Bayesian Networks for Variable Groups

Bayesian networks, and especially their structures, are powerful tools for representing conditional independencies and dependencies between random variables. In applications where related variables form a priori known groups, chosen to represent different “views” to or aspects of the same entities, one may be more interested in modeling dependencies between groups of variables rather than between individual variables. Motivated by this, we study prospects of representing relationships between variable groups using Bayesian network structures. We show that for dependency structures between groups to be expressible exactly, the data have to satisfy the so-called groupwise faithfulness assumption. We also show that one cannot learn causal relations between groups using only groupwise conditional independencies, but also variable-wise relations are needed. Additionally, we present algorithms for finding the groupwise dependency structures.

[1]  Terran Lane,et al.  Improving Bayesian Network Structure Search with Random Variable Aggregation Hierarchies , 2006, ECML.

[2]  Avi Pfeffer,et al.  Object-Oriented Bayesian Networks , 1997, UAI.

[3]  David Maxwell Chickering,et al.  Learning Bayesian Networks: The Combination of Knowledge and Statistical Data , 1994, Machine Learning.

[4]  J. Pearl.  Causality: Models, Reasoning and Inference , 2000 .

[5]  Tom Burr,et al.  Causation, Prediction, and Search , 2003, Technometrics.

[6]  Constantin F. Aliferis,et al.  Local Causal and Markov Blanket Induction for Causal Discovery and Feature Selection for Classification Part I: Algorithms and Empirical Evaluation , 2010, J. Mach. Learn. Res..

[7]  Gregory F. Cooper,et al.  A Bayesian method for the induction of probabilistic networks from data , 1992, Machine Learning.

[8]  James Cussens,et al.  Bayesian network learning with cutting planes , 2011, UAI.

[9]  Peter A. Flach,et al.  Hierarchical Bayesian Networks: A Probabilistic Reasoning Model for Structured Domains , 2002 .

[10]  David Maxwell Chickering,et al.  Optimal Structure Identification With Greedy Search , 2002, J. Mach. Learn. Res..

[11]  David Maxwell Chickering,et al.  Learning Bayesian Networks is NP-Complete , 2016, AISTATS.

[12]  Christopher Meek,et al.  Causal inference and causal explanation with background knowledge , 1995, UAI.

[13]  Patrik O. Hoyer,et al.  Estimating a Causal Order among Groups of Variables in Linear Models , 2012, ICANN.

[14]  Judea Pearl,et al.  Equivalence and Synthesis of Causal Models , 1990, UAI.

[15]  Nir Friedman,et al.  Learning Module Networks , 2002, J. Mach. Learn. Res..

[16]  Tomi Silander,et al.  A Simple Approach for Finding the Globally Optimal Bayesian Network Structure , 2006, UAI.

[17]  Tommi S. Jaakkola,et al.  Learning Bayesian Network Structure using LP Relaxations , 2010, AISTATS.