Generalized Iterative Scaling for Log-Linear Models

@article{citeulike:3019659, 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.}, author = {Darroch, J. N. and Ratcliff, D. }, citeulike-article-id = {3019659}, comment = {should check, but I think this happens to be the original paper to cite for IPF [IPS].}, doi = {http://dx.doi.org/10.2307/2240069}, journal = {The Annals of Mathematical Statistics}, keywords = {mrfs, tribes}, number = {5}, pages = {1470--1480}, posted-at = {2008-07-18 22:04:56}, priority = {2}, publisher = {Institute of Mathematical Statistics}, title = {Generalized Iterative Scaling for Log-Linear Models}, url = {http://dx.doi.org/10.2307/2240069}, volume = {43}, year = {1972} }

TY - JOUR ID - citeulike:3019659 L3 - citeulike-article-id:3019659 TI - Generalized Iterative Scaling for Log-Linear Models JF - The Annals of Mathematical Statistics VL - 43 IS - 5 SP - 1470 EP - 1480 PB - Institute of Mathematical Statistics N2 - 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. KW - mrfs KW - tribes N1 - should check, but I think this happens to be the original paper to cite for IPF [IPS]. AU - Darroch, JN AU - Ratcliff, D PY - 1972/// UR - http://dx.doi.org/10.2307/2240069 ER -