Family Of Conditional Distributions Research Articles

Locally interacting diffusions as Markov random fields on path space

We consider a countable system of interacting (possibly non-Markovian) stochastic differential equations driven by independent Brownian motions and indexed by the vertices of a locally finite graph G=(V,E). The drift of the process at each vertex is influenced by the states of that vertex and its neighbors, and the diffusion coefficient depends on the state of only that vertex. Such processes arise in a variety of applications including statistical physics, neuroscience, engineering and math finance. Under general conditions on the coefficients, we show that if the initial conditions form a second-order Markov random field on d-dimensional Euclidean space, then at any positive time, the collection of histories of the processes at different vertices forms a second-order Markov random field on path space. We also establish a bijection between (second-order) Gibbs measures on (Rd)V (with finite second moments) and a set of (second-order) Gibbs measures on path space, corresponding respectively to the initial law and the law of the solution to the stochastic differential equation. As a corollary, we establish a Gibbs uniqueness property that shows that for infinite graphs the joint distribution of the paths is completely determined by the initial condition and the specifications, namely the family of conditional distributions on finite vertex sets given the configuration on the complement. Along the way, we establish approximation and projection results for Markov random fields on locally finite graphs that may be of independent interest.

Partial or complete characterization of a bivariate distribution based on one conditional distribution and partial specification of the mode function of the other conditional distribution

There are various ways to characterize a bivariate distribution based on given distributional information. For example, information on both families of conditional densities, i.e., of X given Y and of Y given X, is sufficient to characterize the bivariate distribution. On the other hand, knowledge of both regression functions, i.e., E(X|Y=y) and E(Y|X=x), will be inadequate to determine the joint distribution. In this paper, we discuss to what extent we can characterize (either partially or completely) a bivariate distribution on the basis of complete specification of one family of conditional distributions and partial or complete specification of the mode function of the other family of conditional distributions. This problem is related to an open question mentioned in the paper of Arnold, Castillo and Sarabia (2008) [3].

Locally dependent latent trait model and the dutch identity revisited

In this paper, we propose a class of locally dependent latent trait models for responses to psychological and educational tests. Typically, item response models treat an individual's multiple response to stimuli as conditional independent given the individual's latent trait. In this paper, instead the focus is on models based on a family of conditional distributions, or kernel, that describes joint multiple item responses as a function of student latent trait, not assuming conditional independence. Specifically, we examine a hybrid kernel which comprises a component for one-way item response functions and a component for conditional associations between items given latent traits. The class of models allows the extension of item response theory to cover some new and innovative applications in psychological and educational research. An EM algorithm for marginal maximum likelihood of the hybrid kernel model is proposed. Furthermore, we delineate the relationship of the class of locally dependent models and the log-linear model by revisiting the Dutch identity (Holland, 1990).

Exact and near compatibility of discrete conditional distributions

The problem of determining whether two families of conditional distributions are compatible in the finite discrete case may be viewed as a problem involving the solution of linear equations in restricted domains. A general solution is provided. Attention is also directed to the related concepts of near-compatibility and ε-compatibility of conditional distributions.

Asymptotic properties of conditional quantiles of the Cauchy distribution in Hilbert space

In this paper, we consider the conditional distributions that are induced by finite-dimensional projections of a σ-additive Cauchy measure defined in a real Hilbert space. In the case where the dimension of projections tends to infinity, we establish the almost sure convergence of “conforming” sequences of finite-dimensional conditional quantiles and prove the strong law of large numbers for the triangular array scheme applied to a family of conditional distributions.

On a Generalization of the Classical Allocation Problem

In this paper a number of known limit theorems for the classical allocation problem, including the local limit theorem, are generalized to a family of conditional distributions related to a random equiprobable allocation of particles into cells of two-dimensional tables.

Stochastic order in dichotomous item response models for fixed, adaptive, and multidimensional tests

Dichotomous IRT models can be viewed as families of stochastically ordered distributions of responses to test items. This paper explores several properties of such distributions. In particular, it is examined under what conditions stochastic order in families of conditional distributions is transferred to their inverse distributions, from two families of related distributions to a third family, or from multivariate conditional distributions to a marginal distribution. The main results are formulated as a series of theorems and corollaries which apply to dichotomous IRT models. One part of the results holds for unidimensional models with fixed item parameters. The other part holds for models with random item parameters as used, for example, in adaptive testing or for tests with multidimensional abilities.

Characterizations involving conditional specification

Conditional specification models are reviewed. The imposition of additional constraints permits characterization of certain classical bivariate models within the conditionally specified families. Multivariate extensions are described. In addition, conditional survival models and bivariate models characterized by just one family of conditional distributions are discussed.

On the completeness of a family of conditional distributions

In this note, we establish the statistical completeness of a family of conditional distributions of a statistic given another statistic where the first statistic is complete and the second statistic is sufficient for some parametric family of distributions. This resulting method enables us to determine easily the completeness of many families of distributions some of which are well known, and some not so well known. We illustrate our method with a number of examples.

Hidden Markov chains, the forward-backward algorithm, and initial statistics

The objects listed in the title have proven to be useful and practical modeling tools in continuous speech recognition work and elsewhere. Nevertheless, there are natural and simple situations in which the forward-backward algorithm will be inadequate for its intended purpose of finding useful maximum likelihood estimates of the parameters of the distribution of a probabilistic function of a Markov chain (a "hidden Markov model" or "Markov source model"). We observe some difficulties that arise in the case of common (e.g., Gaussian) families of conditional distributions for the observables. These difficulties are due not to the algorithm itself, but to modeling assumptions which introduce singularities into the likelihood function. We also comment on the fact that the parameters of a hidden Markov model cannot, in general, be determined, even if the distribution of the observables is completely known. We close with remarks about some effects of these modeling and estimating difficulties on practical speech recognition, and about the role of initial statistics.

Some empirical Bayes techniques in point estimation

In point estimation with a squared-error loss function the Bayes estimator is the posterior mean. In the empirical Bayes approach we must construct a consistent sequence of estimators for this posterior mean using past experience. This construction is done here for four general families of conditional distributions which include as special cases: the Poisson, the gamma, the normal and the uniform.