Optimal structure identification with greedy search
The Journal of Machine Learning Research
Learning Bayesian Networks
Bayesian network learning algorithms using structural restrictions
International Journal of Approximate Reasoning
Three counter-examples on semi-graphoids
Combinatorics, Probability and Computing
A reconstruction algorithm for the essential graph
International Journal of Approximate Reasoning
Learning locally minimax optimal Bayesian networks
International Journal of Approximate Reasoning
A geometric view on learning Bayesian network structures
International Journal of Approximate Reasoning
Efficient Algorithms for Conditional Independence Inference
The Journal of Machine Learning Research
Probabilistic Conditional Independence Structures
Probabilistic Conditional Independence Structures
Efficient Algorithms for Conditional Independence Inference
The Journal of Machine Learning Research
Characteristic imsets for learning Bayesian network structure
International Journal of Approximate Reasoning
International Journal of Approximate Reasoning
International Journal of Approximate Reasoning
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The basic idea of an algebraic approach to learning Bayesian network (BN) structures is to represent every BN structure by a certain uniquely determined vector, called the standard imset. In a recent paper [18], it was shown that the set S of standard imsets is the set of vertices (=extreme points) of a certain polytope P and natural geometric neighborhood for standard imsets, and, consequently, for BN structures, was introduced. The new geometric view led to a series of open mathematical questions. In this paper, we try to answer some of them. First, we introduce a class of necessary linear constraints on standard imsets and formulate a conjecture that these constraints characterize the polytope P. The conjecture has been confirmed in the case of (at most) 4 variables. Second, we confirm a former hypothesis by Raymond Hemmecke that the only lattice points (=vectors having integers as components) within P are standard imsets. Third, we give a partial analysis of the geometric neighborhood in the case of 4 variables.