8,521 research outputs found
Distributed Bayesian Matrix Factorization with Limited Communication
Bayesian matrix factorization (BMF) is a powerful tool for producing low-rank
representations of matrices and for predicting missing values and providing
confidence intervals. Scaling up the posterior inference for massive-scale
matrices is challenging and requires distributing both data and computation
over many workers, making communication the main computational bottleneck.
Embarrassingly parallel inference would remove the communication needed, by
using completely independent computations on different data subsets, but it
suffers from the inherent unidentifiability of BMF solutions. We introduce a
hierarchical decomposition of the joint posterior distribution, which couples
the subset inferences, allowing for embarrassingly parallel computations in a
sequence of at most three stages. Using an efficient approximate
implementation, we show improvements empirically on both real and simulated
data. Our distributed approach is able to achieve a speed-up of almost an order
of magnitude over the full posterior, with a negligible effect on predictive
accuracy. Our method outperforms state-of-the-art embarrassingly parallel MCMC
methods in accuracy, and achieves results competitive to other available
distributed and parallel implementations of BMF.Comment: 28 pages, 8 figures. The paper is published in Machine Learning
journal. An implementation of the method is is available in SMURFF software
on github (bmfpp branch): https://github.com/ExaScience/smurf
Global consensus Monte Carlo
To conduct Bayesian inference with large data sets, it is often convenient or
necessary to distribute the data across multiple machines. We consider a
likelihood function expressed as a product of terms, each associated with a
subset of the data. Inspired by global variable consensus optimisation, we
introduce an instrumental hierarchical model associating auxiliary statistical
parameters with each term, which are conditionally independent given the
top-level parameters. One of these top-level parameters controls the
unconditional strength of association between the auxiliary parameters. This
model leads to a distributed MCMC algorithm on an extended state space yielding
approximations of posterior expectations. A trade-off between computational
tractability and fidelity to the original model can be controlled by changing
the association strength in the instrumental model. We further propose the use
of a SMC sampler with a sequence of association strengths, allowing both the
automatic determination of appropriate strengths and for a bias correction
technique to be applied. In contrast to similar distributed Monte Carlo
algorithms, this approach requires few distributional assumptions. The
performance of the algorithms is illustrated with a number of simulated
examples
Bayesian nonparametric Plackett-Luce models for the analysis of preferences for college degree programmes
In this paper we propose a Bayesian nonparametric model for clustering
partial ranking data. We start by developing a Bayesian nonparametric extension
of the popular Plackett-Luce choice model that can handle an infinite number of
choice items. Our framework is based on the theory of random atomic measures,
with the prior specified by a completely random measure. We characterise the
posterior distribution given data, and derive a simple and effective Gibbs
sampler for posterior simulation. We then develop a Dirichlet process mixture
extension of our model and apply it to investigate the clustering of
preferences for college degree programmes amongst Irish secondary school
graduates. The existence of clusters of applicants who have similar preferences
for degree programmes is established and we determine that subject matter and
geographical location of the third level institution characterise these
clusters.Comment: Published in at http://dx.doi.org/10.1214/14-AOAS717 the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Random projections for Bayesian regression
This article deals with random projections applied as a data reduction
technique for Bayesian regression analysis. We show sufficient conditions under
which the entire -dimensional distribution is approximately preserved under
random projections by reducing the number of data points from to in the case . Under mild
assumptions, we prove that evaluating a Gaussian likelihood function based on
the projected data instead of the original data yields a
-approximation in terms of the Wasserstein
distance. Our main result shows that the posterior distribution of Bayesian
linear regression is approximated up to a small error depending on only an
-fraction of its defining parameters. This holds when using
arbitrary Gaussian priors or the degenerate case of uniform distributions over
for . Our empirical evaluations involve different
simulated settings of Bayesian linear regression. Our experiments underline
that the proposed method is able to recover the regression model up to small
error while considerably reducing the total running time
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