Locally regularised two-stage learning algorithm for RBF network centre selection

Nonlinear models constructed from radial basis function (RBF) networks can easily be over-fitted due to the noise on the data. While information criteria, such as the final prediction error (FPE), can provide a trade-off between training error and network complexity, the tunable parameters that penalise a large size of network model are hard to determine and are usually network dependent. This article introduces a new locally regularised, two-stage stepwise construction algorithm for RBF networks. The main objective is to produce a parsomous network that generalises well over unseen data. This is achieved by utilising Bayesian learning within a two-stage stepwise construction procedure to penalise centres that are mainly interpreted by the noise.

Efficient Marginal Likelihood Computation for Gaussian Process Regression

In a Bayesian learning setting, the posterior distribution of a predictive model arises from a trade-off between its prior distribution and the conditional likelihood of observed data. Such distribution functions usually rely on additional hyperparameters which need to be tuned in order to achieve optimum predictive performance; this operation can be efficiently performed in an Empirical Bayes fashion by maximizing the posterior marginal likelihood of the observed data. Since the score function of this optimization problem is in general characterized by the presence of local optima, it is necessary to resort to global optimization strategies, which require a large number of function evaluations. Given that the evaluation is usually computationally intensive and badly scaled with respect to the dataset size, the maximum number of observations that can be treated simultaneously is quite limited. In this paper, we consider the case of hyperparameter tuning in Gaussian process regression. A straightforward implementation of the posterior log-likelihood for this model requires O(N^3) operations for every iteration of the optimization procedure, where N is the number of examples in the input dataset. We derive a novel set of identities that allow, after an initial overhead of O(N^3), the evaluation of the score function, as well as the Jacobian and Hessian matrices, in O(N) operations. We prove how the proposed identities, that follow from the eigendecomposition of the kernel matrix, yield a reduction of several orders of magnitude in the computation time for the hyperparameter optimization problem. Notably, the proposed solution provides computational advantages even with respect to state of the art approximations that rely on sparse kernel matrices.

Learning Bayesian networks from data: An information-theory based approach

This paper provides algorithms that use an information-theoretic analysis to learn Bayesian network structures from data. Based on our three-phase learning framework, we develop efficient algorithms that can effectively learn Bayesian networks, requiring only polynomial numbers of conditional independence (CI) tests in typical cases. We provide precise conditions that specify when these algorithms are guaranteed to be correct as well as empirical evidence (from real world applications and simulation tests) that demonstrates that these systems work efficiently and reliably in practice.

Min-BDeu and Max-BDeu Scores for Learning Bayesian Networks

This work presents two new score functions based on the Bayesian Dirichlet equivalent uniform (BDeu) score for learning Bayesian network structures. They consider the sensitivity of BDeu to varying parameters of the Dirichlet prior. The scores take on the most adversary and the most beneficial priors among those within a contamination set around the symmetric one. We build these scores in such way that they are decomposable and can be computed efficiently. Because of that, they can be integrated into any state-of-the-art structure learning method that explores the space of directed acyclic graphs and allows decomposable scores. Empirical results suggest that our scores outperform the standard BDeu score in terms of the likelihood of unseen data and in terms of edge discovery with respect to the true network, at least when the training sample size is small. We discuss the relation between these new scores and the accuracy of inferred models. Moreover, our new criteria can be used to identify the amount of data after which learning is saturated, that is, additional data are of little help to improve the resulting model.

Advances in Learning Bayesian Networks of Bounded Treewidth

This work presents novel algorithms for learning Bayesian networks of bounded treewidth. Both exact and approximate methods are developed. The exact method combines mixed integer linear programming formulations for structure learning and treewidth computation. The approximate method consists in sampling k-trees (maximal graphs of treewidth k), and subsequently selecting, exactly or approximately, the best structure whose moral graph is a subgraph of that k-tree. The approaches are empirically compared to each other and to state-of-the-art methods on a collection of public data sets with up to 100 variables.

Efficient Structure Learning of Bayesian Networks using Constraints

This paper addresses the problem of learning Bayesian network structures from data based on score functions that are decomposable. It describes properties that strongly reduce the time and memory costs of many known methods without losing global optimality guarantees. These properties are derived for different score criteria such as Minimum Description Length (or Bayesian Information Criterion), Akaike Information Criterion and Bayesian Dirichlet Criterion. Then a branch-and-bound algorithm is presented that integrates structural constraints with data in a way to guarantee global optimality. As an example, structural constraints are used to map the problem of structure learning in Dynamic Bayesian networks into a corresponding augmented Bayesian network. Finally, we show empirically the benefits of using the properties with state-of-the-art methods and with the new algorithm, which is able to handle larger data sets than before.

