Showing papers on "Conditional probability distribution published in 1974"
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TL;DR: In this article, the authors consider a group of individuals who must act together as a team or committee, and assume that each individual in the group has his own subjective probability distribution for the unknown value of some parameter.
Abstract: Consider a group of individuals who must act together as a team or committee, and suppose that each individual in the group has his own subjective probability distribution for the unknown value of some parameter. A model is presented which describes how the group might reach agreement on a common subjective probability distribution for the parameter by pooling their individual opinions. The process leading to the consensus is explicitly described and the common distribution that is reached is explicitly determined. The model can also be applied to problems of reaching a consensus when the opinion of each member of the group is represented simply as a point estimate of the parameter rather than as a probability distribution.
3,527 citations
TL;DR: In this article, the conditional distribution of the random measure, given the observations, is no longer that of a simple Dirichlet process, but can be described as being a mixture of DirICHlet processes.
Abstract: process. This paper extends Ferguson's result to cases where the random measure is a mixing distribution for a parameter which determines the distribution from which observations are made. The conditional distribution of the random measure, given the observations, is no longer that of a simple Dirichlet process, but can be described as being a mixture of Dirichlet processes. This paper gives a formal definition for these mixtures and develops several theorems about their properties, the most important of which is a closure property for such mixtures. Formulas for computing the conditional distribution are derived and applications to problems in bio-assay, discrimination, regression, and mixing distributions are given.
2,146 citations
76 citations
TL;DR: A Bayes procedure for classifying an observation consisting of one dichotomous variable (X) and a continuous vector Y is applied to a model assuming that the conditional distribution of Y given X is normal as mentioned in this paper.
Abstract: A Bayes procedure for classifying an observation consisting of one dichotomous variable (X) and a continuous vector Y is applied to a model assuming that the conditional distribution of Y given X is normal. The procedure reduces to two linear discriminant functions, one for each value of X. An example utilizing data on critically ill patients is given. Extension to one polytomous variable or several dichotomous variables is discussed.
71 citations
39 citations
TL;DR: Conditions for tightness and weak convergence of sequences of stochastic processes are given in this article in terms of restrictions on the conditional probabilities of large increments and of large jumps, and the conditions for weak convergence are also given.
Abstract: Conditions for tightness and weak convergence of sequences of stochastic processes are given in terms of restrictions on the conditional probabilities of large increments and of large jumps.
33 citations
TL;DR: Partial rank correlation coefficients are defined as special cases of a generalized product-moment system as discussed by the authors, and the case of two independent variables is examined in detail, and the coefficient is partitioned into components summarizing information lying between and within conditional distributions.
Abstract: Partial rank correlation coefficients are defined as special cases of a generalized product-moment system. The case of two independent variables is examined in detail, and the coefficient is partitioned into components summarizing information lying between and within conditional distributions. Probability interpretations are given for two basic components. Recent developments identifying the structural assumptions needed for a full probability interpretation and bases for a general theory are briefly summarized. The basic components that do not require structural assumptions but cannot be calculated from bivariate data are suggested as alternative measures of partial rank correlation. One of them is closely related to Kendall's partial tau, another to Davis's partial gamma. The manner in which these methods summarize a triple dichotomy is illustrated. The logic of statistically "partialling out" ordinal covariation is discussed and quantified.
27 citations
TL;DR: In this paper, the conditional expectation of a function with probability one in a convex set is shown to be partially measurable with respect to a sub-sub-$sigma-field.
Abstract: If a $n$-dimensional function is with probability one in a convex set, the same holds true for the conditional expectation (with respect to any sub-$\sigma$-field). An extreme point of this convex set can be assumed by the conditional expectation only if it is assumed by the original function and if this function is partially measurable with respect to the conditioning sub-$\sigma$-field. These results are used to prove Jensen's inequality for conditional expectations of $n$-dimensional functions, and to give a condition for strict inequality.
26 citations
TL;DR: Good and Gaskins as discussed by the authors used hypothesis testing to compensate for the lack of information about the higher-order moments of the prior distribution of densities, and synthesized the work of Whittle and of Tarter and Kronmal.
Abstract: Most methods for the estimation of probability densities that have appeared in the literature involve a largely uniform treatment of the data. These include the common histogram, as well as the window method (Rosenblatt, 1956; Parzen, 1962; Bartlett, 1963; Watson & Leadbetter, 1963), the method of Fourier expansion (Tarter and Kronmal, 1968, 1970; Watson, 1969), and the fitting by splines (Boneva, Kendall & Stefanov, 1971). The use of such methods commonly requires the acceptance of a certain amount of roughness in the tails in order to avoid excessive smoothing of the central portion of the estimate. Nonuniform approaches to the estimation of density have been proposed by Whittle (1958) and by Good & Gaskins (1971). Whittle's method requires specification of the firstand second-order moments of the prior distribution of densities. The likelihood approach of Good & Gaskins produces a more general form of estimate, and the authors reduce the problem by constructing a plausible two-parameter family of prior distributions. In this paper, it is assumed that only the first-order moments of the prior distribution of densities are known. The convention of hypothesis testing is used to compensate for the lack of information about the higher-order moments. The method synthesizes the work of Whittle and of Tarter and Kronmal. In ? 2, it is shown that an estimate of the Tarter-Kronmal type is appropriate when the data have been transformed so that the first-order moments of the prior distribution are the uniform (0, 1) density. In ? 3, a heuristic approach related to hypothesis testing is used for the construction of the estimate. Application of the procedure is described in ? 4.
