Learning Markov networks with context-specific independences

Learning the Markov network structure from data is a problem that has received considerable attention in machine learning, and in many other application fields. This work focuses on a particular approach for this purpose called independence-based learning. Such approach guarantees the learning of the correct structure efficiently, whenever data is sufficient for representing the underlying distribution. However, an important issue of such approach is that the learned structures are encoded in an undirected graph. The problem with graphs is that they cannot encode some types of independence relations, such as the context-specific independences. They are a particular case of conditional independences that is true only for a certain assignment of its conditioning set, in contrast to conditional independences that must hold for all its assignments. In this work we present CSPC, an independence-based algorithm for learning structures that encode context-specific independences, and encoding them in a log-linear model, instead of a graph. The central idea of CSPC is combining the theoretical guarantees provided by the independence-based approach with the benefits of representing complex structures by using features in a log-linear model. We present experiments in a synthetic case, showing that CSPC is more accurate than the state-of-the-art IB algorithms when the underlying distribution contains CSIs.Comment: 8 pages, 6 figure

Sparse Nested Markov models with Log-linear Parameters

Hidden variables are ubiquitous in practical data analysis, and therefore modeling marginal densities and doing inference with the resulting models is an important problem in statistics, machine learning, and causal inference. Recently, a new type of graphical model, called the nested Markov model, was developed which captures equality constraints found in marginals of directed acyclic graph (DAG) models. Some of these constraints, such as the so called `Verma constraint', strictly generalize conditional independence. To make modeling and inference with nested Markov models practical, it is necessary to limit the number of parameters in the model, while still correctly capturing the constraints in the marginal of a DAG model. Placing such limits is similar in spirit to sparsity methods for undirected graphical models, and regression models. In this paper, we give a log-linear parameterization which allows sparse modeling with nested Markov models. We illustrate the advantages of this parameterization with a simulation study.Comment: Appears in Proceedings of the Twenty-Ninth Conference on Uncertainty in Artificial Intelligence (UAI2013

The Grow-Shrink strategy for learning Markov network structures constrained by context-specific independences

Markov networks are models for compactly representing complex probability distributions. They are composed by a structure and a set of numerical weights. The structure qualitatively describes independences in the distribution, which can be exploited to factorize the distribution into a set of compact functions. A key application for learning structures from data is to automatically discover knowledge. In practice, structure learning algorithms focused on "knowledge discovery" present a limitation: they use a coarse-grained representation of the structure. As a result, this representation cannot describe context-specific independences. Very recently, an algorithm called CSPC was designed to overcome this limitation, but it has a high computational complexity. This work tries to mitigate this downside presenting CSGS, an algorithm that uses the Grow-Shrink strategy for reducing unnecessary computations. On an empirical evaluation, the structures learned by CSGS achieve competitive accuracies and lower computational complexity with respect to those obtained by CSPC.Comment: 12 pages, and 8 figures. This works was presented in IBERAMIA 201

Concepts and a case study for a flexible class of graphical Markov models

With graphical Markov models, one can investigate complex dependences, summarize some results of statistical analyses with graphs and use these graphs to understand implications of well-fitting models. The models have a rich history and form an area that has been intensively studied and developed in recent years. We give a brief review of the main concepts and describe in more detail a flexible subclass of models, called traceable regressions. These are sequences of joint response regressions for which regression graphs permit one to trace and thereby understand pathways of dependence. We use these methods to reanalyze and interpret data from a prospective study of child development, now known as the Mannheim Study of Children at Risk. The two related primary features concern cognitive and motor development, at the age of 4.5 and 8 years of a child. Deficits in these features form a sequence of joint responses. Several possible risks are assessed at birth of the child and when the child reached age 3 months and 2 years.Comment: 21 pages, 7 figures, 7 tables; invited, refereed chapter in a boo

The IBMAP approach for Markov networks structure learning

In this work we consider the problem of learning the structure of Markov networks from data. We present an approach for tackling this problem called IBMAP, together with an efficient instantiation of the approach: the IBMAP-HC algorithm, designed for avoiding important limitations of existing independence-based algorithms. These algorithms proceed by performing statistical independence tests on data, trusting completely the outcome of each test. In practice tests may be incorrect, resulting in potential cascading errors and the consequent reduction in the quality of the structures learned. IBMAP contemplates this uncertainty in the outcome of the tests through a probabilistic maximum-a-posteriori approach. The approach is instantiated in the IBMAP-HC algorithm, a structure selection strategy that performs a polynomial heuristic local search in the space of possible structures. We present an extensive empirical evaluation on synthetic and real data, showing that our algorithm outperforms significantly the current independence-based algorithms, in terms of data efficiency and quality of learned structures, with equivalent computational complexities. We also show the performance of IBMAP-HC in a real-world application of knowledge discovery: EDAs, which are evolutionary algorithms that use structure learning on each generation for modeling the distribution of populations. The experiments show that when IBMAP-HC is used to learn the structure, EDAs improve the convergence to the optimum

Bayesian optimization of the PC algorithm for learning Gaussian Bayesian networks

The PC algorithm is a popular method for learning the structure of Gaussian Bayesian networks. It carries out statistical tests to determine absent edges in the network. It is hence governed by two parameters: (i) The type of test, and (ii) its significance level. These parameters are usually set to values recommended by an expert. Nevertheless, such an approach can suffer from human bias, leading to suboptimal reconstruction results. In this paper we consider a more principled approach for choosing these parameters in an automatic way. For this we optimize a reconstruction score evaluated on a set of different Gaussian Bayesian networks. This objective is expensive to evaluate and lacks a closed-form expression, which means that Bayesian optimization (BO) is a natural choice. BO methods use a model to guide the search and are hence able to exploit smoothness properties of the objective surface. We show that the parameters found by a BO method outperform those found by a random search strategy and the expert recommendation. Importantly, we have found that an often overlooked statistical test provides the best over-all reconstruction results