Gaussian process (GP) predictors are an important component of many Bayesian approaches to machine learning. However, even a straightforward implementation of Gaussian process regression (GPR) requires O(n^2) space and O(n^3) time for a dataset of n examples. Several approximation methods have been proposed, but there is a lack of understanding of the relative merits of the different approximations, and in what situations they are most useful. We recommend assessing the quality of the predictions obtained as a function of the compute time taken, and comparing to standard baselines (e.g., Subset of Data and FITC). We empirically investigate four different approximation algorithms on four different prediction problems, and make our code available to encourage future comparisons.

We study the mixtures of factorizing probability distributions represented as visible marginal distributions in stochastic layered networks. We take the perspective of kernel transitions of distributions, which gives a unified picture of distributed representations arising from Deep Belief Networks (DBN) and other networks without lateral connections. We describe combinatorial and geometric properties of the set of kernels and products of kernels realizable by DBNs as the network parameters vary. We describe explicit classes of probability distributions, including exponential families, that can be learned by DBNs. We use these submodels to bound the maximal and the expected Kullback-Leibler approximation errors of DBNs from above depending on the number of hidden layers and units that they contain.

Regression, unlike classification, has lacked a comprehensive and effective approach to deal with cost-sensitive problems by the reuse (and not a re-training) of general regression models. In this paper, a wide variety of cost-sensitive problems in regression (such as bids, asymmetric losses and rejection rules) can be solved effectively by a lightweight but powerful approach, consisting of: (1) the conversion of any traditional one-parameter crisp regression model into a two-parameter soft regression model, seen as a normal conditional density estimator, by the use of newly-introduced enrichment methods; and (2) the reframing of an enriched soft regression model to new contexts by an instance-dependent optimisation of the expected loss derived from the conditional normal distribution.

Relational data representations have become an increasingly important topic due to the recent proliferation of network datasets (e.g., social, biological, information networks) and a corresponding increase in the application of Statistical Relational Learning (SRL) algorithms to these domains. In this article, we examine and categorize techniques for transforming graph-based relational data to improve SRL algorithms. In particular, appropriate transformations of the nodes, links, and/or features of the data can dramatically affect the capabilities and results of SRL algorithms. We introduce an intuitive taxonomy for data representation transformations in relational domains that incorporates link transformation and node transformation as symmetric representation tasks. More specifically, the transformation tasks for both nodes and links include (i) predicting their existence, (ii) predicting their label or type, (iii) estimating their weight or importance, and (iv) systematically constructing their relevant features. We motivate our taxonomy through detailed examples and use it to survey competing approaches for each of these tasks. We also discuss general conditions for transforming links, nodes, and features. Finally, we highlight challenges that remain to be addressed.

We propose the Bayesian bridge estimator for regularized regression and classification. Two key mixture representations for the Bayesian bridge model are developed: (1) a scale mixture of normals with respect to an alpha-stable random variable; and (2) a mixture of Bartlett--Fejer kernels (or triangle densities) with respect to a two-component mixture of gamma random variables. Both lead to MCMC methods for posterior simulation, and these methods turn out to have complementary domains of maximum efficiency. The first representation is a well known result due to West (1987), and is the better choice for collinear design matrices. The second representation is new, and is more efficient for orthogonal problems, largely because it avoids the need to deal with exponentially tilted stable random variables. It also provides insight into the multimodality of the joint posterior distribution, a feature of the bridge model that is notably absent under ridge or lasso-type priors. We prove a theorem that extends this representation to a wider class of densities representable as scale mixtures of betas, and provide an explicit inversion formula for the mixing distribution. The connections with slice sampling and scale mixtures of normals are explored. On the practical side, we find that the Bayesian bridge model outperforms its classical cousin in estimation and prediction across a variety of data sets, both simulated and real. We also show that the MCMC for fitting the bridge model exhibits excellent mixing properties, particularly for the global scale parameter. This makes for a favorable contrast with analogous MCMC algorithms for other sparse Bayesian models. All methods described in this paper are implemented in the R package BayesBridge. An extensive set of simulation results are provided in two supplemental files.

This paper introduces new methodology to triangulate dynamic Bayesian networks (DBNs) and dynamic graphical models (DGMs). While most methods to triangulate such networks use some form of constrained elimination scheme based on properties of the underlying directed graph, we find it useful to view triangulation and elimination using properties only of the resulting undirected graph, obtained after the moralization step. We first briefly introduce the Graphical model toolkit (GMTK) and its notion of dynamic graphical models, one that slightly extends the standard notion of a DBN. We next introduce the 'boundary algorithm', a method to find the best boundary between partitions in a dynamic model. We find that using this algorithm, the notions of forward- and backward-interface become moot - namely, the size and fill-in of the best forward- and backward- interface are identical. Moreover, we observe that finding a good partition boundary allows for constrained elimination orders (and therefore graph triangulations) that are not possible using standard slice-by-slice constrained eliminations. More interestingly, with certain boundaries it is possible to obtain constrained elimination schemes that lie outside the space of possible triangulations using only unconstrained elimination. Lastly, we report triangulation results on invented graphs, standard DBNs from the literature, novel DBNs used in speech recognition research systems, and also random graphs. Using a number of different triangulation quality measures (max clique size, state-space, etc.), we find that with our boundary algorithm the triangulation quality can dramatically improve.

We present a new statistical learning paradigm for Boltzmann machines based on a new inference principle we have proposed: the latent maximum entropy principle (LME). LME is different both from Jaynes maximum entropy principle and from standard maximum likelihood estimation.We demonstrate the LME principle BY deriving new algorithms for Boltzmann machine parameter estimation, and show how robust and fast new variant of the EM algorithm can be developed.Our experiments show that estimation based on LME generally yields better results than maximum likelihood estimation, particularly when inferring hidden units from small amounts of data.

Observed associations in a database may be due in whole or part to variations in unrecorded (latent) variables. Identifying such variables and their causal relationships with one another is a principal goal in many scientific and practical domains. Previous work shows that, given a partition of observed variables such that members of a class share only a single latent common cause, standard search algorithms for causal Bayes nets can infer structural relations between latent variables. We introduce an algorithm for discovering such partitions when they exist. Uniquely among available procedures, the algorithm is (asymptotically) correct under standard assumptions in causal Bayes net search algorithms, requires no prior knowledge of the number of latent variables, and does not depend on the mathematical form of the relationships among the latent variables. We evaluate the algorithm on a variety of simulated data sets.

Graphical models with bi-directed edges (<->) represent marginal independence: the absence of an edge between two vertices indicates that the corresponding variables are marginally independent. In this paper, we consider maximum likelihood estimation in the case of continuous variables with a Gaussian joint distribution, sometimes termed a covariance graph model. We present a new fitting algorithm which exploits standard regression techniques and establish its convergence properties. Moreover, we contrast our procedure to existing estimation methods.

There is no known efficient method for selecting k Gaussian features from n which achieve the lowest Bayesian classification error. We show an example of how greedy algorithms faced with this task are led to give results that are not optimal. This motivates us to propose a more robust approach. We present a Branch and Bound algorithm for finding a subset of k independent Gaussian features which minimizes the naive Bayesian classification error. Our algorithm uses additive monotonic distance measures to produce bounds for the Bayesian classification error in order to exclude many feature subsets from evaluation, while still returning an optimal solution. We test our method on synthetic data as well as data obtained from gene expression profiling.