The Cox process is a flexible event model that can account for uncertainty of the intensity function in the Poisson process. However, previous approaches make strong assumptions in terms of time stationarity, potentially failing to generalize when the data do not conform to the assumed stationarity conditions. In this paper we bring up two most popular Cox models representing two extremes, and propose a novel semi-stationary Cox process model that can take benefits from both models. Our model has a set of Gaussian process latent functions governed by a latent stationary Markov process where we provide analytic derivations for the variational inference. Empirical evaluations on several synthetic and real-world events data including the football shot attempts and daily earthquakes, demonstrate that the proposed model is promising, can yield improved generalization performance over existing approaches.

Bayesian probabilistic numerical methods are a set of tools providing posterior distributions on the output of numerical methods. The use of these methods is usually motivated by the fact that they can represent our uncertainty due to incomplete/finite information about the continuous mathematical problem being approximated. In this paper, we demonstrate that this paradigm can provide additional advantages, such as the possibility of transferring information between several numerical methods. This allows users to represent uncertainty in a more faithful manner and, as a by-product, provide increased numerical efficiency. We propose the first such numerical method by extending the well-known Bayesian quadrature algorithm to the case where we are interested in computing the integral of several related functions. We then prove convergence rates for the method in the well-specified and misspecified cases, and demonstrate its efficiency in the context of multi-fidelity models for complex engineering systems and a problem of global illumination in computer graphics.

The generalization properties of Gaussian processes depend heavily on the choice of kernel, and this choice remains a dark art. We present the Neural Kernel Network (NKN), a flexible family of kernels represented by a neural network. The NKN’s architecture is based on the composition rules for kernels, so that each unit of the network corresponds to a valid kernel. It can compactly approximate compositional kernel structures such as those used by the Automatic Statistician (Lloyd et al., 2014), but because the architecture is differentiable, it is end-to-end trainable with gradient- based optimization. We show that the NKN is universal for the class of stationary kernels. Empirically we demonstrate NKN’s pattern discovery and extrapolation abilities on several tasks that depend crucially on identifying the underlying structure, including time series and texture extrapolation, as well as Bayesian optimization.

In order to scale standard Gaussian process (GP) regression to large-scale datasets, aggregation models employ factorized training process and then combine predictions from distributed experts. The state-of-the-art aggregation models, however, either provide inconsistent predictions or require time-consuming aggregation process. We first prove the inconsistency of typical aggregations using disjoint or random data partition, and then present a consistent yet efficient aggregation model for large-scale GP. The proposed model inherits the advantages of aggregations, e.g., closed-form inference and aggregation, parallelization and distributed computing. Furthermore, theoretical and empirical analyses reveal that the new aggregation model performs better due to the consistent predictions that converge to the true underlying function when the training size approaches infinity.