J. N. Darroch

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.

Journal Article Canonical and principal components of shape Get access JOHN N. DARROCH, JOHN N. DARROCH School of Mathematical Sciences, Flinders UniversityAdelaide, S.A., 5042, Australia Search for other works by this author on: Oxford Academic Google Scholar JAMES E. MOSIMANN JAMES E. MOSIMANN Division of Computer Research and Technology, National Institute of HealthBethesda, Maryland 20014, U.S.A. Search for other works by this author on: Oxford Academic Google Scholar Biometrika, Volume 72, Issue 2, August 1985, Pages 241–252, https://doi.org/10.1093/biomet/72.2.241 Published: 01 August 1985 Article history Received: 01 July 1983 Revision received: 01 December 1984 Published: 01 August 1985

We use a close connection between the theory of Markov fields and that of log-linear interaction models for contingency tables to define and investigate a new class of models for such tables, graphical models. These models are hierarchical models that can be represented by a simple, undirected graph on as many vertices as the dimension of the corresponding table. Further all these models can be given an interpretation in terms of conditional independence and the interpretation can be read directly off the graph in the form of a Markov property. The class of graphical models contains that of decomposable models and we give a simple criterion for decomposability of a given graphical model. To some extent we discuss estimation problems and give suggestions for further work.

The time to absorption from the set T of transient states of a Markov chain may be sufficiently long for the probability distribution over T to settle down in some sense to a “quasi-stationary” distribution. Various analogues of the stationary distribution of an irreducible chain are suggested and compared. The reverse process of an absorbing chain is found to be relevant.

Journal Article THE MULTIPLE-RECAPTURE CENSUS: I. ESTIMATION OF A CLOSED POPULATION Get access J. N. DARROCH J. N. DARROCH Department of Mathematics, University of Cape Town Search for other works by this author on: Oxford Academic Google Scholar Biometrika, Volume 45, Issue 3-4, December 1958, Pages 343–359, https://doi.org/10.1093/biomet/45.3-4.343 Published: 01 December 1958