We present a non-factorized variational method for full posterior inference in Bayesian hierarchical models, with the goal of capturing the posterior variable dependencies via efficient and possibly parallel computation. Our approach unifies the integrated nested Laplace approximation (INLA) under the variational framework.

With the advent of modern stimulation techniques in neuroscience, the opportunity arises to map neuron to neuron connectivity. In this work, we develop a method for efficiently inferring posterior distributions over synaptic strengths in neural microcircuits. The input to our algorithm is data from experiments in which action potentials from putative presynaptic neurons can be evoked while a subthreshold recording is made from a single postsynaptic neuron. We present a realistic statistical model which accounts for the main sources of variability in this experiment and allows for significant prior information about the connectivity and neuronal cell types to be incorporated if available. Due to the technical challenges and sparsity of these systems, it is important to focus experimental time stimulating the neurons whose synaptic strength is most ambiguous, therefore we also develop an online optimal design algorithm for choosing which neurons to stimulate at each trial.

Expectation Propagation (EP) is a popular approximate posterior inference algorithm that often provides a fast and accurate alternative to sampling-based methods. However, while the EP framework in theory allows for complex non-Gaussian factors, there is still a significant practical barrier to using them within EP, because doing so requires the implementation of message update operators, which can be difficult and require hand-crafted approximations. In this work, we study the question of whether it is possible to automatically derive fast and accurate EP updates by learning a discriminative model (e.g., a neural network or random forest) to map EP message inputs to EP message outputs. We address the practical concerns that arise in the process, and we provide empirical analysis on several challenging and diverse factors, indicating that there is a space of factors where this approach appears promising.

We introduce a novel variational method that allows to approximately integrate out kernel hyperparameters, such as length-scales, in Gaussian process regression. This approach consists of a novel variant of the variational framework that has been recently developed for the Gaussian process latent variable model which additionally makes use of a standardised representation of the Gaussian process. We consider this technique for learning Mahalanobis distance metrics in a Gaussian process regression setting and provide experimental evaluations and comparisons with existing methods by considering datasets with high-dimensional inputs.

We address the problem of recoverability i.e. deciding whether there exists a consistent estimator of a given relation Q, when data are missing not at random. We employ a formal representation called'Missingness Graphs' to explicitly portray the causal mechanisms responsible for missingness and to encode dependencies between these mechanisms and the variables being measured. Using this representation, we derive conditions that the graph should satisfy to ensure recoverability and devise algorithms to detect the presence of these conditions in the graph.

Stochastic variational inference (SVI) uses stochastic optimization to scale up Bayesian computation to massive data. We present an alternative perspective on SVI as approximate parallel coordinate ascent. SVI trades-off bias and variance to step close to the unknown true coordinate optimum given by batch variational Bayes (VB). We define a model to automate this process.

It was recently proved using graph covers (Ruozzi, 2012) that the Bethe partition function is upper bounded by the true partition function for a binary pairwise model that is attractive. Here we provide a new, arguably simpler proof from first principles. We make use of the idea of clamping a variable to a particular value. For an attractive model, we show that summing over the Bethe partition functions for each sub-model obtained after clamping any variable can only raise (and hence improve) the approximation. In fact, we derive a stronger result that may have other useful implications. Repeatedly clamping until we obtain a model with no cycles, where the Bethe approximation is exact, yields the result. We also provide a related lower bound on a broad class of approximate partition functions of general pairwise multi-label models that depends only on the topology. We demonstrate that clamping a few wisely chosen variables can be of practical value by dramatically reducing approximation error.

In many problem settings, parameter vectors are not merely sparse, but dependent in such a way that non-zero coefficients tend to cluster together. We refer to this form of dependency as "region sparsity". Classical sparse regression methods, such as the lasso and automatic relevance determination (ARD), model parameters as independent a priori, and therefore do not exploit such dependencies. Here we introduce a hierarchical model for smooth, region-sparse weight vectors and tensors in a linear regression setting. Our approach represents a hierarchical extension of the relevance determination framework, where we add a transformed Gaussian process to model the dependencies between the prior variances of regression weights. We combine this with a structured model of the prior variances of Fourier coefficients, which eliminates unnecessary high frequencies. The resulting prior encourages weights to be region-sparse in two different bases simultaneously. We develop efficient approximate inference methods and show substantial improvements over comparable methods (e.g., group lasso and smooth RVM) for both simulated and real datasets from brain imaging.

The central challenge in probabilistic inference is numerical integration, to average over ensembles of models or unknown (hyper-)parameters (for example to compute the marginal likelihood or a partition function).

This paper studies the following problem: given samples from a high dimensional discrete distribution, we want to estimate the leading (δ, ρ)-modes of the underlying distributions. A point is defined to be a (δ, ρ)-mode if it is a local optimum of the density within a δ-neighborhood under metric ρ. As we increase the "scale" parameter δ, the neighborhood size increases and the total number of modes monotonically decreases. The sequence of the (δ, ρ)-modes reveal intrinsic topographical information of the underlying distributions. Though the mode finding problem is generally intractable in high dimensions, this paper unveils that, if the distribution can be approximated well by a tree graphical model, mode characterization is significantly easier. An efficient algorithm with provable theoretical guarantees is proposed and is applied to applications like data analysis and multiple predictions.