451 research outputs found
An Empirical Study of Stochastic Variational Algorithms for the Beta Bernoulli Process
Stochastic variational inference (SVI) is emerging as the most promising
candidate for scaling inference in Bayesian probabilistic models to large
datasets. However, the performance of these methods has been assessed primarily
in the context of Bayesian topic models, particularly latent Dirichlet
allocation (LDA). Deriving several new algorithms, and using synthetic, image
and genomic datasets, we investigate whether the understanding gleaned from LDA
applies in the setting of sparse latent factor models, specifically beta
process factor analysis (BPFA). We demonstrate that the big picture is
consistent: using Gibbs sampling within SVI to maintain certain posterior
dependencies is extremely effective. However, we find that different posterior
dependencies are important in BPFA relative to LDA. Particularly,
approximations able to model intra-local variable dependence perform best.Comment: ICML, 12 pages. Volume 37: Proceedings of The 32nd International
Conference on Machine Learning, 201
A Tutorial on Bayesian Nonparametric Models
A key problem in statistical modeling is model selection, how to choose a
model at an appropriate level of complexity. This problem appears in many
settings, most prominently in choosing the number ofclusters in mixture models
or the number of factors in factor analysis. In this tutorial we describe
Bayesian nonparametric methods, a class of methods that side-steps this issue
by allowing the data to determine the complexity of the model. This tutorial is
a high-level introduction to Bayesian nonparametric methods and contains
several examples of their application.Comment: 28 pages, 8 figure
Gamma Processes, Stick-Breaking, and Variational Inference
While most Bayesian nonparametric models in machine learning have focused on
the Dirichlet process, the beta process, or their variants, the gamma process
has recently emerged as a useful nonparametric prior in its own right. Current
inference schemes for models involving the gamma process are restricted to
MCMC-based methods, which limits their scalability. In this paper, we present a
variational inference framework for models involving gamma process priors. Our
approach is based on a novel stick-breaking constructive definition of the
gamma process. We prove correctness of this stick-breaking process by using the
characterization of the gamma process as a completely random measure (CRM), and
we explicitly derive the rate measure of our construction using Poisson process
machinery. We also derive error bounds on the truncation of the infinite
process required for variational inference, similar to the truncation analyses
for other nonparametric models based on the Dirichlet and beta processes. Our
representation is then used to derive a variational inference algorithm for a
particular Bayesian nonparametric latent structure formulation known as the
infinite Gamma-Poisson model, where the latent variables are drawn from a gamma
process prior with Poisson likelihoods. Finally, we present results for our
algorithms on nonnegative matrix factorization tasks on document corpora, and
show that we compare favorably to both sampling-based techniques and
variational approaches based on beta-Bernoulli priors
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