Showing papers on "Conditional probability distribution published in 1988"

Random truncation models and Markov processes

Abstract: Random left truncation is modelled by the conditional distribution of the random variable $X$ of interest, given that it is larger than the truncating random variable $Y$; usually $X$ and $Y$ are assumed independent. The present paper is based on a simple reparametrization of the left truncation model as a three-state Markov process. The derivation of a nonparametric estimator is a distribution function under random truncation is then a special case of results on the statistical theory of counting processes by Aalen and Johansen. This framework also clarifies the status of the estimator as a nonparametric maximum likelihood estimator, and consistency, asymptotic normality and efficiency may be derived directly as special cases of Aalen and Johansen's general theorems and later work. Although we do not carry through these here, we note that the present framework also allows several generalizations: censoring may be incorporated; the independence hypothesis underlying the truncation models may be tested; ties (occurring when the distributions of $F$ and $G$ have discrete components) may be handled.

Bivariate Distributions with Exponential Conditionals

Abstract: It is frequently easier to visualize conditional distributions of experimental variables rather than joint distributions. In this article we consider the most general class of bivariate distributions such that both sets of conditional densities are exponential. The class proves to be remarkably simple to describe: The joint density must be proportional to exp(- λx - μy - νxy), where the constant of proportionality depends on the classical exponential integral. The joint distribution has marginals that are not exponential and a negative correlation coefficient, except in the special case of independence. After deriving some distributional results, we develop methods for parameter estimation and simulation. A simple method-of-moments estimator appears to give reasonable results. We also briefly discuss generalizations to higher dimensions and to distributions with conditionals in a general exponential family.

A Bayesian statistical theory of the phase problem. I. A multichannel maximum-entropy formalism for constructing generalized joint probability distributions of structure factors

Abstract: In this first of three papers on a full Bayesian theory of crystal structure determination, it is shown that all currently used sources of phase information can be represented and combined through a universal expression for the joint probability distribution of structure factors Particular attention is given to situations arising in macromolecular crystallography, where the proper treatment of non-uniform distributions of atoms is absolutely essential A procedure is presented, in stages of gradually increasing complexity, for constructing the joint probability distribution of an arbitrary collection of structure factors These structure factors may be gathered from one or several crystal forms of an unknown molecule, each comprising one or several isomorphous structures related by substitution operations, possibly containing solvent regions and known fragments, and/or obeying a set of non-crystallographic symmetries This universal joint probability distribution can be effectively approximated by the saddlepoint method, using maximum-entropy distributions of atoms [Bricogne (1984) Acta Cryst A40, 410-445] and a generalization of structure-factor algebra Atomic scattering factors may assume arbitrary complex values, so that this formalism applies to neutron as well as to X-ray diffraction methods This unified procedure will later be extended by the construction of conditional distributions allowing phase extension, and of likelihood functions capable of detecting and characterizing all potential sources of phase information considered so far, thus completing the formulation of a full Bayesian inference scheme for crystal structure determination

Unique Characterization of Conditional Distributions in Nonlinear Filtering

Abstract: Let $(X, Y)$ solve the martingale problem for a given generator $A$. This paper studies the problem of uniquely characterizing the conditional distribution of $X(t)$ given observations $\{Y(s)\mid 0 \leq s \leq t\}$. We define a filtered martingale problem for $A$ and we show, given appropriate hypotheses on $A$, that the conditional distribution is the unique solution to the filtered martingale problem for $A$. Using these results, we then prove that the solutions to the Kushner-Stratonovich and Zakai equations for filtering Markov processes in additive white noise are unique under fairly general circumstances.

Strong Uniform Consistency Rates for Estimators of Conditional Functionals

Abstract: Strong uniform consistency rates are established for kernel type estimators of functionals of the conditional distribution function, under general conditions. The present treatment unifies a number of specific problems previously studied separately in the literature. Some of these applications we treat in detail, including regression curve estimation, density estimation, estimation of conditional df's, $L$-smoothing and $M$-smoothing. Various previous results in the literature are extended and/or sharpened.

