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Graphical models, regression graphs, and recursive linear regression in a unified way

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Abstract

This versatile topic goes back to the inventions of Gauss, Markov, and Gibbs, whose ideas are incorporated in graphical models and regression graphs. Later, the geneticist S. Wright (1923–1934) and the philosopher and computer scientist J. Pearl (1986–1987) developed the tools, but their notation is too complicated to formulate the mathematical background. Here we mainly follow the up-to-date discussion of statisticians S. Lauritzen and N. Wermuth, and try to juxtapose the directed–undirected and discrete–continuous cases.

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Abbreviations

An(A):

ancestral set of the vertex-set A, which is the smallest possible vertex-set (including A) containing all vertices from where a directed path emanates to vertices of A in a directed graph

ant(i):

anteriors of the vertex i in a directed graphs (non-descendants except its parents)

bd(i):

boundary of the vertex i (its neighbors in the undirected, and its parents in the directed case)

BN:

Bayesian Network

CG:

Conditional Gaussian

cl(i):

closure of the vertex i (it and its biundary)

DAG:

Directed Acyclic Graph

DF:

Directed Factorization Property

DG:

Directed Global Markov Property

DL:

Directed Local Markov Property

DP:

Directed Pairwise Markov Property

EDHS:

Egypt Demographic and Health Survey

iid:

independent identically distributed

IPS:

Iterative Proportional Scaling

JT:

Junction Tree

MCS:

Maximal Cardinality Search

ML:

Maximum Likelihood

MRF:

Markov Random Field

par(i):

parents of the vertex i (from where directed edge shows to it) in a directed graph

pdf:

Probability Density Function

pmf:

Probability Mass Function

RCF:

Recursive Casual Factorization

RIP:

Running Intersection Property

rv:

random variable

RZP:

Reducible Zero Pattern

SEM:

Structural Equation Modeling

UF:

Undirected Factorization Property

UG:

Undirected Global Markov Property

UL:

Undirected Local Markov Property

UP:

Undirected Pairwise Markov Property

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Authors and Affiliations

  1. Institute of Mathematics, Budapest University of Technology and Economics, H-1111, Budapest, Muegyetem rkp. 3, Hungary

    Marianna Bolla, Fatma Abdelkhalek & Máté Baranyi

Authors
  1. Marianna Bolla
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  2. Fatma Abdelkhalek
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  3. Máté Baranyi
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Correspondence to Marianna Bolla.

Additional information

Dedicated to the memory of András Krámli

Communicated by Gy. Pap

Acknowledgment.

The first author is indebted to NannyWermuth for her valuable explanations.

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Bolla, M., Abdelkhalek, F. & Baranyi, M. Graphical models, regression graphs, and recursive linear regression in a unified way. ActaSci.Math. 85, 9–57 (2019). https://doi.org/10.14232/actasm-018-331-4

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