Contextual multi-armed bandit problems arise frequently in important industrial applications. Existing solutions model the context either linearly, which enables uncertainty driven (principled) exploration, or non-linearly, by using epsilon-greedy exploration policies. Here we present a deep learning framework for contextual multi-armed bandits that is both non-linear and enables principled exploration at the same time. We tackle the exploration vs. exploitation trade-off through Thompson sampling by exploiting the connection between inference time dropout and sampling from the posterior over the weights of a Bayesian neural network. In order to adjust the level of exploration automatically as more data is made available to the model, the dropout rate is learned rather than considered a hyperparameter. We demonstrate that our approach substantially reduces regret on two tasks (the UCI Mushroom task and the Casino Parity task) when compared to 1) non-contextual bandits, 2) epsilon-greedy deep contextual bandits, and 3) fixed dropout rate deep contextual bandits. Our approach is currently being applied to marketing optimization problems at HubSpot.

A recently-introduced class of probabilistic (uncertainty-aware) solvers for ordinary differential equations (ODEs) applies Gaussian (Kalman) filtering to initial value problems. These methods model the true solution $x$ and its first $q$ derivatives a priori as a Gauss--Markov process $\boldsymbol{X}$, which is then iteratively conditioned on information about $\dot{x}$. We prove worst-case local convergence rates of order $h^{q+1}$ for a wide range of versions of this Gaussian ODE filter, as well as global convergence rates of order $h^q$ in the case of $q=1$ and an integrated Brownian motion prior, and analyze how inaccurate information on $\dot{x}$ coming from approximate evaluations of $f$ affects these rates. Moreover, we present explicit formulas for the steady states and show that the posterior confidence intervals are well calibrated in all considered cases that exhibit global convergence---in the sense that they globally contract at the same rate as the truncation error.

Probabilistic representations, such as Bayesian and Markov networks, are fundamental to much of statistical machine learning. Thus, learning probabilistic representations directly from data is a deep challenge, the main computational bottleneck being inference that is intractable. Tractable learning is a powerful new paradigm that attempts to learn distributions that support efficient probabilistic querying. By leveraging local structure, representations such as sum-product networks (SPNs) can capture high tree-width models with many hidden layers, essentially a deep architecture, while still admitting a range of probabilistic queries to be computable in time polynomial in the network size. The leaf nodes in SPNs, from which more intricate mixtures are formed, are tractable univariate distributions, and so the literature has focused on Bernoulli and Gaussian random variables. This is clearly a restriction for handling mixed discrete-continuous data, especially if the continuous features are generated from non-parametric and non-Gaussian distribution families. In this work, we present a framework that systematically integrates SPN structure learning with weighted model integration, a recently introduced computational abstraction for performing inference in hybrid domains, by means of piecewise polynomial approximations of density functions of arbitrary shape. Our framework is instantiated by exploiting the notion of propositional abstractions, thus minimally interfering with the SPN structure learning module, and supports a powerful query interface for conditioning on interval constraints. Our empirical results show that our approach is effective, and allows a study of the trade off between the granularity of the learned model and its predictive power.

We propose a Multi-vAlue Rule Set (MRS) model for in-hospital predicting patient mortality. Compared to rule sets built from single-valued rules, MRS adopts a more generalized form of association rules that allows multiple values in a condition. Rules of this form are more concise than classical single-valued rules in capturing and describing patterns in data. Our formulation also pursues a higher efficiency of feature utilization, which reduces possible cost in data collection and storage. We propose a Bayesian framework for formulating a MRS model and propose an efficient inference method for learning a maximum \emph{a posteriori}, incorporating theoretically grounded bounds to iteratively reduce the search space and improve the search efficiency. Experiments show that our model was able to achieve better performance than baseline method including the current system used by the hospital.

While imitation learning is often used in robotics, this approach often suffers from data mismatch and compounding errors. DAgger is an iterative algorithm that addresses these issues by aggregating training data from both the expert and novice policies, but does not consider the impact of safety. We present a probabilistic extension to DAgger, which attempts to quantify the confidence of the novice policy as a proxy for safety. Our method, EnsembleDAgger, approximates a GP using an ensemble of neural networks. Using the variance as a measure of confidence, we compute a decision rule that captures how much we doubt the novice, thus determining when it is safe to allow the novice to act. With this approach, we aim to maximize the novice's share of actions, while constraining the probability of failure. We demonstrate improved safety and learning performance compared to other DAgger variants and classic imitation learning on an inverted pendulum and in the MuJoCo HalfCheetah environment.

