Two problems with variational expectation maximisation for time series models

Abstract

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 requiring fewer computational resources 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 wellknown compactness property 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.

Markov Chain Monte Carlo Methods for State-Space Models with Point Process Observations

“…the uncertainty. VB methods, on the other hand, often underestimate the uncertainty within the state transition process due to the independent assumption of the mean field approximation (Turner & Sahani, 2010). Note that the number of data points is large in this data set, and given the fact that the posteriors are unimodal, it is reasonable to expect MCMC and VB to show similar performance.…”

Semi-Supervised Video Segmentation Using Tree Structured Graphical Models

“…The desire to capture short and long range correlations inherently leads to a "loopy" graphical model, which does not permit exact inference. Variational inference methods such as the popular mean-field approximations [9] fail to propagate meaningful uncertainty information in timeseries models [20]. Therefore, we avoid such approximations and instead estimate a tree structured graphical model for videos, which permits exact inference.…”

Variational Bayesian tracking: Whole track convergence for large-scale ecological video monitoring

“…Turner and Sahani have pointed out that the mean field approximation fails to correctly propagate the state uncertainty from one time step to the next [4]. That is, the uncertainty represented by p(x t−1 | Y t−1 ) is not properly factored into the q(x t ) which means that q(x t ) is too narrow.…”

“…As Turner and Sahani have shown [4], the independence assumptions inherent in the mean field models appears to break the temporal dependencies that drive the filtering and This work was supported by a grant for the Natural Environment Research Council of the UK smoothing algorithms, preventing the propagation of uncertainty through time and leading to unrealistically precise estimates of the hidden state. In this paper we show that the mean field approximation may be used to successfully track, via filtering and smoothing, provided that the variational updates are made in an appropriate order.…”