1. On the Number of Samples Needed to Learn the Correct Structure of a Bayesian Network
by Or Zuk, Shiri Margel and Eytan Domany
Bayesian Networks (BNs) are useful tools giving a natural and compact representation of joint probability distributions. In many applications one needs to learn a Bayesian Network (BN) from data. In this context, it is important to understand the number of samples needed in order to guarantee a successful learning. Previous works have studied BNs sample complexity, yet they mainly focused on the requirement that the learned distribution will be close to the original distribution which generated the data. In this work, we study a different aspect of the learning task, namely the number of samples needed in order to learn the correct structure of the network. We give both asymptotic results (lower and upper-bounds) on the probability of learning a wrong structure, valid in the large sample limit, and experimental results, demonstrating the learning behavior for feasible sample sizes.
2. Non-Minimal Triangulations for Mixed Stochastic/Deterministic Graphical Models
by Chris D. Bartels and Jeff A. Bilmes
We observe that certain large-clique graph triangulations can be useful for reducing computational requirements when making queries on mixed stochastic/deterministic graphical models. We demonstrate that many of these large clique triangulations are non-minimal and are thus unattainable via the elimination algorithm. We introduce ancestral pairs as the basis for novel triangulation heuristics and prove that no more than the addition of edges between ancestral pairs need be considered when searching for state space optimal triangulations in such graphs. Empirical results on random and real world graphs are given. We also present an algorithm and correctness proof for determining if a triangulation can be obtained via elimination, and we show that the decision problem associated with finding optimal state space triangulations in this mixed setting is NP-complete.
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