On open questions in the geometric approach to structural learning Bayesian nets

0358907 - ÚTIA 2012 RIV US eng J - Journal Article
Studený, Milan - Vomlel, Jiří
On open questions in the geometric approach to structural learning Bayesian nets.
International Journal of Approximate Reasoning. Roč. 52, č. 5 (2011), s. 627-640. ISSN 0888-613X. E-ISSN 1873-4731.
[Workshop on Uncertainty Processing WUPES'09 /8./. Liblice, 19.09.2009-23.09.2009]
R&D Projects: GA MŠMT(CZ) 1M0572; GA ČR GA201/08/0539; GA ČR GEICC/08/E010
Grant - others:GA MŠk(CZ) 2C06019
Institutional research plan: CEZ:AV0Z10750506
Keywords : structural learning Bayesian nets * standard imset * polytope * geometric neighborhood * differential imset
Subject RIV: BA - General Mathematics
Impact factor: 1.948, year: 2011
http://library.utia.cas.cz/separaty/2011/MTR/studeny-0358907.pdf

The basic idea of an algebraic approach to learning a Bayesian network (BN) structure is to represent it by a certain uniquely determined vector, called the standard imset. In a recent paper, it was shown that the set of standard imsets is the set of vertices of a certain polytope 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. 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 within the polytope are standard imsets. Third, we give a partial analysis of the geometric neighborhood in the case of 4 variables.
Permanent Link: http://hdl.handle.net/11104/0196817