10 research outputs found
Structured Sparse Modelling with Hierarchical GP
In this paper a new Bayesian model for sparse linear regression with a
spatio-temporal structure is proposed. It incorporates the structural
assumptions based on a hierarchical Gaussian process prior for spike and slab
coefficients. We design an inference algorithm based on Expectation Propagation
and evaluate the model over the real data.Comment: SPARS 201
CATVI: conditional and adaptively truncated variational inference for hierarchical Bayesian nonparametric models
Current variational inference methods for hierarchical Bayesian nonparametric models can neither characterize the correlation struc- ture among latent variables due to the mean- eld setting, nor infer the true posterior dimension because of the universal trunca- tion. To overcome these limitations, we pro- pose the conditional and adaptively trun- cated variational inference method (CATVI) by maximizing the nonparametric evidence lower bound and integrating Monte Carlo into the variational inference framework. CATVI enjoys several advantages over tra- ditional methods, including a smaller diver- gence between variational and true posteri- ors, reduced risk of undertting or overt- ting, and improved prediction accuracy. Em- pirical studies on three large datasets re- veal that CATVI applied in Bayesian non- parametric topic models substantially out- performs competing models, providing lower perplexity and clearer topic-words clustering
Horseshoe priors for edge-preserving linear Bayesian inversion
In many large-scale inverse problems, such as computed tomography and image
deblurring, characterization of sharp edges in the solution is desired. Within
the Bayesian approach to inverse problems, edge-preservation is often achieved
using Markov random field priors based on heavy-tailed distributions. Another
strategy, popular in statistics, is the application of hierarchical shrinkage
priors. An advantage of this formulation lies in expressing the prior as a
conditionally Gaussian distribution depending of global and local
hyperparameters which are endowed with heavy-tailed hyperpriors. In this work,
we revisit the shrinkage horseshoe prior and introduce its formulation for
edge-preserving settings. We discuss a sampling framework based on the Gibbs
sampler to solve the resulting hierarchical formulation of the Bayesian inverse
problem. In particular, one of the conditional distributions is
high-dimensional Gaussian, and the rest are derived in closed form by using a
scale mixture representation of the heavy-tailed hyperpriors. Applications from
imaging science show that our computational procedure is able to compute sharp
edge-preserving posterior point estimates with reduced uncertainty
Practical Hilbert space approximate Bayesian Gaussian processes for probabilistic programming
Gaussian processes are powerful non-parametric probabilistic models for
stochastic functions. However they entail a complexity that is computationally
intractable when the number of observations is large, especially when estimated
with fully Bayesian methods such as Markov chain Monte Carlo. In this paper, we
focus on a novel approach for low-rank approximate Bayesian Gaussian processes,
based on a basis function approximation via Laplace eigenfunctions for
stationary covariance functions. The main contribution of this paper is a
detailed analysis of the performance and practical implementation of the method
in relation to key factors such as the number of basis functions, domain of the
prediction space, and smoothness of the latent function. We provide intuitive
visualizations and recommendations for choosing the values of these factors,
which make it easier for users to improve approximation accuracy and
computational performance. We also propose diagnostics for checking that the
number of basis functions and the domain of the prediction space are adequate
given the data. The proposed approach is simple and exhibits an attractive
computational complexity due to its linear structure, and it is easy to
implement in probabilistic programming frameworks. Several illustrative
examples of the performance and applicability of the method in the
probabilistic programming language Stan are presented together with the
underlying Stan model code.Comment: 33 pages, 19 figure
Bayesian inference for spatio-temporal spike-and-slab priors
In this work, we address the problem of solving a series of underdetermined linear inverse problemblems subject to a sparsity constraint. We generalize the spike-and-slab prior distribution to encode a priori correlation of the support of the solution in both space and time by imposing a transformed Gaussian process on the spike-and-slab probabilities. An expectation propagation (EP) algorithm for posterior inference under the proposed model is derived. For large scale problems, the standard EP algorithm can be prohibitively slow. We therefore introduce three different approximation schemes to reduce the computational complexity. Finally, we demonstrate the proposed model using numerical experiments based on both synthetic and real data sets.Peer reviewe