A note on the IPF algorithm when the marginal problem is unsolvable

Claudio Asci; Mauro Piccioni

Kybernetika (2003)

  • Volume: 39, Issue: 6, page [731]-737
  • ISSN: 0023-5954

Abstract

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In this paper we analyze the asymptotic behavior of the IPF algorithm for the problem of finding a 2x2x2 contingency table whose pair marginals are all equal to a specified 2x2 table, depending on a parameter. When this parameter lies below a certain threshold the marginal problem has no solution. We show that in this case the IPF has a “period three limit cycle” attracting all positive initial tables, and a bifurcation occur when the parameter crosses the threshold.

How to cite

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Asci, Claudio, and Piccioni, Mauro. "A note on the IPF algorithm when the marginal problem is unsolvable." Kybernetika 39.6 (2003): [731]-737. <http://eudml.org/doc/33677>.

@article{Asci2003,
abstract = {In this paper we analyze the asymptotic behavior of the IPF algorithm for the problem of finding a 2x2x2 contingency table whose pair marginals are all equal to a specified 2x2 table, depending on a parameter. When this parameter lies below a certain threshold the marginal problem has no solution. We show that in this case the IPF has a “period three limit cycle” attracting all positive initial tables, and a bifurcation occur when the parameter crosses the threshold.},
author = {Asci, Claudio, Piccioni, Mauro},
journal = {Kybernetika},
keywords = {contingency tables; hierarchical models; partial maximization algorithms; contingency tables; hierarchical model; partial maximization algorithm},
language = {eng},
number = {6},
pages = {[731]-737},
publisher = {Institute of Information Theory and Automation AS CR},
title = {A note on the IPF algorithm when the marginal problem is unsolvable},
url = {http://eudml.org/doc/33677},
volume = {39},
year = {2003},
}

TY - JOUR
AU - Asci, Claudio
AU - Piccioni, Mauro
TI - A note on the IPF algorithm when the marginal problem is unsolvable
JO - Kybernetika
PY - 2003
PB - Institute of Information Theory and Automation AS CR
VL - 39
IS - 6
SP - [731]
EP - 737
AB - In this paper we analyze the asymptotic behavior of the IPF algorithm for the problem of finding a 2x2x2 contingency table whose pair marginals are all equal to a specified 2x2 table, depending on a parameter. When this parameter lies below a certain threshold the marginal problem has no solution. We show that in this case the IPF has a “period three limit cycle” attracting all positive initial tables, and a bifurcation occur when the parameter crosses the threshold.
LA - eng
KW - contingency tables; hierarchical models; partial maximization algorithms; contingency tables; hierarchical model; partial maximization algorithm
UR - http://eudml.org/doc/33677
ER -

References

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  1. Csiszár I., 10.1214/aop/1176996454, Ann. Probab. 3 (1975), 146–158 (1975) MR0365798DOI10.1214/aop/1176996454
  2. Deming W. E., Stephan F. F., 10.1214/aoms/1177731829, Ann. Math. Statist. 11 (1940), 427–444 (1940) MR0003527DOI10.1214/aoms/1177731829
  3. Haberman S. J., The analysis of frequency data, The University of Chicago Press, Chicago 1974 Zbl0325.62017MR0408098
  4. Jensen S. T., Johansen, S., Lauritzen S. L., Globally convergent algorithms for maximizing a likelihood function, Biometrika 78 (1991), 867–877 (1991) Zbl0752.62031MR1147024
  5. Jiroušek R., Solution of the marginal problem and decomposable distributions, Kybernetika 27 (1991), 403–412 (1991) Zbl0752.60009MR1132602
  6. Lauritzen S. L., Graphical Models, Clarendon Press, Oxford 1996 Zbl1055.62126MR1419991

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