M. Studeny, J. Vomlel, R. Hemmecke:
Geometric view on learning Bayesian network structures.
International Journal of Approximate Reasoning
51 (2010), n. 5, pp. 578-586.
We recall the basic idea of an algebraic approach to learning
Bayesian network (BN) structure, namely to represent every BN
structure by a certain (uniquely determined) vector, called
The main result of the paper is that the set of standard imsets is
the set of vertices (= extreme points) of a certain polytope.
Motivated by the geometric view, we introduce the concept of the
geometric neighborhood for standard imsets, and,
consequently, for BN structures. Then we show it always includes
the inclusion neighborhood, which was introduced earlier
in connection with the greedy equivalence search (GES) algorithm.
The third result is that the global optimum of an affine function
over the polytope coincides with the local optimum relative to the
To illustrate the new concept by an example, we describe the
geometric neighborhood in the case of three variables and show it
differs from the inclusion neighborhood. This leads to a simple
example of the failure of the GES algorithm if data are not ``generated"
from a perfectly Markovian distribution. The point is that one can
avoid this failure if the search technique is based on the geometric
neighborhood instead. We also found out what is the geomteric neigborhood
in the case of four and five variables.
- AMS classification 68T30, 62H05
- learning Bayesian networks
- standard imset
- inclusion neighborhood
- geometric neighborhood
- GES algorithm
pdf version of the published paper (402kB) is already open-access available.
The paper comes from (and extends substantially) the observations in the conference paper:
Thus, like the original conference paper, the paper refers to results from the following publications:
- M. Studeny (2005). Probabilistic Conditional Independence
Structures. London: Springer-Verlag.
- D. M. Chickering:
Optimal structure indentification with greedy search.
Journal of Machine Learning Research 3 (2002), pp. 507-554.