Geodesic
A note on the IPF algorithm when the marginal problem is unsolvable
Kybernetika, Tome 39 (2003) no. 6, pp. 731-737 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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.
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.
Classification : 62H17, 65C60
Keywords: contingency tables; hierarchical models; partial maximization algorithms 
@article{KYB_2003_39_6_a4,
     author = {Asci, Claudio and Piccioni, Mauro},
     title = {A note on the {IPF} algorithm when the marginal problem is unsolvable},
     journal = {Kybernetika},
     pages = {731--737},
     year = {2003},
     volume = {39},
     number = {6},
     mrnumber = {2035647},
     zbl = {1245.62070},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/KYB_2003_39_6_a4/}
}
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
SP  - 731
EP  - 737
VL  - 39
IS  - 6
UR  - http://geodesic.mathdoc.fr/item/KYB_2003_39_6_a4/
LA  - en
ID  - KYB_2003_39_6_a4
ER  - 
%0 Journal Article
%A Asci, Claudio
%A Piccioni, Mauro
%T A note on the IPF algorithm when the marginal problem is unsolvable
%J Kybernetika
%D 2003
%P 731-737
%V 39
%N 6
%U http://geodesic.mathdoc.fr/item/KYB_2003_39_6_a4/
%G en
%F KYB_2003_39_6_a4
Asci, Claudio; Piccioni, Mauro. A note on the IPF algorithm when the marginal problem is unsolvable. Kybernetika, Tome 39 (2003) no. 6, pp. 731-737. http://geodesic.mathdoc.fr/item/KYB_2003_39_6_a4/

[1] Csiszár I.: I-divergence geometry of probability distributions and minimization problems. Ann. Probab. 3 (1975), 146–158 | DOI | MR

[2] Deming W. E., Stephan F. F.: On a least square adjustment of a sampled frequency table when the expected marginal totals are known. Ann. Math. Statist. 11 (1940), 427–444 | DOI | MR

[3] Haberman S. J.: The analysis of frequency data. The University of Chicago Press, Chicago 1974 | MR | Zbl

[4] Jensen S. T., Johansen, S., Lauritzen S. L.: Globally convergent algorithms for maximizing a likelihood function. Biometrika 78 (1991), 867–877 | MR | Zbl

[5] Jiroušek R.: Solution of the marginal problem and decomposable distributions. Kybernetika 27 (1991), 403–412 | MR | Zbl

[6] Lauritzen S. L.: Graphical Models. Clarendon Press, Oxford 1996 | MR | Zbl