- (Darroch & Ratcliff, 1972) ⇒ John N. Darroch, and Douglas Ratcliff. (1972). “Generalized Iterative Scaling for Log-Linear Models.” In: The Annals of Mathematical Statistics, 43(5).
Subject Headings: Generalized Iterative Scaling Algorithm.
Say that a probability distribution (pi; i ∈ I) over a finite set I is in "product form" if (1) $p_i = π_iμ ∏^d_s=1 μ_s^b_si$ 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..
The theory of iterative scaling is generalized and, at the same time, the existing theory is formulated in a general setting and the proof of convergence is shortened. The application to maximum likelihood estimation is discussed, with special reference to truncated distributions, contingency tables and /^-independence.
|1972 GeneralizedIterativeScaling||John N. Darroch|
|Generalized Iterative Scaling for Log-Linear Models||The Annals of Mathematical Statistics||http://www.csri.toronto.edu/~roweis/csc412-2006/extras/gis.pdf||1972|