Markov network structure discovery using independence tests

We investigate efficient algorithms for learning the structure of a Markov network from data using the independence-based approach. Such algorithms conduct a series of conditional independence tests on data, successively restricting the set of possible structures until there is only a single structure consistent with the outcomes of the conditional independence tests executed (if possible). As Pearl has shown, the instances of the conditional independence relation in any domain are theoretically interdependent, made explicit in his well-known conditional independence axioms. The first couple of algorithms we discuss, GSMN and GSIMN, exploit Pearl\u27s independence axioms to reduce the number of tests required to learn a Markov network. This is useful in domains where independence tests are expensive, such as cases of very large data sets or distributed data. Subsequently, we explore how these axioms can be exploited to correct the outcome of unreliable statistical independence tests, such as in applications where little data is available. We show how the problem of incorrect tests can be mapped to inference in inconsistent knowledge bases, a problem studied extensively in the field of non-monotonic logic. We present an algorithm for inferring independence values based on a sub-class of non-monotonic logics: the argumentation framework. Our results show the advantage of using our approach in the learning of structures, with improvements in the accuracy of learned networks of up to 20%. As an alternative to logic-based interdependence among independence tests, we also explore probabilistic interdependence. Our algorithm, called PFMN, takes a Bayesian particle filtering approach, using a population of Markov network structures to maintain the posterior probability distribution over them given the outcomes of the tests performed. The result is an approximate algorithm (due to the use of particle filtering) that is useful in domains where independence tests are expensive

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

Efficient comparison of independence structures of log-linear models

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

Improving the Reliability of Causal Discovery from Small Data Sets using the Argumentation Framework

We address the problem of reliability of independence-based causal discovery algorithms that results from unreliable statistical independence tests. We model the problem as a knowledge base containing a set of independences that are related through the well-known Pearl\u27s axioms. Statistical tests on finite data sets may result in errors in these tests and inconsistencies in the knowledge base. Our approach uses an instance of the class of defeasible logics called argumentation, augmented with a preference function that is used to reason and possibly correct errors in these tests, thereby resolving the corresponding inconsistencies. This results in a more robust conditional independence test, called argumentative independence test. We evaluated our approach on data sets sampled from randomly generated causal models as well as real-world data sets. Our experiments show a clear advantage of argumentative over purely statistical tests, with improvements in accuracy of up to 17%, measured as the ratio of independence tests correct as evaluated on data. We also conducted experiments to measure the impact of these improvements on the problem of causal structure discovery. Comparisons of the networks output by the PC algorithm using argumentative tests versus using purely statistical ones show significant improvements of up to 15%

Efficient Markov Network Structure Discovery Using Independence Tests

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