In this paper we present a method for simultaneously segmenting brain tumors and an extensive set of organs-at-risk for radiation therapy planning of glioblastomas. The method combines a contrast-adaptive generative model for whole-brain segmentation with a new spatial regularization model of tumor shape using convolutional restricted Boltzmann machines. We demonstrate experimentally that the method is able to adapt to image acquisitions that differ substantially from any available training data, ensuring its applicability across treatment sites; that its tumor segmentation accuracy is comparable to that of the current state of the art; and that it captures most organs-at-risk sufficiently well for radiation therapy planning purposes. The proposed method may be a valuable step towards automating the delineation of brain tumors and organs-at-risk in glioblastoma patients undergoing radiation therapy.

It is the focus of this work to extend and study the previously proposed quantum-like Bayesian networks to deal with decision-making scenarios by incorporating the notion of maximum expected utility in influence diagrams. The general idea is to take advantage of the quantum interference terms produced in the quantum-like Bayesian Network to influence the probabilities used to compute the expected utility of some action. This way, we are not proposing a new type of expected utility hypothesis. On the contrary, we are keeping it under its classical definition. We are only incorporating it as an extension of a probabilistic graphical model in a compact graphical representation called an influence diagram in which the utility function depends on the probabilistic influences of the quantum-like Bayesian network. Our findings suggest that the proposed quantum-like influence digram can indeed take advantage of the quantum interference effects of quantum-like Bayesian Networks to maximise the utility of a cooperative behaviour in detriment of a fully rational defect behaviour under the prisoner's dilemma game.

This paper introduces a new sparse spatio-temporal structured Gaussian process regression framework for online and offline Bayesian inference. This is the first framework that gives a time-evolving representation of the interdependencies between the components of the sparse signal of interest. A hierarchical Gaussian process describes such structure and the interdependencies are represented via the covariance matrices of the prior distributions. The inference is based on the expectation propagation method and the theoretical derivation of the posterior distribution is provided in the paper. The inference framework is thoroughly evaluated over synthetic, real video and electroencephalography (EEG) data where the spatio-temporal evolving patterns need to be reconstructed with high accuracy. It is shown that it achieves 15% improvement of the F-measure compared with the alternating direction method of multipliers, spatio-temporal sparse Bayesian learning method and one-level Gaussian process model. Additionally, the required memory for the proposed algorithm is less than in the one-level Gaussian process model. This structured sparse regression framework is of broad applicability to source localisation and object detection problems with sparse signals.

Evidence in probabilistic reasoning may be `hard' or `soft', that is, it may be of yes/no form, or it may involve a strength of belief, in the unit interval [0,1]. Reasoning with soft, $[0,1]$-valued evidence is important in many situations but may lead to different, confusing interpretations. This paper intends to bring more mathematical clarity to the field by shifting the existing focus from specification of soft evidence to accomodation of soft evidence. There are two main approaches, known as Jeffrey's rule and Pearl's method, which give different outcomes on soft evidence. This paper describes these two approaches as different ways of updating with soft evidence, highlighting their differences, similarities and applications. This account is based on a novel channel-based approach to Bayesian probability. Proper understanding of these two update mechanisms is highly relevant for inference, decision tools and probabilistic programming languages.

We introduce a methodology for nonlinear inverse problems using a variational Bayesian approach where the unknown quantity is a spatial field. A structured Bayesian Gaussian process latent variable model is used both to construct a low-dimensional generative model of the sample-based stochastic prior as well as a surrogate for the forward evaluation. Its Bayesian formulation captures epistemic uncertainty introduced by the limited number of input and output examples, automatically selects an appropriate dimensionality for the learned latent representation of the data, and rigorously propagates the uncertainty of the data-driven dimensionality reduction of the stochastic space through the forward model surrogate. The structured Gaussian process model explicitly leverages spatial information for an informative generative prior to improve sample efficiency while achieving computational tractability through Kronecker product decompositions of the relevant kernel matrices. Importantly, the Bayesian inversion is carried out by solving a variational optimization problem, replacing traditional computationally-expensive Monte Carlo sampling. The methodology is demonstrated on an elliptic PDE and is shown to return well-calibrated posteriors and is tractable with latent spaces with over 100 dimensions.