Robust Nonparametric Regression with Simultaneous Scale Curve Estimation

Abstract: Let $\{X_i, Y_i\}^n_{i=1} \subset \mathbb{R}^d \times \mathbb{R}$ be independent identically distributed random variables. If the conditional distribution $F(y \mid x)$ can be parametrized by $F(y \mid x) = F_0((y - m(x))/\sigma(x))$ with a fixed and known distribution $F_0$, the regression curve $m(x)$ and scale curve $\sigma(x)$ could be estimated by some parametric method. More generally, we assume that $F$ is unknown and consider nonparametric simultaneous $M$-type estimates of the unknown functions $m(x)$ and $\sigma(x)$, using kernel estimators for the conditional distribution function $F(y \mid x)$. We show pointwise consistency and asymptotic normality of these estimates. The rate of convergence is optimal in the sense of Stone (1980). The asymptotic bias term of this robust estimate turns out to be the same as for the linear Nadaraya-Watson kernel estimate.

Inference about the mean of a poisson distribution in the presence of a nuisance parameter

Abstract: Summary In this note we examine the problem of estimating the mean of a Poisson distribution when a nuisance parameter is present. Using a condition of Cox (1958) about ancillarity in the presence of a nuisance parameter, we justify that inference about the parameter should be carried out using the conditional distribution given the appropriate ancillary statistics. A small simulation study has been done to compare the performance of the conditional likelihood approach and the standard likelihood approach.

Probability distributions and their partitioning

Abstract: The partitioned multi-objective risk method (PMRM) was developed for solving risk-based multi-objective decision making problems Based on the premise that the expected value concept is not sufficient for proper decision making, the PMRM generates a number of conditional expected value functions (or risk functions) by partitioning the probability axis into probability ranges The goal of partitioning the probability axis is to have better information on extreme events for decision making purposes These conditional expectations are dependent on the chosen partitioning points This paper analyzes how conditional expectations are sensitive to variations in partitioning One of the risk functions is a measure of extreme and catastrophic events By using the relationship between this particular risk function and the statistics of extremes, the sensitivity analysis is simplified In many practical applications, it is difficult to determine which type of distribution function best represents the random process Conditional expectations also depend on the choice of distribution, and the impact of this selection is discussed

Distributed estimation: constraints on the choice of the local models

Abstract: The following estimation problem is considered: a coordinator must reconstruct the (global) probability density of a nonlinear random process, conditioned on the noise-corrupted observation history. The coordinator can only access the (local) probability density produced by local processing of the observation history using a (local) model different from the process model. It is shown that if the local model satisfies an algebraic constraint, the coordinator can reconstruct the same conditional density of the state process as the one obtained if the observations were processed using the coordinator (process) model. It is assumed that the random process is a nonlinear stochastic differential equation driven by a Brownian motion, and the observation process is corrupted by additive Brownian motion, which is identically modeled by the coordinator and the local processor. >

On conditional inference for a real parameter: A differential approach on the sample space

Abstract: SUMMARY A one-dimensional conditional procedure defines a partition of the sample space into curves which can be represented by means of a unit vector field. A formula is given for the conditional distribution in terms of local properties of the vector field. Conditions are developed for reducing the first-order effects of nuisance parameters and reproducing to higher order the likelihood change for the parameter of interest. The emphasis is on extending exponential family methods after locally approximating the statistical model by an exponential family.

Likelihood inference for linear regression models

Abstract: SUMMARY Approximate conditional inference based on large-sample likelihood ratio methods is considered for the parameters of linear regression models. Mean and variance adjustments that improve the standard normal approximation to the conditional distribution of the signed square root of the log-likelihood-ratio statistic for a scalar parameter of interest are given. A Bartlett adjustment factor that improves the chi-squared approximation to the conditional distribution of the log-likelihood-ratio statistic for a vector parameter of interest is also presented. The accuracy of approximate confidence limits obtained by using the adjustments is demonstrated for a location-scale analysis of Darwin's data.

