Representable Models Over N={1,2,3,4}

This page summarizes results on probabilistic representability and contains several lists of representable models over N={1,2,3,4} with respect to different distribution frameworks. A description of the data file format is at the end of the page.

Gaussian distribution framework

Results:

53 types of regularly Gaussian representable models
80 types of (generally) Gaussian representable models
27 types that are generally but not regularly representable

Literature:

F. Matus (2004): Conditional independences in Gaussian vectors and rings of polynomials. Springer-Proceedings of Workshop ``Conditionals, Information, and Inference'' Hagen, Germany, May 13-15, 2002.
R. Lnenicka (2005): On Gaussian Conditional Independence Structures. Technical report - DAR project. UTIA AV CR, Prague.
P. Simecek (2006): Gaussian Representation of Independence Models over Four Random Variables. COMPSTAT 2006, Rome.

Yest - R package for fitting regular Gaussian models:

version 0.4 [ZIP] [TAR.GZ]
version 0.3 [ZIP] [TAR.GZ]
version 0.2 [ZIP] [TAR.GZ]
version 0.1 [ZIP] [TAR.GZ]

Discrete distribution framework

Results:

18478 models of 1098 types representable by a (general) discrete distribution (generated by 13 types by the intersection)
356 types of discretely representable models meeting graphoid and Spohn property (generated by 23 types by the intersection)
remaining 57 types undecided with respect to positive discrete representability

Literature:

F. Matus and M. Studeny (1995): Conditional independences among four random variables I. Combinatorics, Probability & Computing 4.
F. Matus (1995): Conditional independences among four random variables II. Combinatorics, Probability & Computing 4.
F. Matus (1999): Conditional independences among four random variables III. Combinatorics, Probability & Computing 8.
M. Studeny and P. Bocek (1994): CI-models arising among 4 random variables. In Proceedings of the 3rd workshop WUPES, Trest, Czech Republic.
F. Matus (1997): Conditional independence structures examined via minors. The Annals of Mathematics and Artificial Intelligence 21
P. Simecek (2006): A Short Note on Discrete Representability of Independence Models. PGM workshop, Prague.

Binary distribution framework

Results:

Independence models corresponding to randomly generated positive binary distributions

Utilities for manipulation of data files

All sources contain no comments and they are quite messy (sorry). Originally, they were written only for my own purposes. The first four source codes have been written for GNU Pascal, the last one should be compiled in Borland Pascal.

int file1 [file2]

Finds all types that can be derived from types in file1 by the operation of intersection.
If file2 is input then saves resulting this into file2. Otherwise, only report number of types.
Win32 binaries, Cygwin DLL

rot file1 [file2]

Find all independence models for types given in file1.
If file2 is input then saves resulting this into file2. Otherwise, only report number of models.
Win32 binaries, Cygwin DLL

dif file1 file2 file3

Substract list of types in file2 from file1 and report the result to file3.
Win32 binaries, Cygwin DLL

lst file1

Return a list of models in file1.
Win32 binaries, Cygwin DLL

vis file1

Visualise models in file1 using Lnenicka's diagrams.
Win32 binaries

Data file format

Each model is coded by 24 bytes, i.e. the presence of each independence triple is coded by 1 byte (Yes, I know, it is wasting of disk space :-). If the triple (a,b|C) is present, then the corresponding byte is 1, otherwise it equals 0. The order of triples is as follows

(1,2|0)
(2,3|0)
(3,4|0)
(1,4|0)
(1,3|0)
(2,4|0)

(1,2|3)
(1,2|4)
(2,3|1)
(2,3|4)
(3,4|2)
(3,4|1)

(1,4|2)
(1,4|3)
(2,4|1)
(2,4|3)
(1,3|2)
(1,3|4)

(1,2|34)
(2,3|14)
(3,4|12)
(1,4|23)
(1,3|24)
(2,4|13)