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Generalized Iterative Scaling for Log-Linear Models

J. N. Darroch and D. Ratcliff
The Annals of Mathematical Statistics
Vol. 43, No. 5 (Oct., 1972), pp. 1470-1480
Stable URL: http://www.jstor.org/stable/2240069
Page Count: 11
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Generalized Iterative Scaling for Log-Linear Models
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Abstract

Say that a probability distribution {pi; i ∈ 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 πi and {bsi} are given constants and where μ and {μs} are determined from the equations (2) ∑i ∈ I bsi pi = ks, s = 1, 2, ⋯, d; (3) ∑i ∈ I pi = 1. Probability distributions in product form arise from minimizing the discriminatory information ∑i ∈ I pi log pi/πi 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 bsi = 0, 1. In this paper the method is generalized to allow the bsi 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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