Log-linear models are a family of probability distributions which capture a variety of relationships between variables, including context-specific independencies. There are a number of approaches for automatic learning of their independence structures from data, although to date, no efficient method exists for evaluating these approaches directly in terms of the structures of the models. The only known methods evaluate these approaches indirectly through the complete model produced, that includes not only the structure but also the model parameters, introducing potential distortions in the comparison. This work presents such a method, that is, a measure for the direct comparison of the independence structures of log-linear models, inspired by the Hamming distance comparison method used in undirected graphical models. The measure presented can be efficiently computed in terms of the number of variables of the domain, and is proven to be a distance metric.

Markov networks are extensively used to model complex sequential, spatial, and relational interactions in a wide range of fields. By learning the structure of independences of a domain, more accurate joint probability distributions can be obtained for inference tasks or, more directly, for interpreting the most significant relations among the variables. Recently, several researchers have investigated techniques for automatically learning the structure from data by obtaining the probabilistic maximum-a-posteriori structure given the available data. However, all the approximations proposed decompose the posterior of the whole structure into local sub-problems, by assuming that the posteriors of the Markov blankets of all the variables are mutually independent. In this work, we propose a scoring function for relaxing such assumption. The Blankets Joint Posterior score computes the joint posterior of structures as a joint distribution of the collection of its Markov blankets. Essentially, the whole posterior is obtained by computing the posterior of the blanket of each variable as a conditional distribution that takes into account information from other blankets in the network. We show in our experimental results that the proposed approximation can improve the sample complexity of state-of-the-art scores when learning complex networks, where the independence assumption between blanket variables is clearly incorrect.

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.

We present two algorithms for learning the structure of a Markov network from data: GSMN* and GSIMN. Both algorithms use statistical independence tests to infer the structure by successively constraining the set of structures consistent with the results of these tests. Until very recently, algorithms for structure learning were based on maximum likelihood estimation, which has been proved to be NP-hard for Markov networks due to the difficulty of estimating the parameters of the network, needed for the computation of the data likelihood. The independence-based approach does not require the computation of the likelihood, and thus both GSMN* and GSIMN can compute the structure efficiently (as shown in our experiments). GSMN* is an adaptation of the Grow-Shrink algorithm of Margaritis and Thrun for learning the structure of Bayesian networks. GSIMN extends GSMN* by additionally exploiting Pearls well-known properties of the conditional independence relation to infer novel independences from known ones, thus avoiding the performance of statistical tests to estimate them. To accomplish this efficiently GSIMN uses the Triangle theorem, also introduced in this work, which is a simplified version of the set of Markov axioms. Experimental comparisons on artificial and real-world data sets show GSIMN can yield significant savings with respect to GSMN*, while generating a Markov network with comparable or in some cases improved quality. We also compare GSIMN to a forward-chaining implementation, called GSIMN-FCH, that produces all possible conditional independences resulting from repeatedly applying Pearls theorems on the known conditional independence tests. The results of this comparison show that GSIMN, by the sole use of the Triangle theorem, is nearly optimal in terms of the set of independences tests that it infers.

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.