Iterative proportional scaling via decomposable submodels for contingency tables

We propose iterative proportional scaling (IPS) via decomposable submodels for maximizing the likelihood function of a hierarchical model for contingency tables. In ordinary IPS the proportional scaling is performed by cycling through the members of the generating class of a hierarchical model. We propose the adjustment of more marginals at each step. This is accomplished by expressing the generating class as a union of decomposable submodels and cycling through the decomposable models. We prove the convergence of our proposed procedure, if the amount of scaling is adjusted properly at each step. We also analyze the proposed algorithms around the maximum likelihood estimate (MLE) in detail. The faster convergence of our proposed procedure is illustrated by numerical examples.

[1]  Hisayuki Hara,et al.  Improving on the maximum likelihood estimators of the means in Poisson decomposable graphical models , 2007 .

[2]  S. Fienberg.  An Iterative Procedure for Estimation in Contingency Tables , 1970 .

[3]  Radim Jirousek,et al.  Solution of the marginal problem and decomposable distributions , 1991, Kybernetika (Praha).

[4]  R. Jirousek,et al.  On the effective implementation of the iterative proportional fitting procedure , 1995 .

[5]  Francesco M. Malvestuto,et al.  An implementation of the iterative proportional fitting procedure by propagation trees , 2001 .

[6]  P. Sreenivasa Kumar,et al.  Minimal Vertex Separators of Chordal Graphs , 1998, Discret. Appl. Math..

[7]  I. Csiszár.  $I$-Divergence Geometry of Probability Distributions and Minimization Problems , 1975 .

[8]  Jue Xue,et al.  Edge-maximal triangulated subgraphs and heuristics for the maximum clique problem , 1994, Networks.

[9]  Pinar Heggernes,et al.  A Vertex Incremental Approach for Dynamically Maintaining Chordal Graphs , 2003, ISAAC.

[10]  S. Kullback.  Probability Densities with Given Marginals , 1968 .

[11]  W. Deming,et al.  On a Least Squares Adjustment of a Sampled Frequency Table When the Expected Marginal Totals are Known , 1940 .

[12]  L. Rüschendorf.  Convergence of the iterative proportional fitting procedure , 1995 .

[13]  Dee Denteneer,et al.  A fast algorithm for iterative proportional fitting in log-linear models , 1985 .

[14]  Robert E. Tarjan,et al.  Simple Linear-Time Algorithms to Test Chordality of Graphs, Test Acyclicity of Hypergraphs, and Selectively Reduce Acyclic Hypergraphs , 1984, SIAM J. Comput..

[15]  Akimichi Takemura,et al.  Bayes admissible estimation of the means in Poisson decomposable graphical models , 2009 .

[16]  J. Darroch,et al.  Generalized Iterative Scaling for Log-Linear Models , 1972 .

[17]  Blair J R S,et al.  Introduction to Chordal Graphs and Clique Trees, in Graph Theory and Sparse Matrix Computation , 1997 .

[18]  Michael I. Jordan.  Graphical Models , 1998 .

[19]  Thomas M. Cover,et al.  Elements of Information Theory , 2005 .

[20]  Douglas R. Shier,et al.  Maximal chordal subgraphs , 1988, Discret. Appl. Math..

[21]  F. M. Malvestuto.  Computing the maximum-entropy extension of given discrete probability distributions , 1989 .

[22]  I. Csiszár.  A geometric interpretation of Darroch and Ratcliff's generalized iterative scaling , 1989 .

[23]  B. Peyton,et al.  An Introduction to Chordal Graphs and Clique Trees , 1993 .

[24]  S. Kullback,et al.  Contingency tables with given marginals. , 1968, Biometrika.

[25]  P. Sreenivasa Kumar,et al.  Clique tree generalization and new subclasses of chordal graphs , 2002, Discret. Appl. Math..