Abstract
The R package sns implements the stochastic Newton sampler (SNS), a MetropolisHastings Markov chain Monte Carlo (MCMC) algorithm where the proposal density function is a multivariate Gaussian based on a local, second-order Taylor-series expansion of log-density. The mean of the proposal function is the full Newton step in the NewtonRaphson optimization algorithm. Taking advantage of the local, multivariate geometry captured in log-density Hessian allows SNS to be more efficient than univariate samplers, approaching independent sampling as the density function increasingly resembles a multivariate Gaussian. SNS requires the log-density Hessian to be negative-definite everywhere in order to construct a valid proposal function. This property holds, or can be easily checked, for many GLM-like models. When the initial point is far from density peak, running SNS in non-stochastic mode by taking the Newton step - augmented with line search - allows the MCMC chain to converge to high-density areas faster. For high-dimensional problems, partitioning the state space into lower-dimensional subsets, and applying SNS to the subsets within a Gibbs sampling framework can significantly improve the mixing of SNS chains. In addition to the above strategies for improving convergence and mixing, sns offers utilities for diagnostics and visualization, sample-based calculation of Bayesian predictive posterior distributions, numerical differentiation, and log-density validation.
Highlights
In most real-world applications of Monte Carlo Markov Chain (MCMC) sampling, the probability density function (PDF) being sampled is multidimensional
When the Gaussian approximation is sufficiently close to the true PDF, this can lead to a nearly-uncorrelated chain of samples, with the extreme case of perfectly uncorrelated samples for a multivariate Gaussian distribution
In this paper we presented sns, an R package for Stochastic Newton Sampling of twicedifferentiable, log-concave PDFs, where a multivariate Gaussian resulting from second-order Taylor series expansion of log-density is used as proposal function in a Metropolis-Hastings framework
Summary
In most real-world applications of Monte Carlo Markov Chain (MCMC) sampling, the probability density function (PDF) being sampled is multidimensional. Univariate samplers generally have few tuning parameters, making them ideal candidates for black-box MCMC software such as JAGS (Plummer 2013) and OpenBUGS (Thomas, O’Hara, Ligges, and Sturtz 2006). They become less effective as PDF dimensionality rises and dimensions become more correlated (Girolami and Calderhead 2011). Development - and software implementation - of efficient, black-box multivariate MCMC algorithms is of great importance to widespread application of probabilistic models in statistics and machine learning.
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