Stochastic and deterministic algorithms for MAP texture segmentation

Abstract: A model-based approach is proposed for the problem of texture segmentation using a maximum a posteriori (MAP) estimation technique. A Gauss-Markov random field (GMRF) is used for the conditional density of the intensity array, given the unobserved texture class and a second-order Ising distribution for the prior distribution over the texture classes. The GMRF model for the conditional density allows a closed-form expression for the density to be written, so that the dependence of the density on the label parameters can be expressed. This expression is used here to derive the joint distribution of intensity and label arrays. The joint distribution is maximized using the stochastic relaxation method and the deterministic iterated conditional mode (ICM) technique. The ICM algorithm can be implemented efficiently on a neural net with local connectivity and regular structure. Comparisons of these two methods are given using real textured images. >

Conditional expectations, conditional distributions, and a posteriori ensembles in generalized probability theory

Abstract: A general probabilistic framework containing the essential mathematical structure of any statistical physical theory is reviewed and enlarged to enable the generalization of some concepts of classical probability theory. In particular, generalized conditional probabilities of effects and conditional distributions of observables are introduced and their interpretation is discussed in terms of successive measurements. The existence of generalized conditional distributions is proved, and the relation to M. Ozawa'sa posteriori states is investigated. Examples concerning classical as well as quantum probability are discussed.

Probability densities for conditional statistics in the cubic sensor problem

Abstract: This paper applies the techniques of Malliavin’s stochastic calculus of variations to Zakai’s equation for the one-dimensional cubic sensor problem in order to study the existence of densities of conditional statistics. Let {Xt} be a Brownian motion observed by a cubic sensor corrupted by white noise, and let\(\hat \phi \) denote the unnormalized conditional estimate of φ(Xi). If φ1,...,φn are linearly independent, and if\(\hat \Phi = (\hat \phi _1 ,...,\hat \phi _n )\), it is shown that the probability distribution of\(\hat \Phi \) admits a density with respect to Lebesgue measure for anyn. This implies that, at any fixed time, the unnormalized conditional density cannot be characterized by a finite set of sufficient statistics.

Conditional even point estimation for bivariate discrete distributions

Abstract: This paper presents a simple procedure for estimating the parameters of bivariate discrete distributions. The procedure uses the marginal means and certain observed frequencies in one or more conditional distributions. The bivariate Poisson and Negative Binomial distributions are used as illustrative examples, Parameter estimators are derived and asymptotic efficiencies are examined for various parameter values.

Factorization of measures and normal conditional distributions

Abstract: On etudie l'existence de factorisations qui sont additives de facon finie pour des mesures de probabilite sur un espace produit et l'existence de distributions conditionnelles normales

The Probability of War in the N-Crises Problem: Modeling New Alternatives to Wright's Solution

Abstract: In his "Study of War," Q. Wright considered a model for the probability of war P "during a period of n crises", and proposed the equation , where p is the probability of war escalating at each individual crisis. This probability measure was formally derived recently by Cioffi-Revilla (1987), using the general theory of political reliability and an interpretation of the "n-crises problem" as a branching process. Two new, alternate solutions are presented here, one using D. Bernoulli's St. Petersburg Paradox as an analogue, the other based on the logic of conditional probabilities. Analysis shows that, while Wright's solution is robust with regard to the general overall behavior of p and n, some significant qualitative and quantitative differences emerge from the alternative solutions. In particular, P converges to 1 only in a special case (Wright's) and not generally.

Comment on "Small-signal analysis in high-energy physics: A Bayesian approach"

Abstract: Prosper's recent paper postulates that, in the absence of any prior information, the distribution of prior probabilities for an experiment should have the same symmetries as the conditional probabilities for the experiment. From this assumption, he derives a unique form for the prior probability distribution in the case of an experiment to measure a small signal above a background. His derivation does not adequately take into account the arbitrary nature of the normalization of the prior probability distribution. Allowing for changes in the normalization, the form of the prior probability distribution is no longer uniquely determined.

A generalization of the Erlang formula of traffic engineering

Abstract: Calls arrive at a switch, where they are assigned to any one of the available idle outgoing links. A call is blocked if all the links are busy. A call assigned to an idle link may be immediately lost with a probability which depends on the link. For exponential holding times and an arbitrary arrival process we show that the conditional distribution of the time to reach the blocked state from any state, given the sequence of arrivals, is independent of the policy used to route the calls. Thus the law of overflow traffic is independent of the assignment policy. An explicit formula for the stationary probability that an arriving call sees the node blocked is given for Poisson arrivals. We also give a simple asymptotic formula in this case.