We consider the problem of estimating the differences between two causal directed acyclic graph (DAG) models with a shared topological order given i.i.d. samples from each model. This is of interest for example in genomics, where changes in the structure or edge weights of the underlying causal graphs reflect alterations in the gene regulatory networks. We here provide the first provably consistent method for directly estimating the differences in a pair of causal DAGs without separately learning two possibly large and dense DAG models and computing their difference. Our two-step algorithm first uses invariance tests between regression coefficients of the two data sets to estimate the skeleton of the difference graph and then orients some of the edges using invariance tests between regression residual variances. We demonstrate the properties of our method through a simulation study and apply it to the analysis of gene expression data from ovarian cancer and during T-cell activation.

To date, research effort has largely focussed on the development of Bayesian cubature, whose distributional output provides uncertainty quantification for the integral. However, the point estimators associated to Bayesian cubature can be inaccurate and acutely sensitive to the prior when the domain is high-dimensional. To address these drawbacks we introduce Bayes-Sard cubature, a probabilistic framework that combines the flexibility of Bayesian cubature with the robustness of classical cubatures which are well-established. This is achieved by considering a Gaussian process model for the integrand whose mean is a parametric regression model, with an improper prior on each regression coefficient. The features in the regression model consist of test functions which are guaranteed to be exactly integrated, with remaining degrees of freedom afforded to the non-parametric part. The asymptotic convergence of the Bayes-Sard cubature method is established and the theoretical results are numerically verified. In particular, we report two orders of magnitude reduction in error compared to Bayesian cubature in the context of a high-dimensional financial integral.

We investigate the problem of learning Bayesian networks in a robust model where an $\epsilon$-fraction of the samples are adversarially corrupted. In this work, we study the fully observable discrete case where the structure of the network is given. Even in this basic setting, previous learning algorithms either run in exponential time or lose dimension-dependent factors in their error guarantees. We provide the first computationally efficient robust learning algorithm for this problem with dimension-independent error guarantees. Our algorithm has near-optimal sample complexity, runs in polynomial time, and achieves error that scales nearly-linearly with the fraction of adversarially corrupted samples. Finally, we show on both synthetic and semi-synthetic data that our algorithm performs well in practice.

We introduce the variational filtering EM algorithm, a simple, general-purpose method for performing variational inference in dynamical latent variable models using information from only past and present variables, i.e. filtering. The algorithm is derived from the variational objective in the filtering setting and consists of an optimization procedure at each time step. By performing each inference optimization procedure with an iterative amortized inference model, we obtain a computationally efficient implementation of the algorithm, which we call amortized variational filtering. We present experiments demonstrating that this general-purpose method improves inference performance across several recent deep dynamical latent variable models.

Estimating the structure of directed acyclic graphs (DAGs, also known as Bayesian networks) is a challenging problem since the search space of DAGs is combinatorial and scales superexponentially with the number of nodes. Existing approaches rely on various local heuristics for enforcing the acyclicity constraint. In this paper, we introduce a fundamentally different strategy: we formulate the structure learning problem as a purely continuous optimization problem over real matrices that avoids this combinatorial constraint entirely. This is achieved by a novel characterization of acyclicity that is not only smooth but also exact. The resulting problem can be efficiently solved by standard numerical algorithms, which also makes implementation effortless. The proposed method outperforms existing ones, without imposing any structural assumptions on the graph such as bounded treewidth or in-degree.

In this paper, we propose a novel sampling method, the thermostat-assisted continuously-tempered Hamiltonian Monte Carlo, for the purpose of multimodal Bayesian learning. It simulates a noisy dynamical system by incorporating both a continuously-varying tempering variable and the Nos\'e-Hoover thermostats. A significant benefit is that it is not only able to efficiently generate i.i.d. samples when the underlying posterior distributions are multimodal, but also capable of adaptively neutralising the noise arising from the use of mini-batches. While the properties of the approach have been studied using synthetic datasets, our experiments on three real datasets have also shown its performance gains over several strong baselines for Bayesian learning with various types of neural networks plunged in.

We present an approach for simultaneously separating and localizing multiple sound sources using recorded microphone data. Inspired by topic models, our approach is based on a probabilistic model of inter-microphone phase differences, and poses separation and localization as a Bayesian inference problem. We assume sound activity is locally smooth across time, frequency, and location, and use the known position of the microphones to obtain a consistent separation. We compare the performance of our method against existing algorithms on simulated anechoic voice data and find that it obtains high performance across a variety of input conditions.

We consider testing and learning problems on causal Bayesian networks as defined by Pearl (Pearl, 2009). Given a causal Bayesian network M on a graph with n discrete variables and bounded in-degree and bounded ``confounded components'', we show that O(log n) interventions on an unknown causal Bayesian network X on the same graph, and O(n/epsilon^2) samples per intervention, suffice to efficiently distinguish whether X=M or whether there exists some intervention under which X and M are farther than epsilon in total variation distance. We also obtain sample/time/intervention efficient algorithms for: (i) testing the identity of two unknown causal Bayesian networks on the same graph; and (ii) learning a causal Bayesian network on a given graph. Although our algorithms are non-adaptive, we show that adaptivity does not help in general: Omega(log n) interventions are necessary for testing the identity of two unknown causal Bayesian networks on the same graph, even adaptively. Our algorithms are enabled by a new subadditivity inequality for the squared Hellinger distance between two causal Bayesian networks.

Combining Bayesian nonparametrics and a forward model selection strategy, we construct parsimonious Bayesian deep networks (PBDNs) that infer capacity-regularized network architectures from the data and require neither cross-validation nor fine-tuning when training the model. One of the two essential components of a PBDN is the development of a special infinite-wide single-hidden-layer neural network, whose number of active hidden units can be inferred from the data. The other one is the construction of a greedy layer-wise learning algorithm that uses a forward model selection criterion to determine when to stop adding another hidden layer. We develop both Gibbs sampling and stochastic gradient descent based maximum a posteriori inference for PBDNs, providing state-of-the-art classification accuracy and interpretable data subtypes near the decision boundaries, while maintaining low computational complexity for out-of-sample prediction.

Bayesian optimization is a powerful tool for global optimization of expensive functions. One of its key components is the underlying probabilistic model used for the objective function f. In practice, however, it is often unclear how one should appropriately choose a model, especially when gathering data is expensive. In this work, we introduce a novel automated Bayesian optimization approach that dynamically selects promising models for explaining the observed data using Bayesian Optimization in the model space. Crucially, we account for the uncertainty in the choice of model; our method is capable of using multiple models to represent its current belief about f and subsequently using this information for decision making. We argue, and demonstrate empirically, that our approach automatically finds suitable models for the objective function, which ultimately results in more-efficient optimization.