Improving parameter learning of Bayesian nets from incomplete data

This paper addresses the estimation of parameters of a Bayesian network from incomplete data. The task is usually tackled by running the Expectation-Maximization (EM) algorithm several times in order to obtain a high log-likelihood estimate. We argue that choosing the maximum log-likelihood estimate (as well as the maximum penalized log-likelihood and the maximum a posteriori estimate) has severe drawbacks, being affected both by overfitting and model uncertainty. Two ideas are discussed to overcome these issues: a maximum entropy approach and a Bayesian model averaging approach. Both ideas can be easily applied on top of EM, while the entropy idea can be also implemented in a more sophisticated way, through a dedicated non-linear solver. A vast set of experiments shows that these ideas produce significantly better estimates and inferences than the traditional and widely used maximum (penalized) log-likelihood and maximum a posteriori estimates. In particular, if EM is adopted as optimization engine, the model averaging approach is the best performing one; its performance is matched by the entropy approach when implemented using the non-linear solver. The results suggest that the applicability of these ideas is immediate (they are easy to implement and to integrate in currently available inference engines) and that they constitute a better way to learn Bayesian network parameters.

Learning Bayesian Networks with Thousands of Variables

We present a method for learning Bayesian networks from data sets containing thousands of variables without the need for structure constraints. Our approach is made of two parts. The first is a novel algorithm that effectively explores the space of possible parent sets of a node. It guides the exploration towards the most promising parent sets on the basis of an approximated score function that is computed in constant time. The second part is an improvement of an existing ordering-based algorithm for structure optimization. The new algorithm provably achieves a higher score compared to its original formulation. Our novel approach consistently outperforms the state of the art on very large data sets.

Learning Bounded Tree-Width Bayesian Networks via Sampling

Learning Bayesian networks with bounded tree-width has attracted much attention recently, because low tree-width allows exact inference to be performed efficiently. Some existing methods [12, 14] tackle the problem by using k-trees to learn the optimal Bayesian network with tree-width up to k. In this paper, we propose a sampling method to efficiently find representative k-trees by introducing an Informative score function to characterize the quality of a k-tree. The proposed algorithm can efficiently learn a Bayesian network with tree-width at most k. Experiment results indicate that our approach is comparable with exact methods, but is much more computationally efficient.

Learning Bayesian Networks with Bounded Tree-Width via Guided Search

Bounding the tree-width of a Bayesian network can reduce the chance of overfitting, and allows exact inference to be performed efficiently. Several existing algorithms tackle the problem of learning bounded tree-width Bayesian networks by learning from k-trees as super-structures, but they do not scale to large domains and/or large tree-width. We propose a guided search algorithm to find k-trees with maximum Informative scores, which is a measure of quality for the k-tree in yielding good Bayesian networks. The algorithm achieves close to optimal performance compared to exact solutions in small domains, and can discover better networks than existing approximate methods can in large domains. It also provides an optimal elimination order of variables that guarantees small complexity for later runs of exact inference. Comparisons with well-known approaches in terms of learning and inference accuracy illustrate its capabilities.

Efficient Learning of Bayesian Networks with Bounded Tree-width

Learning Bayesian networks with bounded tree-width has attracted much attention recently, because low tree-width allows exact inference to be performed efficiently. Some existing methods \cite{korhonen2exact, nie2014advances} tackle the problem by using $k$-trees to learn the optimal Bayesian network with tree-width up to $k$. Finding the best $k$-tree, however, is computationally intractable. In this paper, we propose a sampling method to efficiently find representative $k$-trees by introducing an informative score function to characterize the quality of a $k$-tree. To further improve the quality of the $k$-trees, we propose a probabilistic hill climbing approach that locally refines the sampled $k$-trees. The proposed algorithm can efficiently learn a quality Bayesian network with tree-width at most $k$. Experimental results demonstrate that our approach is more computationally efficient than the exact methods with comparable accuracy, and outperforms most existing approximate methods.

Learning Treewidth-Bounded Bayesian Networks with Thousands of Variables

We present a method for learning treewidth-bounded Bayesian networks from data sets containing thousands of variables. Bounding the treewidth of a Bayesian network greatly reduces the complexity of inferences. Yet, being a global property of the graph, it considerably increases the difficulty of the learning process. Our novel algorithm accomplishes this task, scaling both to large domains and to large treewidths. Our novel approach consistently outperforms the state of the art on experiments with up to thousands of variables.