This paper presents a novel approach for approximate integration over the uncertainty of noise and signal variances in Gaussian process (GP) regression. Our efficient and straightforward approach can also be applied to integration over input dependent noise variance (heteroscedasticity) and input dependent signal variance (nonstationarity) by setting independent GP priors for the noise and signal variances. We use expectation propagation (EP) for inference and compare results to Markov chain Monte Carlo in two simulated data sets and three empirical examples. The results show that EP produces comparable results with less computational burden.

Recently, there has been a growing interest in modeling planning with information constraints. Accordingly, an agent maximizes a regularized expected utility known as the free energy, where the regularizer is given by the information divergence from a prior to a posterior policy. While this approach can be justified in various ways, including from statistical mechanics and information theory, it is still unclear how it relates to decision-making against adversarial environments. This connection has previously been suggested in work relating the free energy to risk-sensitive control and to extensive form games. Here, we show that a single-agent free energy optimization is equivalent to a game between the agent and an imaginary adversary. The adversary can, by paying an exponential penalty, generate costs that diminish the decision maker's payoffs. It turns out that the optimal strategy of the adversary consists in choosing costs so as to render the decision maker indifferent among its choices, which is a definining property of a Nash equilibrium, thus tightening the connection between free energy optimization and game theory.

When making decisions under uncertainty, the optimal choices are often difficult to discern, especially if not enough information has been gathered. Two key questions in this regard relate to whether one should stop the information gathering process and commit to a decision (stopping criterion), and if not, what information to gather next (selection criterion). In this paper, we show that the recently introduced notion, Same-Decision Probability (SDP), can be useful as both a stopping and a selection criterion, as it can provide additional insight and allow for robust decision making in a variety of scenarios. This query has been shown to be highly intractable, being PP^PP-complete, and is exemplary of a class of queries which correspond to the computation of certain expectations. We propose the first exact algorithm for computing the SDP, and demonstrate its effectiveness on several real and synthetic networks. Finally, we present new complexity results, such as the complexity of computing the SDP on models with a Naive Bayes structure. Additionally, we prove that computing the non-myopic value of information is complete for the same complexity class as computing the SDP.

Massive Open Online Courses (MOOCs) bring together thousands of people from different geographies and demographic backgrounds -- but to date, little is known about how they learn or communicate. We introduce a new content-analysed MOOC dataset and use Bayesian Non-negative Matrix Factorization (BNMF) to extract communities of learners based on the nature of their online forum posts. We see that BNMF yields a superior probabilistic generative model for online discussions when compared to other models, and that the communities it learns are differentiated by their composite students' demographic and course performance indicators. These findings suggest that computationally efficient probabilistic generative modelling of MOOCs can reveal important insights for educational researchers and practitioners and help to develop more intelligent and responsive online learning environments.

Stable random variables are motivated by the central limit theorem for densities with (potentially) unbounded variance and can be thought of as natural generalizations of the Gaussian distribution to skewed and heavy-tailed phenomenon. In this paper, we introduce stable graphical (SG) models, a class of multivariate stable densities that can also be represented as Bayesian networks whose edges encode linear dependencies between random variables. One major hurdle to the extensive use of stable distributions is the lack of a closed-form analytical expression for their densities. This makes penalized maximum-likelihood based learning computationally demanding. We establish theoretically that the Bayesian information criterion (BIC) can asymptotically be reduced to the computationally more tractable minimum dispersion criterion (MDC) and develop StabLe, a structure learning algorithm based on MDC. We use simulated datasets for five benchmark network topologies to empirically demonstrate how StabLe improves upon ordinary least squares (OLS) regression. We also apply StabLe to microarray gene expression data for lymphoblastoid cells from 727 individuals belonging to eight global population groups. We establish that StabLe improves test set performance relative to OLS via ten-fold cross-validation. Finally, we develop SGEX, a method for quantifying differential expression of genes between different population groups.

This paper studies graphical model selection, i.e., the problem of estimating a graph of statistical relationships among a collection of random variables. Conventional graphical model selection algorithms are passive, i.e., they require all the measurements to have been collected before processing begins. We propose an active learning algorithm that uses junction tree representations to adapt future measurements based on the information gathered from prior measurements. We prove that, under certain conditions, our active learning algorithm requires fewer scalar measurements than any passive algorithm to reliably estimate a graph. A range of numerical results validate our theory and demonstrates the benefits of active learning.

We have observed an interesting, yet unexplained, phenomenon: Semidefinite programming (SDP) based relaxations of maximum likelihood estimators (MLE) tend to be tight in recovery problems with noisy data, even when MLE cannot exactly recover the ground truth. Several results establish tightness of SDP based relaxations in the regime where exact recovery from MLE is possible. However, to the best of our knowledge, their tightness is not understood beyond this regime. As an illustrative example, we focus on the generalized Procrustes problem.

In recent work on both generative and discriminative score to log-likelihood-ratio calibration, it was shown that linear transforms give good accuracy only for a limited range of operating points. Moreover, these methods required tailoring of the calibration training objective functions in order to target the desired region of best accuracy. Here, we generalize the linear recipes to non-linear ones. We experiment with a non-linear, non-parametric, discriminative PAV solution, as well as parametric, generative, maximum-likelihood solutions that use Gaussian, Student's T and normal-inverse-Gaussian score distributions. Experiments on NIST SRE'12 scores suggest that the non-linear methods provide wider ranges of optimal accuracy and can be trained without having to resort to objective function tailoring.

Predicting an individual's risk of experiencing a future clinical outcome is a statistical task with important consequences for both practicing clinicians and public health experts. Modern observational databases such as electronic health records (EHRs) provide an alternative to the longitudinal cohort studies traditionally used to construct risk models, bringing with them both opportunities and challenges. Large sample sizes and detailed covariate histories enable the use of sophisticated machine learning techniques to uncover complex associations and interactions, but observational databases are often ``messy,'' with high levels of missing data and incomplete patient follow-up. In this paper, we propose an adaptation of the well-known Naive Bayes (NB) machine learning approach for classification to time-to-event outcomes subject to censoring. We compare the predictive performance of our method to the Cox proportional hazards model which is commonly used for risk prediction in healthcare populations, and illustrate its application to prediction of cardiovascular risk using an EHR dataset from a large Midwest integrated healthcare system.

Models for predicting the risk of cardiovascular events based on individual patient characteristics are important tools for managing patient care. Most current and commonly used risk prediction models have been built from carefully selected epidemiological cohorts. However, the homogeneity and limited size of such cohorts restricts the predictive power and generalizability of these risk models to other populations. Electronic health data (EHD) from large health care systems provide access to data on large, heterogeneous, and contemporaneous patient populations. The unique features and challenges of EHD, including missing risk factor information, non-linear relationships between risk factors and cardiovascular event outcomes, and differing effects from different patient subgroups, demand novel machine learning approaches to risk model development. In this paper, we present a machine learning approach based on Bayesian networks trained on EHD to predict the probability of having a cardiovascular event within five years. In such data, event status may be unknown for some individuals as the event time is right-censored due to disenrollment and incomplete follow-up. Since many traditional data mining methods are not well-suited for such data, we describe how to modify both modelling and assessment techniques to account for censored observation times. We show that our approach can lead to better predictive performance than the Cox proportional hazards model (i.e., a regression-based approach commonly used for censored, time-to-event data) or a Bayesian network with {\em{ad hoc}} approaches to right-censoring. Our techniques are motivated by and illustrated on data from a large U.S. Midwestern health care system.