12 research outputs found
Flexible Variational Bayes based on a Copula of a Mixture of Normals
Variational Bayes methods approximate the posterior density by a family of
tractable distributions and use optimisation to estimate the unknown parameters
of the approximation. Variational approximation is useful when exact inference
is intractable or very costly. Our article develops a flexible variational
approximation based on a copula of a mixture of normals, which is implemented
using the natural gradient and a variance reduction method. The efficacy of the
approach is illustrated by using simulated and real datasets to approximate
multimodal, skewed and heavy-tailed posterior distributions, including an
application to Bayesian deep feedforward neural network regression models. Each
example shows that the proposed variational approximation is much more accurate
than the corresponding Gaussian copula and a mixture of normals variational
approximations.Comment: 39 page
Monotonic Alpha-divergence Minimisation
In this paper, we introduce a novel iterative algorithm which carries out
-divergence minimisation by ensuring a systematic decrease in the
-divergence at each step. In its most general form, our framework
allows us to simultaneously optimise the weights and components parameters of a
given mixture model. Notably, our approach permits to build on various methods
previously proposed for -divergence minimisation such as gradient or
power descent schemes. Furthermore, we shed a new light on an integrated
Expectation Maximization algorithm. We provide empirical evidence that our
methodology yields improved results, all the while illustrating the numerical
benefits of having introduced some flexibility through the parameter
of the -divergence
Statistical and Computational Trade-offs in Variational Inference: A Case Study in Inferential Model Selection
Variational inference has recently emerged as a popular alternative to the
classical Markov chain Monte Carlo (MCMC) in large-scale Bayesian inference.
The core idea of variational inference is to trade statistical accuracy for
computational efficiency. It aims to approximate the posterior, reducing
computation costs but potentially compromising its statistical accuracy. In
this work, we study this statistical and computational trade-off in variational
inference via a case study in inferential model selection. Focusing on Gaussian
inferential models (a.k.a. variational approximating families) with diagonal
plus low-rank precision matrices, we initiate a theoretical study of the
trade-offs in two aspects, Bayesian posterior inference error and frequentist
uncertainty quantification error. From the Bayesian posterior inference
perspective, we characterize the error of the variational posterior relative to
the exact posterior. We prove that, given a fixed computation budget, a
lower-rank inferential model produces variational posteriors with a higher
statistical approximation error, but a lower computational error; it reduces
variances in stochastic optimization and, in turn, accelerates convergence.
From the frequentist uncertainty quantification perspective, we consider the
precision matrix of the variational posterior as an uncertainty estimate. We
find that, relative to the true asymptotic precision, the variational
approximation suffers from an additional statistical error originating from the
sampling uncertainty of the data. Moreover, this statistical error becomes the
dominant factor as the computation budget increases. As a consequence, for
small datasets, the inferential model need not be full-rank to achieve optimal
estimation error. We finally demonstrate these statistical and computational
trade-offs inference across empirical studies, corroborating the theoretical
findings.Comment: 56 pages, 8 figure
Bayesian Inversion, Uncertainty Analysis and Interrogation Using Boosting Variational Inference
Geoscientists use observed data to estimate properties of the Earth's interior. This often requires non-linear inverse problems to be solved and uncertainties to be estimated. Bayesian inference solves inverse problems under a probabilistic framework, in which uncertainty is represented by a so-called posterior probability distribution. Recently, variational inference has emerged as an efficient method to estimate Bayesian solutions. By seeking the closest approximation to the posterior distribution within any chosen family of distributions, variational inference yields a fully probabilistic solution. It is important to define expressive variational families so that the posterior distribution can be represented accurately. We introduce boosting variational inference (BVI) as a computationally efficient means to construct a flexible approximating family comprising all possible finite mixtures of simpler component distributions. We use Gaussian mixture components due to their fully parametric nature and the ease with which they can be optimized. We apply BVI to seismic travel time tomography and full waveform inversion, comparing its performance with other methods of solution. The results demonstrate that BVI achieves reasonable efficiency and accuracy while enabling the construction of a fully analytic expression for the posterior distribution. Samples that represent major components of uncertainty in the solution can be obtained analytically from each mixture component. We demonstrate that these samples can be used to solve an interrogation problem: to assess the size of a subsurface target structure. To the best of our knowledge, this is the first method in geophysics that provides both analytic and reasonably accurate probabilistic solutions to fully non-linear, high-dimensional Bayesian full waveform inversion problems