is needed. HMC, in the form of the MC-Stan package, is effective for jointly refreshing the ensemble under a high-dimensional model. MC-Stan uses automatic differentiation to compute the gradients of the likelihood that HMC requires in about the same amount of time as it computes the likelihood function itself, easing the programming burden compared to implementations of HMC that require explicitly specified gradient functions. We present a description of the overall TI-Stan procedure and results for representative example problems.
Highlights
TI is a numerical technique for evaluating model evidence integrals
Various improvements and changes have been made over the decades, and the incarnation of the technique that our method is based on is the adaptively-annealed, importance sampling-based method described by Goggans and Chi [2]
We have found instead that the biggest problem in implementing model evidence estimation methods lies in devising a method to generate new samples within an equi-likelihood contour that are sufficiently independent for nested sampling to proceed
Summary
TI is a numerical technique for evaluating model evidence integrals. The technique was originally developed [1] to estimate the free energy of a fluid. Various improvements and changes have been made over the decades, and the incarnation of the technique that our method is based on is the adaptively-annealed, importance sampling-based method described by Goggans and Chi [2]. Their implementation follows John Skilling’s BayeSys [3], and both make use of BSS and the Hilbert curve to complete the implementation. This article proposes a modification of this method that uses PyStan [4,5] and the NUTS [6] instead of BSS and the Hilbert curve. Inference and Maximum Entropy Methods in Science and Engineering [7]
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