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October, 1972 Generalized Iterative Scaling for Log-Linear Models
J. N. Darroch, D. Ratcliff
Ann. Math. Statist. 43(5): 1470-1480 (October, 1972). DOI: 10.1214/aoms/1177692379


Say that a probability distribution $\{p_i; i \in I\}$ over a finite set $I$ is in "product form" if (1) $p_i = \pi_i\mu \prod^d_{s=1} \mu_s^{b_si}$ where $\pi_i$ and $\{b_{si}\}$ are given constants and where $\mu$ and $\{\mu_s\}$ are determined from the equations (2) $\sum_{i \in I} b_{si} p_i = k_s, s = 1, 2, \cdots, d$; (3) $\sum_{i \in I} p_i = 1$. Probability distributions in product form arise from minimizing the discriminatory information $\sum_{i \in I} p_i \log p_i/\pi_i$ subject to (2) and (3) or from maximizing entropy or maximizing likelihood. The theory of the iterative scaling method of determining (1) subject to (2) and (3) has, until now, been limited to the case when $b_{si} = 0, 1$. In this paper the method is generalized to allow the $b_{si}$ to be any real numbers. This expands considerably the list of probability distributions in product form which it is possible to estimate by maximum likelihood.


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J. N. Darroch. D. Ratcliff. "Generalized Iterative Scaling for Log-Linear Models." Ann. Math. Statist. 43 (5) 1470 - 1480, October, 1972.


Published: October, 1972
First available in Project Euclid: 27 April 2007

zbMATH: 0251.62020
MathSciNet: MR345337
Digital Object Identifier: 10.1214/aoms/1177692379

Rights: Copyright © 1972 Institute of Mathematical Statistics

Vol.43 • No. 5 • October, 1972
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