Article

# Computing Posterior Probabilities of Structural Features in Bayesian Networks

05/2012;
Source: arXiv

ABSTRACT We study the problem of learning Bayesian network structures from data.
Koivisto and Sood (2004) and Koivisto (2006) presented algorithms that can
compute the exact marginal posterior probability of a subnetwork, e.g., a
single edge, in O(n2n) time and the posterior probabilities for all n(n-1)
potential edges in O(n2n) total time, assuming that the number of parents per
node or the indegree is bounded by a constant. One main drawback of their
algorithms is the requirement of a special structure prior that is non uniform
and does not respect Markov equivalence. In this paper, we develop an algorithm
that can compute the exact posterior probability of a subnetwork in O(3n) time
and the posterior probabilities for all n(n-1) potential edges in O(n3n) total
time. Our algorithm also assumes a bounded indegree but allows general
structure priors. We demonstrate the applicability of the algorithm on several
data sets with up to 20 variables.

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##### Article: Being Bayesian About Network Structure: A Bayesian Approach to Structure Discovery in Bayesian Networks
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ABSTRACT: . In many multivariate domains, we are interested in analyzing the dependency structure of the underlying distribution, e.g., whether two variables are in direct interaction. We can represent dependency structures using Bayesian network models. To analyze a given data set, Bayesian model selection attempts to find the most likely (MAP) model, and uses its structure to answer these questions. However, when the amount of available data is modest, there might be many models that have non-negligible posterior. Thus, we want compute the Bayesian posterior of a feature, i.e., the total posterior probability of all models that contain it. In this paper, we propose a new approach for this task. We first show how to efficiently compute a sum over the exponential number of networks that are consistent with a fixed order over network variables. This allows us to compute, for a given order, both the marginal probability of the data and the posterior of a feature. We then use this result as the basis for an algorithm that approximates the Bayesian posterior of a feature. Our approach uses a Markov Chain Monte Carlo (MCMC) method, but over orders rather than over network structures. The space of orders is smaller and more regular than the space of structures, and has much a smoother posterior "landscape". We present empirical results on synthetic and reallife datasets that compare our approach to full model averaging (when possible), to MCMC over network structures, and to a non-Bayesian bootstrap approach.
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##### Article: Exact Bayesian Structure Discovery in Bayesian Networks.
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##### Article: Advances in exact Bayesian structure discovery in Bayesian networks
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ABSTRACT: We consider a Bayesian method for learning the Bayesian network structure from complete data. Recently, Koivisto and Sood (2004) presented an algorithm that for any single edge computes its marginal posterior probability in O(n 2^n) time, where n is the number of attributes; the number of parents per attribute is bounded by a constant. In this paper we show that the posterior probabilities for all the n (n - 1) potential edges can be computed in O(n 2^n) total time. This result is achieved by a forward-backward technique and fast Moebius transform algorithms, which are of independent interest. The resulting speedup by a factor of about n^2 allows us to experimentally study the statistical power of learning moderate-size networks. We report results from a simulation study that covers data sets with 20 to 10,000 records over 5 to 25 discrete attributes
06/2012;

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