arXiv subjects

Akihiko Nishimura

Publications and source records attributed to Akihiko Nishimura.

24 records · Page 2Linked to original sources

arXiv2019

Discontinuous Hamiltonian Monte Carlo for discrete parameters and discontinuous likelihoods

Hamiltonian Monte Carlo has emerged as a standard tool for posterior computation. In this article, we present an extension that can efficiently explore target distributions with discontinuous densities. Our extension in particular enables efficient sampling from ordinal parameters though embedding of probability mass functions into continuous spaces. We motivate our approach through a theory of discontinuous Hamiltonian dynamics and develop a corresponding numerical solver. The proposed solver is the first of its kind, with a remarkable ability to exactly preserve the Hamiltonian. We apply our algorithm to challenging posterior inference problems to demonstrate its wide applicability and competitive performance.

stat.CO↗

arXiv2019

Recycling intermediate steps to improve Hamiltonian Monte Carlo

Hamiltonian Monte Carlo (HMC) and related algorithms have become routinely used in Bayesian computation. In this article, we present a simple and provably accurate method to improve the efficiency of HMC and related algorithms with essentially no extra computational cost. This is achieved by {recycling the intermediate states along simulated trajectories of Hamiltonian dynamics. Standard algorithms use only the end points of trajectories, wastefully discarding all the intermediate states. Compared to the alternative methods for utilizing the intermediate states, our algorithm is simpler to apply in practice and requires little programming effort beyond the usual implementations of HMC and related algorithms. Our algorithm applies straightforwardly to the no-U-turn sampler, arguably the most popular variant of HMC. Through a variety of experiments, we demonstrate that our recycling algorithm yields substantial computational efficiency gains.

stat.CO↗

arXiv2019

Gradients do grow on trees: a linear-time ${\cal O}\hspace{-0.2em}\left( N \right)$-dimensional gradient for statistical phylogenetics

Calculation of the log-likelihood stands as the computational bottleneck for many statistical phylogenetic algorithms. Even worse is its gradient evaluation, often used to target regions of high probability. Order ${\cal O}\hspace{-0.2em}\left( N \right)$-dimensional gradient calculations based on the standard pruning algorithm require ${\cal O}\hspace{-0.2em}\left( N^2 \right)$ operations where N is the number of sampled molecular sequences. With the advent of high-throughput sequencing, recent phylogenetic studies have analyzed hundreds to thousands of sequences, with an apparent trend towards even larger data sets as a result of advancing technology. Such large-scale analyses challenge phylogenetic reconstruction by requiring inference on larger sets of process parameters to model the increasing data heterogeneity. To make this tractable, we present a linear-time algorithm for ${\cal O}\hspace{-0.2em}\left( N \right)$-dimensional gradient evaluation and apply it to general continuous-time Markov processes of sequence substitution on a phylogenetic tree without a need to assume either stationarity or reversibility. We apply this approach to learn the branch-specific evolutionary rates of three pathogenic viruses: West Nile virus, Dengue virus and Lassa virus. Our proposed algorithm significantly improves inference efficiency with a 126- to 234-fold increase in maximum-likelihood optimization and a 16- to 33-fold computational performance increase in a Bayesian framework.

stat.CO↗

arXiv2018

Bayesian Constraint Relaxation

Prior information often takes the form of parameter constraints. Bayesian methods include such information through prior distributions having constrained support. By using posterior sampling algorithms, one can quantify uncertainty without relying on asymptotic approximations. However, sharply constrained priors are (a) not necessary in some settings; and (b) tend to limit modeling scope to a narrow set of distributions that are tractable computationally. Inspired by the vast literature that replaces the slab-and-spike prior with a continuous approximation, we propose to replace the sharp indicator function of the constraint with an exponential kernel, thereby creating a close-to-constrained neighborhood within the Euclidean space in which the constrained subspace is embedded. This kernel decays with distance from the constrained space at a rate depending on a relaxation hyperparameter. By avoiding the sharp constraint, we enable use of off-the-shelf posterior sampling algorithms, such as Hamiltonian Monte Carlo, facilitating automatic computation in broad models. We study the constrained and relaxed distributions under multiple settings, and theoretically quantify their differences. We illustrate the method through multiple novel modeling examples.

stat.ME↗

arXiv2017

Geometrically Tempered Hamiltonian Monte Carlo

Hamiltonian Monte Carlo (HMC) has become routinely used for sampling from posterior distributions. Its extension Riemann manifold HMC (RMHMC) modifies the proposal kernel through distortion of local distances by a Riemannian metric. The performance depends critically on the choice of metric, with the Fisher information providing the standard choice. In this article, we propose a new class of metrics aimed at improving HMC's performance on multi-modal target distributions. We refer to the proposed approach as geometrically tempered HMC (GTHMC) due to its connection to other tempering methods. We establish a geometric theory behind RMHMC to motivate GTHMC and characterize its theoretical properties. Moreover, we develop a novel variable step size integrator for simulating Hamiltonian dynamics to improve on the usual Störmer-Verlet integrator which suffers from numerical instability in GTHMC settings. We illustrate GTHMC through simulations, demonstrating generality and substantial gains over standard HMC implementations in terms of effective sample sizes.

stat.CO↗

arXiv2016

Variable length trajectory compressible hybrid Monte Carlo

Hybrid Monte Carlo (HMC) generates samples from a prescribed probability distribution in a configuration space by simulating Hamiltonian dynamics, followed by the Metropolis (-Hastings) acceptance/rejection step. Compressible HMC (CHMC) generalizes HMC to a situation in which the dynamics is reversible but not necessarily Hamiltonian. This article presents a framework to further extend the algorithm. Within the existing framework, each trajectory of the dynamics must be integrated for the same amount of (random) time to generate a valid Metropolis proposal. Our generalized acceptance/rejection mechanism allows a more deliberate choice of the integration time for each trajectory. The proposed algorithm in particular enables an effective application of variable step size integrators to HMC-type sampling algorithms based on reversible dynamics. The potential of our framework is further demonstrated by another extension of HMC which reduces the wasted computations due to unstable numerical approximations and corresponding rejected proposals.

physics.comp-ph↗