22 citations
01 Jan 1974
14 citations
TL;DR: In this paper, it was shown that the independent random vectors x and y have multinomial (negative mUltinomial) distributions with the same parameter vector o, and the other parameters being respectively m and n if and only if the conditional distribution of x given x + y is multivariate hypergeometric (multivariate inverse hypergeometry) distribution with parameters m + n = N and x+ y = N
Abstract: The intent of this paper is to show that the independent random vectors x and y have multinomial (negative mUltinomial) distributions with the same parameter vector o, and the other parameters being respectively m and n if and only if the conditional distribution of x given x + y is multivariate hypergeometric (multivariate inverse hypergeometric) distribution with parameters m + n = N and x + y = N
TL;DR: In this paper, the authors remove all cards except aces and kings from a deck, so that only eight cards remain, of which four are aces, four are spades, and four are kings.
Abstract: Remove all cards except aces and kings from a deck, so that only eight cards remain, of which four are aces and four are kings. From this abbreviated deck, deal two cards to a friend. If he looks at his cards and announces (truthfully) that his hand contains an ace, what is the probability that both his cards are aces? If he announces instead that one of his cards is the ace of spades, what is the probability then that both his cards are aces? (These two probabilities are not the same!) ([1], p. 433)
01 Jan 1974
TL;DR: In this paper, the authors argue that power considerations are not relevant in choosing between conditional and unconditional tests, and that the conditional test is the appropriate one for measuring the significance of Poisson-distributed observations.
Abstract: Some logical aspects of tests of significance are illustrated using the example of equality of means of Poisson-distributed observations. Specifically, in the conventional test for the significance of a difference between two Poisson-distributed observations, the significance level is computed from the conditional distribution given their observed total. However, an unconditional test has greater power and so is sometimes advocated in place of the conditional test. The present paper argues that power considerations are not relevant in choosing between conditional and unconditional tests, and that the conditional test is the appropriate one. The example is extended to include tests of equality of means of two Poisson samples (pointing out an error in a formula that is sometimes used) and also to include tests concerning the ratio of Poisson means. There is a general discussion of significance tests with reference to the above examples.
TL;DR: The triangular model of chance-constrained programming with stochastic A-matrix and deterministic right-hand side is considered and the use of conditional probabilities makes it possible to solve this problem for any type of distribution function of the elements of the A-Matrix.
Abstract: The triangular model of chance-constrained programming with stochastic A-matrix and deterministic right-hand side is considered. The use of conditional probabilities makes it possible to solve this problem for any type of distribution function of the elements of the A-matrix provided that there is only one decision variable at each stage. The extension of the model to several decision variables per stage is possible under certain conditions and for special distribution (stable distributions) of the elements of A.
TL;DR: In this paper, a characterization of the exponential and geometric distribution in terms of conditional expectations is given, and a property of bivariate distributions is given by generalizing Kotlarski's result in the univariate case.
TL;DR: In this paper, the inference problems of an extension of Fisher's bivariate exponential model, using his method of conditional inference based on the observed value of ancillary statistics, are studied.
Abstract: This paper studies the inference problems of an extension of Fisher's bivariate exponential model, using his method of conditional inference based on the observed value of ancillary statistics. In the process we discover a simple method of obtaining the values of maximum likelihood estimates and their distributions conditional on the ancillary statistics. An application in reliability estimation is given. An error in Fisher's derivation of conditional information related to his bivariate model is also indicated and corrected.
TL;DR: In this article, it was shown that the set of conditional probability measures generated by the differential system in response to a class of bounded measurable controls with values in a compact convex subset U of a finite dimensional Euclidean space Rv, is equivalent to those corresponding to controls taking values only on the boundary of the set U.
Abstract: In this paper we consider a class of stochastic linear Ito differential equations with control parameters. It is shown (theorem 2.1) that the set of conditional probability measures, generated by the differential system in response to a class of bounded measurable controls with values in a compact convex subset U of a finite dimensional Euclidean space Rv , is equivalent to those corresponding to controls taking values only on the boundary of the set U. Further this set of conditional probability measures is shown (theorem 2.2) to be weakly compact. These results are then used to prove the existence (theorem 2.4) of an optimal control that satisfies the so called ’ bang-bang ’ principle (Hermes and LaSalle, p. 46).
01 Jan 1974
TL;DR: In this paper, the distribution of a scalar random variable is studied in detail, and the probability that the variable will exceed some value, say α, or that it will assume a value in the interval (α, (β) and so on).
Abstract: In elementary mathematical statistics, one studies in some detail various characteristics of the distribution of a scalar random variable. Thus its density and various parameters are considered and statements are made regarding this variable. For example, given the information above, we can compute the probability that the variable will exceed some value, say α, or that it will assume a value in the interval (α, (β) and so on.
30 Sep 1974
TL;DR: In this article, the authors studied the conditional distribution properties of one-parameter power series distributions truncated on the left at several known or unknown points via exponential generating functions and showed how they can be used in confidence interval estimation of the reliability of multicomponent attribute failure models.
Abstract: : Convolutions of one-parameter power series distributions (PSD) truncated on the left at several known or unknown points are studied via exponential generating functions. The special cases of the logarithmic series, the Poisson and the binomial and negative binomial distributions lead to multiparameter Stirling numbers of the first and second type and C-numbers respectively. Minimum variance unbiased estimators are found for certain functions of the parameters, including the probability functions themselves. Some conditional distribution properties are given and it is indicated how they can be used in confidence interval estimation of the reliability of multicomponent attribute failure models.