TY - JOUR

T1 - Self-adhesivity in lattices of abstract conditional independence models

AU - Boege, Tobias

AU - Bolt, Janneke H.

AU - Studený, Milan

N1 - Publisher Copyright:
© 2024 The Author(s)

PY - 2025/1/30

Y1 - 2025/1/30

N2 - We introduce an algebraic concept of the frame for abstract conditional independence (CI) models, together with basic operations with respect to which such a frame should be closed: copying and marginalization. Three standard examples of such frames are (discrete) probabilistic CI structures, semi-graphoids and structural semi-graphoids. We concentrate on those frames which are closed under the operation of set-theoretical intersection because, for these, the respective families of CI models are lattices. This allows one to apply the results from lattice theory and formal concept analysis to describe such families in terms of implications among CI statements. The central concept of this paper is that of self-adhesivity defined in algebraic terms, which is a combinatorial reflection of the self-adhesivity concept studied earlier in context of polymatroids and information theory. The generalization also leads to a self-adhesivity operator defined on the meta-level of CI frames. We answer some of the questions related to this approach and raise other open questions. The core of the paper is in computations. The combinatorial approach to computation might overcome some memory and space limitation of software packages based on polyhedral geometry, in particular, if SAT solvers are utilized. We characterize some basic CI families over 4 variables in terms of canonical implications among CI statements. We apply our method in information-theoretical context to the task of entropic region demarcation over 5 variables.

AB - We introduce an algebraic concept of the frame for abstract conditional independence (CI) models, together with basic operations with respect to which such a frame should be closed: copying and marginalization. Three standard examples of such frames are (discrete) probabilistic CI structures, semi-graphoids and structural semi-graphoids. We concentrate on those frames which are closed under the operation of set-theoretical intersection because, for these, the respective families of CI models are lattices. This allows one to apply the results from lattice theory and formal concept analysis to describe such families in terms of implications among CI statements. The central concept of this paper is that of self-adhesivity defined in algebraic terms, which is a combinatorial reflection of the self-adhesivity concept studied earlier in context of polymatroids and information theory. The generalization also leads to a self-adhesivity operator defined on the meta-level of CI frames. We answer some of the questions related to this approach and raise other open questions. The core of the paper is in computations. The combinatorial approach to computation might overcome some memory and space limitation of software packages based on polyhedral geometry, in particular, if SAT solvers are utilized. We characterize some basic CI families over 4 variables in terms of canonical implications among CI statements. We apply our method in information-theoretical context to the task of entropic region demarcation over 5 variables.

KW - Boolean satisfiability

KW - Conditional independence

KW - Lattice (order theory)

KW - Pseudo-closed element

KW - Self-adhesivity

KW - Semi-graphoid

UR - http://www.scopus.com/inward/record.url?scp=85207029715&partnerID=8YFLogxK

U2 - 10.1016/j.dam.2024.10.006

DO - 10.1016/j.dam.2024.10.006

M3 - Article

AN - SCOPUS:85207029715

SN - 0166-218X

VL - 361

SP - 196

EP - 225

JO - Discrete Applied Mathematics

JF - Discrete Applied Mathematics

ER -