We provide a comprehensive overview and tooling for GP modeling with non-Gaussian likelihoods using state space methods. The state space formulation allows for solving one-dimensional GP models in $\mathcal{O}(n)$ time and memory complexity. While existing literature has focused on the connection between GP regression and state space methods, the computational primitives allowing for inference using general likelihoods in combination with the Laplace approximation (LA), variational Bayes (VB), and assumed density filtering (ADF) / expectation propagation (EP) schemes has been largely overlooked. We present means of combining the efficient $\mathcal{O}(n)$ state space methodology with existing inference methods. We also further extend existing methods, and provide unifying code implementing all approaches.

Quadrature is the problem of estimating intractable integrals, a problem that arises in many Bayesian machine learning settings. We present an improved Bayesian framework for estimating intractable integrals of specific kinds of constrained integrands. We derive the necessary approximation scheme for a specific and especially useful instantiation of this framework: the use of a log transformation to model non-negative integrands. We also propose a novel method for optimizing the hyperparameters associated with this framework; we optimize the hyperparameters in the original space of the integrand as opposed to in the transformed space, resulting in a model that better explains the actual data. Experiments on both synthetic and real-world data demonstrate that the proposed framework achieves more-accurate estimates using less wall-clock time than previously preposed Bayesian quadrature procedures for non-negative integrands.

Learning algorithms that aggregate predictions from an ensemble of diverse base classifiers consistently outperform individual methods. Many of these strategies have been developed in a supervised setting, where the accuracy of each base classifier can be empirically measured and this information is incorporated in the training process. However, the reliance on labeled data precludes the application of ensemble methods to many real world problems where labeled data has not been curated. To this end we developed a new theoretical framework for binary classification, the Strategy for Unsupervised Multiple Method Aggregation (SUMMA), to estimate the performances of base classifiers and an optimal strategy for ensemble learning from unlabeled data.

In meta-learning an agent extracts knowledge from observed tasks, aiming to facilitate learning of novel future tasks. Under the assumption that future tasks are 'related' to previous tasks, representations should be learned in a way which captures the common structure across learned tasks, while allowing the learner sufficient flexibility to adapt to novel aspects of new tasks. We present a framework for meta-learning that is based on generalization error bounds, allowing us to extend various PAC-Bayes bounds to meta-learning. Learning takes place through the construction of a distribution over hypotheses based on the observed tasks, and its utilization for learning a new task. Thus, prior knowledge is incorporated through setting an experience-dependent prior for novel tasks. We develop a gradient-based algorithm which minimizes an objective function derived from the bounds and demonstrate its effectiveness numerically with deep neural networks. In addition to establishing the improved performance available through meta-learning, we demonstrate the intuitive way by which prior information is manifested at different levels of the network.

Recent work in unsupervised representation learning has focused on learning deep directed latent-variable models. Fitting these models by maximizing the marginal likelihood or evidence is typically intractable, thus a common approximation is to maximize the evidence lower bound (ELBO) instead. However, maximum likelihood training (whether exact or approximate) does not necessarily result in a good latent representation, as we demonstrate both theoretically and empirically. In particular, we derive variational lower and upper bounds on the mutual information between the input and the latent variable, and use these bounds to derive a rate-distortion curve that characterizes the tradeoff between compression and reconstruction accuracy. Using this framework, we demonstrate that there is a family of models with identical ELBO, but different quantitative and qualitative characteristics. Our framework also suggests a simple new method to ensure that latent variable models with powerful stochastic decoders do not ignore their latent code.

A parametric point process model is developed, with modeling based on the assumption that sequential observations often share latent phenomena, while also possessing idiosyncratic effects. An alternating optimization method is proposed to learn a "registered" point process that accounts for shared structure, as well as "warping" functions that characterize idiosyncratic aspects of each observed sequence. Under reasonable constraints, in each iteration we update the sample-specific warping functions by solving a set of constrained nonlinear programming problems in parallel, and update the model by maximum likelihood estimation. The justifiability, complexity and robustness of the proposed method are investigated in detail, and the influence of sequence stitching on the learning results is examined empirically. Experiments on both synthetic and real-world data demonstrate that the method yields explainable point process models, achieving encouraging results compared to state-of-the-art methods.

