39 research outputs found
Subsampling MCMC - An introduction for the survey statistician
The rapid development of computing power and efficient Markov Chain Monte
Carlo (MCMC) simulation algorithms have revolutionized Bayesian statistics,
making it a highly practical inference method in applied work. However, MCMC
algorithms tend to be computationally demanding, and are particularly slow for
large datasets. Data subsampling has recently been suggested as a way to make
MCMC methods scalable on massively large data, utilizing efficient sampling
schemes and estimators from the survey sampling literature. These developments
tend to be unknown by many survey statisticians who traditionally work with
non-Bayesian methods, and rarely use MCMC. Our article explains the idea of
data subsampling in MCMC by reviewing one strand of work, Subsampling MCMC, a
so called pseudo-marginal MCMC approach to speeding up MCMC through data
subsampling. The review is written for a survey statistician without previous
knowledge of MCMC methods since our aim is to motivate survey sampling experts
to contribute to the growing Subsampling MCMC literature.Comment: Accepted for publication in Sankhya A. Previous uploaded version
contained a bug in generating the figures and reference
Variational Hamiltonian Monte Carlo via Score Matching
Traditionally, the field of computational Bayesian statistics has been
divided into two main subfields: variational methods and Markov chain Monte
Carlo (MCMC). In recent years, however, several methods have been proposed
based on combining variational Bayesian inference and MCMC simulation in order
to improve their overall accuracy and computational efficiency. This marriage
of fast evaluation and flexible approximation provides a promising means of
designing scalable Bayesian inference methods. In this paper, we explore the
possibility of incorporating variational approximation into a state-of-the-art
MCMC method, Hamiltonian Monte Carlo (HMC), to reduce the required gradient
computation in the simulation of Hamiltonian flow, which is the bottleneck for
many applications of HMC in big data problems. To this end, we use a {\it
free-form} approximation induced by a fast and flexible surrogate function
based on single-hidden layer feedforward neural networks. The surrogate
provides sufficiently accurate approximation while allowing for fast
exploration of parameter space, resulting in an efficient approximate inference
algorithm. We demonstrate the advantages of our method on both synthetic and
real data problems
Hamiltonian Monte Carlo Acceleration Using Surrogate Functions with Random Bases
For big data analysis, high computational cost for Bayesian methods often
limits their applications in practice. In recent years, there have been many
attempts to improve computational efficiency of Bayesian inference. Here we
propose an efficient and scalable computational technique for a
state-of-the-art Markov Chain Monte Carlo (MCMC) methods, namely, Hamiltonian
Monte Carlo (HMC). The key idea is to explore and exploit the structure and
regularity in parameter space for the underlying probabilistic model to
construct an effective approximation of its geometric properties. To this end,
we build a surrogate function to approximate the target distribution using
properly chosen random bases and an efficient optimization process. The
resulting method provides a flexible, scalable, and efficient sampling
algorithm, which converges to the correct target distribution. We show that by
choosing the basis functions and optimization process differently, our method
can be related to other approaches for the construction of surrogate functions
such as generalized additive models or Gaussian process models. Experiments
based on simulated and real data show that our approach leads to substantially
more efficient sampling algorithms compared to existing state-of-the art
methods
Hamiltonian monte carlo with energy conserving subsampling
© 2019 Khue-Dung Dang, Matias Quiroz, Robert Kohn, Minh-Ngoc Tran, Mattias Villani. Hamiltonian Monte Carlo (HMC) samples efficiently from high-dimensional posterior distributions with proposed parameter draws obtained by iterating on a discretized version of the Hamiltonian dynamics. The iterations make HMC computationally costly, especially in problems with large data sets, since it is necessary to compute posterior densities and their derivatives with respect to the parameters. Naively computing the Hamiltonian dynamics on a subset of the data causes HMC to lose its key ability to generate distant parameter proposals with high acceptance probability. The key insight in our article is that efficient subsampling HMC for the parameters is possible if both the dynamics and the acceptance probability are computed from the same data subsample in each complete HMC iteration. We show that this is possible to do in a principled way in a HMC-within-Gibbs framework where the subsample is updated using a pseudo marginal MH step and the parameters are then updated using an HMC step, based on the current subsample. We show that our subsampling methods are fast and compare favorably to two popular sampling algorithms that use gradient estimates from data subsampling. We also explore the current limitations of subsampling HMC algorithms by varying the quality of the variance reducing control variates used in the estimators of the posterior density and its gradients