Data on high-dimensional spheres arise frequently in many disciplines either naturally or as a consequence of preliminary processing and can have intricate dependence structure that needs to be understood. We develop exploratory factor analysis of the projected normal distribution to explain the variability in such data using a few easily interpreted latent factors. Our methodology provides maximum likelihood estimates through a novel fast alternating expectation profile conditional maximization algorithm. Results on simulation experiments on a wide range of settings are uniformly excellent. Our methodology provides interpretable and insightful results when applied to tweets with the $\#MeToo$ hashtag in early December 2018, to time-course functional Magnetic Resonance Images of the average pre-teen brain at rest, to characterize handwritten digits, and to gene expression data from cancerous cells in the Cancer Genome Atlas.

Variable selection in Gaussian processes (GPs) is typically undertaken by thresholding the inverse lengthscales of `automatic relevance determination' kernels, but in high-dimensional datasets this approach can be unreliable. A more probabilistically principled alternative is to use spike and slab priors and infer a posterior probability of variable inclusion. However, existing implementations in GPs are extremely costly to run in both high-dimensional and large-$n$ datasets, or are intractable for most kernels. As such, we develop a fast and scalable variational inference algorithm for the spike and slab GP that is tractable with arbitrary differentiable kernels. We improve our algorithm's ability to adapt to the sparsity of relevant variables by Bayesian model averaging over hyperparameters, and achieve substantial speed ups using zero temperature posterior restrictions, dropout pruning and nearest neighbour minibatching. In experiments our method consistently outperforms vanilla and sparse variational GPs whilst retaining similar runtimes (even when $n=10^6$) and performs competitively with a spike and slab GP using MCMC but runs up to $1000$ times faster.

Reconstructing missing information in epidemic spreading on contact networks can be essential in prevention and containment strategies. For instance, identifying and warning infective but asymptomatic individuals (e.g., manual contact tracing) helped contain outbreaks in the COVID-19 pandemic. The number of possible epidemic cascades typically grows exponentially with the number of individuals involved. The challenge posed by inference problems in the epidemics processes originates from the difficulty of identifying the almost negligible subset of those compatible with the evidence (for instance, medical tests). Here we present a new generative neural networks framework that can sample the most probable infection cascades compatible with observations. Moreover, the framework can infer the parameters governing the spreading of infections. The proposed method obtains better or comparable results with existing methods on the patient zero problem, risk assessment, and inference of infectious parameters in synthetic and real case scenarios like spreading infections in workplaces and hospitals.

A Neyman-Scott process is a special case of a Cox process. The latent and observable stochastic processes are both Poisson processes. We consider a deep Neyman-Scott process in this paper, for which the building components of a network are all Poisson processes. We develop an efficient posterior sampling via Markov chain Monte Carlo and use it for likelihood-based inference. Our method opens up room for the inference in sophisticated hierarchical point processes. We show in the experiments that more hidden Poisson processes brings better performance for likelihood fitting and events types prediction. We also compare our method with state-of-the-art models for temporal real-world datasets and demonstrate competitive abilities for both data fitting and prediction, using far fewer parameters.

Bayesian optimization (BO) is an efficient method to optimize expensive black-box functions. It has been generalized to scenarios where objective function evaluations return stochastic binary feedback, such as success/failure in a given test, or preference between different parameter settings. In many real-world situations, the objective function can be evaluated in controlled 'contexts' or 'environments' that directly influence the observations. For example, one could directly alter the 'difficulty' of the test that is used to evaluate a system's performance. With binary feedback, the context determines the information obtained from each observation. For example, if the test is too easy/hard, the system will always succeed/fail, yielding uninformative binary outputs. Here we combine ideas from Bayesian active learning and optimization to efficiently choose the best context and optimization parameter on each iteration. We demonstrate the performance of our algorithm and illustrate how it can be used to tackle a concrete application in visual psychophysics: efficiently improving patients' vision via corrective lenses, using psychophysics measurements.

Model speed can be a deal breaker on large datasets. Leveraging an empirical study, we will look at two dimension reduction techniques and how they can be applied to a Gaussian Processes. Regarding implementation of the method, anyone familiar with the basics of conditional probability can develop a Gaussian Process model. However, to fully leverage the capabilities of the framework, a fair amount of in-depth knowledge is required. Gaussian processes also are not very computationally efficient, but their flexibility is makes them a common choice for niche regression problems.

It is difficult to use subsampling with variational inference in hierarchical models since the number of local latent variables scales with the dataset. Thus, inference in hierarchical models remains a challenge at large scale. It is helpful to use a variational family with structure matching the posterior, but optimization is still slow due to the huge number of local distributions. Instead, this paper suggests an amortized approach where shared parameters simultaneously represent all local distributions. This approach is similarly accurate as using a given joint distribution (e.g., a full-rank Gaussian) but is feasible on datasets that are several orders of magnitude larger. It is also dramatically faster than using a structured variational distribution.

We consider the problem of discovering $K$ related Gaussian directed acyclic graphs (DAGs), where the involved graph structures share a consistent causal order and sparse unions of supports. Under the multi-task learning setting, we propose a $l_1/l_2$-regularized maximum likelihood estimator (MLE) for learning $K$ linear structural equation models. We theoretically show that the joint estimator, by leveraging data across related tasks, can achieve a better sample complexity for recovering the causal order (or topological order) than separate estimations. Moreover, the joint estimator is able to recover non-identifiable DAGs, by estimating them together with some identifiable DAGs. Lastly, our analysis also shows the consistency of union support recovery of the structures. To allow practical implementation, we design a continuous optimization problem whose optimizer is the same as the joint estimator and can be approximated efficiently by an iterative algorithm. We validate the theoretical analysis and the effectiveness of the joint estimator in experiments.

Energy-efficient navigation constitutes an important challenge in electric vehicles, due to their limited battery capacity. We employ a Bayesian approach to model the energy consumption at road segments for efficient navigation. In order to learn the model parameters, we develop an online learning framework and investigate several exploration strategies such as Thompson Sampling and Upper Confidence Bound. We then extend our online learning framework to multi-agent setting, where multiple vehicles adaptively navigate and learn the parameters of the energy model. We analyze Thompson Sampling and establish rigorous regret bounds on its performance in the single-agent and multi-agent settings, through an analysis of the algorithm under batched feedback. Finally, we demonstrate the performance of our methods via experiments on several real-world city road networks.

We formulate natural gradient variational inference (VI), expectation propagation (EP), and posterior linearisation (PL) as extensions of Newton's method for optimising the parameters of a Bayesian posterior distribution. This viewpoint explicitly casts inference algorithms under the framework of numerical optimisation. We show that common approximations to Newton's method from the optimisation literature, namely Gauss-Newton and quasi-Newton methods (e.g., the BFGS algorithm), are still valid under this'Bayes-Newton' framework. This leads to a suite of novel algorithms which are guaranteed to result in positive semi-definite covariance matrices, unlike standard VI and EP. Our unifying viewpoint provides new insights into the connections between various inference schemes. All the presented methods apply to any model with a Gaussian prior and non-conjugate likelihood, which we demonstrate with (sparse) Gaussian processes and state space models. Keywords: Approximate Bayesian inference, optimisation, variational inference, expectation propagation, Gaussian processes.