414 research outputs found
Slice sampling covariance hyperparameters of latent Gaussian models
The Gaussian process (GP) is a popular way to specify dependencies between
random variables in a probabilistic model. In the Bayesian framework the
covariance structure can be specified using unknown hyperparameters.
Integrating over these hyperparameters considers different possible
explanations for the data when making predictions. This integration is often
performed using Markov chain Monte Carlo (MCMC) sampling. However, with
non-Gaussian observations standard hyperparameter sampling approaches require
careful tuning and may converge slowly. In this paper we present a slice
sampling approach that requires little tuning while mixing well in both strong-
and weak-data regimes.Comment: 9 pages, 4 figures, 4 algorithms. Minor corrections to previous
version. This version to appear in Advances in Neural Information Processing
Systems (NIPS) 23, 201
Incorporating Side Information in Probabilistic Matrix Factorization with Gaussian Processes
Probabilistic matrix factorization (PMF) is a powerful method for modeling
data associated with pairwise relationships, finding use in collaborative
filtering, computational biology, and document analysis, among other areas. In
many domains, there is additional information that can assist in prediction.
For example, when modeling movie ratings, we might know when the rating
occurred, where the user lives, or what actors appear in the movie. It is
difficult, however, to incorporate this side information into the PMF model. We
propose a framework for incorporating side information by coupling together
multiple PMF problems via Gaussian process priors. We replace scalar latent
features with functions that vary over the space of side information. The GP
priors on these functions require them to vary smoothly and share information.
We successfully use this new method to predict the scores of professional
basketball games, where side information about the venue and date of the game
are relevant for the outcome.Comment: 18 pages, 4 figures, Submitted to UAI 201
Learning the Structure of Deep Sparse Graphical Models
Deep belief networks are a powerful way to model complex probability
distributions. However, learning the structure of a belief network,
particularly one with hidden units, is difficult. The Indian buffet process has
been used as a nonparametric Bayesian prior on the directed structure of a
belief network with a single infinitely wide hidden layer. In this paper, we
introduce the cascading Indian buffet process (CIBP), which provides a
nonparametric prior on the structure of a layered, directed belief network that
is unbounded in both depth and width, yet allows tractable inference. We use
the CIBP prior with the nonlinear Gaussian belief network so each unit can
additionally vary its behavior between discrete and continuous representations.
We provide Markov chain Monte Carlo algorithms for inference in these belief
networks and explore the structures learned on several image data sets.Comment: 20 pages, 6 figures, AISTATS 2010, Revise
Elliptical slice sampling
Many probabilistic models introduce strong dependencies between variables
using a latent multivariate Gaussian distribution or a Gaussian process. We
present a new Markov chain Monte Carlo algorithm for performing inference in
models with multivariate Gaussian priors. Its key properties are: 1) it has
simple, generic code applicable to many models, 2) it has no free parameters,
3) it works well for a variety of Gaussian process based models. These
properties make our method ideal for use while model building, removing the
need to spend time deriving and tuning updates for more complex algorithms.Comment: 8 pages, 6 figures, appearing in AISTATS 2010 (JMLR: W&CP volume 6).
Differences from first submission: some minor edits in response to feedback
Gaussian Process Kernels for Pattern Discovery and Extrapolation
Gaussian processes are rich distributions over functions, which provide a Bayesian nonparametric approach to smoothing and interpolation. We introduce simple closed form kernels that can be used with Gaussian processes to discover patterns and enable extrapolation. These kernels are derived by modeling a spectral density – the Fourier transform of a kernel – with a Gaussian mixture. The proposed kernels support a broad class of stationary covariances, but Gaussian process inference remains simple and analytic. We demonstrate the proposed kernels by discovering patterns and performing long range extrapolation on synthetic examples, as well as atmospheric CO2 trends and airline passenger data. We also show that it is possible to reconstruct several popular standard covariances within our framework.Engineering and Applied Science
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Message Passing Inference with Chemical Reaction Networks
Recent work on molecular programming has explored new possibilities for computational abstractions with biomolecules, including logic gates, neural networks, and linear systems. In the future such abstractions might enable nanoscale devices that can sense and control the world at a molecular scale. Just as in macroscale robotics, it is critical that such devices can learn about their environment and reason under uncertainty. At this small scale, systems are typically modeled as chemical reaction networks. In this work, we develop a procedure that can take arbitrary probabilistic graphical models, represented as factor graphs over discrete random variables, and compile them into chemical reaction networks that implement inference. In particular, we show that marginalization based on sum-product message passing can be implemented in terms of reactions between chemical species whose concentrations represent probabilities. We show algebraically that the steady state concentration of these species correspond to the marginal distributions of the random variables in the graph and validate the results in simulations. As with standard sum-product inference, this procedure yields exact results for tree-structured graphs, and approximate solutions for loopy graphs.Engineering and Applied SciencesOther Research Uni
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