## The Annals of Mathematical Statistics

### Generalized Iterative Scaling for Log-Linear Models

#### Abstract

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.

#### Article information

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

Dates
First available in Project Euclid: 27 April 2007

https://projecteuclid.org/euclid.aoms/1177692379

Digital Object Identifier
doi:10.1214/aoms/1177692379

Mathematical Reviews number (MathSciNet)
MR345337

Zentralblatt MATH identifier
0251.62020

JSTOR