The Annals of Mathematical Statistics

Generalized Iterative Scaling for Log-Linear Models

J. N. Darroch and D. Ratcliff

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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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Ann. Math. Statist., Volume 43, Number 5 (1972), 1470-1480.

First available in Project Euclid: 27 April 2007

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Darroch, J. N.; Ratcliff, D. Generalized Iterative Scaling for Log-Linear Models. Ann. Math. Statist. 43 (1972), no. 5, 1470--1480. doi:10.1214/aoms/1177692379.

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