Two problems with variational expectation maximisation for time-series models

Variational methods are a key component of the approximate inference and learning toolbox. These methods fill an important middle ground, retaining distributional information about uncertainty in latent variables, unlike maximum a posteriori methods (MAP), and yet generally requiring less computational time than Monte Carlo Markov Chain methods. In particular the variational Expectation Maximisation (vEM) and variational Bayes algorithms, both involving variational optimisation of a free-energy, are widely used in time-series modelling. Here, we investigate the success of vEM in simple probabilistic time-series models. First we consider the inference step of vEM, and show that a consequence of the well-known compactness property of variational inference is a failure to propagate uncertainty in time, thus limiting the usefulness of the retained distributional information. In particular, the uncertainty may appear to be smallest precisely when the approximation is poorest. Second, we consider parameter learning and analytically reveal systematic biases in the parameters found by vEM. Surprisingly, simpler variational approximations (such a mean-field) can lead to less bias than more complicated structured approximations.

Bayesian Structured Prediction Using Gaussian Processes

We introduce a conceptually novel structured prediction model, GPstruct, which is kernelized, non-parametric and Bayesian, by design. We motivate the model with respect to existing approaches, among others, conditional random fields (CRFs), maximum margin Markov networks (M3N), and structured support vector machines (SVMstruct), which embody only a subset of its properties. We present an inference procedure based on Markov Chain Monte Carlo. The framework can be instantiated for a wide range of structured objects such as linear chains, trees, grids, and other general graphs. As a proof of concept, the model is benchmarked on several natural language processing tasks and a video gesture segmentation task involving a linear chain structure. We show prediction accuracies for GPstruct which are comparable to or exceeding those of CRFs and SVMstruct.

Bayesian parameter identification for vibro-acoustic models with nonparametric uncertainty

Vibration and acoustic analysis at higher frequencies faces two challenges: computing the response without using an excessive number of degrees of freedom, and quantifying its uncertainty due to small spatial variations in geometry, material properties and boundary conditions. Efficient models make use of the observation that when the response of a decoupled vibro-acoustic subsystem is sufficiently sensitive to uncertainty in such spatial variations, the local statistics of its natural frequencies and mode shapes saturate to universal probability distributions. This holds irrespective of the causes that underly these spatial variations and thus leads to a nonparametric description of uncertainty. This work deals with the identification of uncertain parameters in such models by using experimental data. One of the difficulties is that both experimental errors and modeling errors, due to the nonparametric uncertainty that is inherent to the model type, are present. This is tackled by employing a Bayesian inference strategy. The prior probability distribution of the uncertain parameters is constructed using the maximum entropy principle. The likelihood function that is subsequently computed takes the experimental information, the experimental errors and the modeling errors into account. The posterior probability distribution, which is computed with the Markov Chain Monte Carlo method, provides a full uncertainty quantification of the identified parameters, and indicates how well their uncertainty is reduced, with respect to the prior information, by the experimental data. © 2013 Taylor & Francis Group, London.

A reversible infinite HMM using normalised random measures

Copyright 2014 by the author(s). We present a nonparametric prior over reversible Markov chains. We use completely random measures, specifically gamma processes, to construct a countably infinite graph with weighted edges. By enforcing symmetry to make the edges undirected we define a prior over random walks on graphs that results in a reversible Markov chain. The resulting prior over infinite transition matrices is closely related to the hierarchical Dirichlet process but enforces reversibility. A reinforcement scheme has recently been proposed with similar properties, but the de Finetti measure is not well characterised. We take the alternative approach of explicitly constructing the mixing measure, which allows more straightforward and efficient inference at the cost of no longer having a closed form predictive distribution. We use our process to construct a reversible infinite HMM which we apply to two real datasets, one from epigenomics and one ion channel recording.

Bayesian inference and learning in Gaussian process state-space models with Particle MCMC

State-space models are successfully used in many areas of science, engineering and economics to model time series and dynamical systems. We present a fully Bayesian approach to inference and learning (i.e. state estimation and system identification) in nonlinear nonparametric state-space models. We place a Gaussian process prior over the state transition dynamics, resulting in a flexible model able to capture complex dynamical phenomena. To enable efficient inference, we marginalize over the transition dynamics function and, instead, infer directly the joint smoothing distribution using specially tailored Particle Markov Chain Monte Carlo samplers. Once a sample from the smoothing distribution is computed, the state transition predictive distribution can be formulated analytically. Our approach preserves the full nonparametric expressivity of the model and can make use of sparse Gaussian processes to greatly reduce computational complexity.