Compatibility of discrete conditional distributions with structural zeros

A general algorithm is provided for determining the compatibility among full conditionals of discrete random variables with structural zeros. The algorithm is scalable and it can be implemented in a fairly straightforward manner. A MATLAB program is included in the Appendix and therefore, it is now feasible to check the compatibility of multi-dimensional conditional distributions with constrained supports. Rather than the linear equations in the restricted domain of Arnold et al. (2002) [11] Tian et al. (2009) [16], the approach is odds-oriented and it is a discrete adaptation of the compatibility check of Besag (1994) [17]. The method naturally leads to the calculation of a compatible joint distribution or, in the absence of compatibility, a nearly compatible joint distribution. Besag's [5] factorization of a joint density in terms of conditional densities is used to justify the algorithm.Consecutive site Full conditionals Geometric average Incidence set Nearly compatible Odds Path

A simple algorithm for checking compatibility among discrete conditional distributions

A distribution is said to be conditionally specified when only its conditional distributions are known or available. The very first issue is always compatibility: does there exist a joint distribution capable of reproducing all of the conditional distributions? We review five methods-mostly for two or three variables-published since 2002, and we conclude that these methods are either mathematically too involved and/or are too difficult (and in many cases impossible) to generalize to a high dimension. The purpose of this paper is to propose a general algorithm that can efficiently verify compatibility in a straightforward fashion. Our method is intuitively simple and general enough to deal with any full-conditional specifications. Furthermore, we illustrate the phenomenon that two theoretically equivalent conditional models can be different in terms of compatibilities, or can result in different joint distributions. The implications of this phenomenon are also discussed.Connected site Consecutive site Full conditionals Support Odds Path

Canonical representation of conditionally specified multivariate discrete distributions

Most work on conditionally specified distributions has focused on approaches that operate on the probability space, and the constraints on the probability space often make the study of their properties challenging. We propose decomposing both the joint and conditional discrete distributions into characterizing sets of canonical interactions, and we prove that certain interactions of a joint distribution are shared with its conditional distributions. This invariance opens the door for checking the compatibility between conditional distributions involving the same set of variables. We formulate necessary and sufficient conditions for the existence and uniqueness of discrete conditional models, and we show how a joint distribution can be easily computed from the pool of interactions collected from the conditional distributions. Hence, the methods can be used to calculate the exact distribution of a Gibbs sampler. Furthermore, issues such as how near compatibility can be reconciled are also discussed. Using mixed parametrization, we show that the proposed approach is based on the canonical parameters, while the conventional approaches are based on the mean parameters. Our advantage is partly due to the invariance that holds only for the canonical parameters.primary, 62E10, 62E15 secondary, 62E17, 62H05 Canonical parameter Characterizing set of interactions Compatibility check Exponential family Near-compatible Pseudo-Gibbs sampler

A note on cuts for contingency tables

In this note, we propose a general method to find cuts for a contingency table. Useful cuts are, in many cases, statistics S-sufficient for the nuisance parameter and S-ancillary for the parameter of interest. In general, cuts facilitate a strong form of parameter separation known to be useful for conditional inference [E.L. Lehmann, Testing Statistical Hypotheses, 2nd ed., Springer, New York, 1997, pp. 546-548]. Cuts also achieve significant dimension reduction, hence, increase computational efficiency. This is particularly true for the inference about cross-tabulated data, usually with a large number of parameters. Depending on the parameter of interest, we propose a flexible transformation to reparameterize the discrete multivariate response distribution. Inference on cell probabilities or odds ratios will require different parameterizations. The reparameterized distribution is not sum-symmetric. Thus, the finding in this paper expands the results in Barndorff-Nielsen [O.E. Barndorff-Nielsen, Information and Exponential Families in Statistical Theory, John Wiley, New York, 1978, pp. 202-206].62B05 62H17 62E10 Dimension reduction Iterative proportional fitting algorithm Likelihood factorization Power series family Separation of inference S-ancillary

Conditionally specified continuous distributions

A distribution is conditionally specified when its model constraints are expressed conditionally. For example, Besag's (1974) spatial model was specified conditioned on the neighbouring states, and pseudolikelihood is intended to approximate the likelihood using conditional likelihoods. There are three issues of interest: existence, uniqueness and computation of a joint distribution. In the literature, most results and proofs are for discrete probabilities; here we exclusively study distributions with continuous state space. We examine all three issues using the dependence functions derived from decomposition of the conditional densities. We show that certain dependence functions of the joint density are shared with its conditional densities. Therefore, two conditional densities involving the same set of variables are compatible if their overlapping dependence functions are identical. We prove that the joint density is unique when the set of dependence functions is both compatible and complete. In addition, a joint density, apart from a constant, can be computed from the dependence functions in closed form. Since all of the results are expressed in terms of dependence functions, we consider our approach to be dependence-based, whereas methods in the literature are generally density-based. Applications of the dependence-based formulation are discussed. Copyright 2008, Oxford University Press.

Structural decompositions of multivariate distributions with applications in moment and cumulant

We provide lattice decompositions for multivariate distributions. The lattice decompositions reveal the structural relationship between the Lancaster/Bahadur model and the model of Streitberg (Ann. Statist. 18 (1990) 1878). For multivariate categorical data, the decompositions allows modeling strategy for marginal inference. The theory discussed in this paper illustrates the concept of reproducibility, which was discussed in Liang et al. (J. Roy. Statist. Soc. Ser. B 54 (1992) 3). For the purpose of delineating the relationship between the various types of decompositions of distributions, we develop a theory of polytypefication, the generality of which is exploited to prove results beyond interaction.Lattice decomposition Lancaster model Bahadur model Streitberg's interaction Cumulant Polytypefication

Gibbs ensembles for nearly compatible and incompatible conditional models

The Gibbs sampler has been used exclusively for compatible conditionals that converge to a unique invariant joint distribution. However, conditional models are not always compatible. In this paper, a Gibbs sampling-based approach-using the Gibbs ensemble-is proposed for searching for a joint distribution that deviates least from a prescribed set of conditional distributions. The algorithm can be easily scalable, such that it can handle large data sets of high dimensionality. Using simulated data, we show that the proposed approach provides joint distributions that are less discrepant from the incompatible conditionals than those obtained by other methods discussed in the literature. The ensemble approach is also applied to a data set relating to geno-polymorphism and response to chemotherapy for patients with metastatic colorectal cancer.Gibbs sampler Conditionally specified distribution Linear programming Ensemble method Odds ratio

Some equivalence results concerning multiplicative lattice decompositions of multivariate densities

This note provides some equivalence results across the partition lattice, the monotypic lattice, and the subset lattice, for decomposing a multivariate density.Lancaster model Bahadur model Partition lattice Combinatorics