Recently, increased computational power and data availability, as well as algorithmic advances, have led machine learning techniques to impressive results in regression, classification, data-generation and reinforcement learning tasks. Despite these successes, the proximity to the physical limits of chip fabrication alongside the increasing size of datasets are motivating a growing number of researchers to explore the possibility of harnessing the power of quantum computation to speed-up classical machine learning algorithms. Here we review the literature in quantum machine learning and discuss perspectives for a mixed readership of classical machine learning and quantum computation experts. Particular emphasis will be placed on clarifying the limitations of quantum algorithms, how they compare with their best classical counterparts and why quantum resources are expected to provide advantages for learning problems. Learning in the presence of noise and certain computationally hard problems in machine learning are identified as promising directions for the field. Practical questions, like how to upload classical data into quantum form, will also be addressed.

Bayesian matrix factorization (BMF) is a powerful tool for producing low-rank representations of matrices and for predicting missing values and their confidence intervals. Scaling up the posterior inference for massive-scale matrices is challenging and requires distributing both data and computation over many workers, making communication the main computational bottleneck. Embarrassingly parallel inference would remove the communication needed, by using completely independent computations on different data subsets, but suffers from the inherent unidentifiability of BMF solutions. We introduce a hierarchical decomposition of the joint posterior distribution, which couples the subset inferences, allowing for embarrassingly parallel computations in a sequence of at most three stages. Using an efficient approximate implementation, we show empirically on both real and simulated data that our distributed approach is able to achieve a speed-up of almost an order of magnitude, with a negligible effect on predictive accuracy.

Imagine a situation where a group of adversaries is preparing an attack on the United States or U.S. interests. An intelligence analyst has observed some signals, but the situation is rapidly changing. The analyst faces the decision to alert a principal decision maker that an attack is imminent, or to wait until more is known about the situation. This warning decision is based on the analyst's observation and evaluation of signals, independent or correlated, and on her updating of the prior probabilities of possible scenarios and their outcomes. The warning decision also depends on the analyst's assessment of the crisis' dynamics and perception of the preferences of the principal decision maker, as well as the lead time needed for an appropriate response. This article presents a model to support this analyst's dynamic warning decision. As with most problems involving warning, the key is to manage the tradeoffs between false positives and false negatives given the probabilities and the consequences of intelligence failures of both types. The model is illustrated by revisiting the case of the attack on Pearl Harbor in December 1941. It shows that the radio silence of the Japanese fleet carried considerable information (Sir Arthur Conan Doyle's "dog in the night" problem), which was misinterpreted at the time. Even though the probabilities of different attacks were relatively low, their consequences were such that the Bayesian dynamic reasoning described here may have provided valuable information to key decision makers.

Their joint effects give rise to the variation in responses to the same question by different crowdworkers. This variation is low when the question is easy to answer and objective, and high when it is difficult and subjective. Unfortunately, current quality control methods for crowdsourcing consider only the question difficulty to account for the variation. As a result, these methods cannot distinguish workers' personal preferences for different correct answers of a partially subjective question from their ability/expertise to avoid objectively wrong answers for that question. To address this issue, we present a probabilistic model which (i) explicitly encodes question difficulty as a model parameter and (ii) implicitly encodes question subjectivity via latent preference factors for crowd-workers. We show that question subjectivity induces grouping of crowd-workers, revealed through clustering of their latent preferences. Moreover, we develop a quantitative measure of the subjectivity of a question. Experiments show that our model (1) improves the performance of both quality control for crowdsourced answers and next answer prediction for crowd-workers, and (2) can potentially provide coherent rankings of questions in terms of their difficulty and subjectivity, so that task providers can refine their designs of the crowdsourcing tasks, e.g. by removing highly subjective questions or inappropriately difficult questions.