Abstract: A measure of uncertainly and information for possibility theory is introduced in this paper The measure is called the U-uncertainty or, alternatively, the U-information. Due to its properties, the U-uncertainty/information can be viewed as a possibilistic counterpart or the Shannon entropy and, at the same time, a generalization or the Hartley uncertainty/information. A conditional U-uncertainty is also derived in this paper, it depends on the U-uncertainties or the joint and marginal possibility distributions in exactly the same way as the conditional Shannon entropy depends on the entropies or the joint and marginal probability distributions. The conditional U-uncertainty is derived without the use of the notion of conditional possibilities, thus avoiding a current controversy in possibility theory. The proposed measures of U-uncertainty and conditional U-uncertainty provide a foundation for developing an alternative theory of information, one based on possibility theory rather than probability...

Abstract: Mathematical modeling of groundwater flow in heterogeneous porous formations of large extent is investigated. The formation properties (hydraulic conductivity, transmissivity) as well as flow variables (head, specific discharge, solute concentration) are regarded as random variables subjected to uncertainty. The main aim of the study is to analyze the influence of conditional probability of input variables upon the statistical structure of the dependent variables. The unconditional probability density functions are supposed to be stationary multivariate normal, while conditioning accounts for the measured values at a few points of the formation. Two problems of groundwater flow are discussed in part 1: conditional simulation and the direct problem for steady flow. Analytical results are obtained by using perturbation approximations. Average uniform head gradient flows as well as recharge and flow to wells are discussed. In unconditional modeling the input variable (the conductivity or transmissivity log) is regarded as stationary and is represented by its constant mean and variogram. The ensemble of formations on which statistical calculations are carried out represents all aquifers with same probability density distributions. In conditional modeling the measured values at a few points are kept fixed and uncertainty prevails only at other points. Consequently, statistical computations are performed for the subensemble of aquifers which preserve the measured values, and as a result, both input and output variables are nonstationary. The main effect of conditioning is to reduce the variance, i.e., the uncertainty, of these variables. This effect is particularly important in the case of flow toward wells. Application of the modeling method to field problems is outlined.

Abstract: SUMMARY Differential geometry is applied to the problems of defining higher-order asymptotic ancillarity and of obtaining the asymptotic conditional distribution of an efficient estimator in multiparameter curved exponential families. It is shown that a fundamental role is played in the asymptotic theory of estimation by a one-parameter family of affine connexions and curvatures of subspaces. Asymptotic ancillary statistics of higher order are explicitly constructed with the help of the geometry. The conditional distribution of an estimator is given in terms of the exponential curvature of the model and the mixture curvature of the ancillary subspaces associated with the estimator.

Abstract: The recently secured mathematical formalism of direct methods is here generalized to the case that the atomic scattering factors are arbitrary complex numbers, thus including the special case that one or more anomalous scatterers are present. Once again the neighborhood concept plays the central role. Final results from the probabilistic theory of the two- and three-phase structure invariants are briefly summarized. In particular, the conditional probability distribution of the three-phase structure invariant, given the six magnitudes |E| in its first neighborhood, is described. The distribution yields an estimate for the three-phase structure invariant which is particularly good in the favorable case that the variance of the distribution happens to be small (the neighborhood principle). Particularly noteworthy is the fact that, in sharp contrast to all earlier work, the estimate is unique in the whole range 0 to 2π. An example shows that the method is capable of yielding unique estimates for tens of thousands of three-phase structure invariants with unprecedented accuracy, even in the macromolecular case. The clear implication is that the fusion of the traditional techniques of direct methods with anomalous dispersion, which is described here, will facilitate the solution of those crystal structures which contain one or more anomalous scatterers.

Abstract: Consider N objects on which two correlated measurements X and Y can be made and where the measurement of X is easier or less expensive to make than the one of Y. Suppose that the probability that the Y measurement meets a certain specification is γ. We present a method, which uses the conditional probabilities that Y will meet the specification given the observed value of the x's, for choosing a maximal subset of m out of the N objects, based on the observed X measurements, so that there is a high probability that a large proportion (Π ≥ γ) of the selected subset will meet the desired specification. The procedure is useful for helping to meet a guaranteed proportion of satisfactory items. It compares favorably with one suggested by Owen, Li, and Chou (1981).

Abstract: The present paper is concerned with optimal estimation of the 1st and 2nd order structural moments appearing in credibility formulas. In a recent paper De Vylder has treated the problem in the case of multinormal conditional distributions under quite restrictive assumptions. He minimizes, within a certain restricted class of unbiased estimators, the variance (or the sum of variances if the estimand is a matrix) and next replaces all structural moments (up to fourth order) in the solution by estimates based on the data. This paper is an attempt to simplify the method and extend it so as to make it applicable in more general situations. By suitable choice of a (sufficient) set of statistics and a suitable parametrization, the powerful theory of estimation in linear models can be employed, which makes cumbersome minimization procedures superfluous. The theory is applied to the cases with binomial. Poisson, compound Poisson, and multinormal conditional distributions. Some simulation studies have been performed to assess the performance of the estimators.

Abstract: In a previous paper (Suppes and Zanotti, 1976) we gave simple necessary and sufficient qualitative axioms for the existence of a unique expectation function for the set of extended indicator functions. As we defined this set of functions earlier, it is the closure of the set of indicator functions of events under function addition. In the present paper we extend the same approach to conditional probability. One of the more troublesome aspects of the qualitative theory of conditional probability is that A [ B is not an object in particular it is not a new event composed somehow from events A and B. Thus the qualitative theory rests on a quaternary relation A [ B 2 CI D, which is read: event A given event B is at least as probable as event C given event D. There have been a number of attempts to axiomatize this quaternary relation (Koopman, 1940a, 1940b; Aczel, 1961, 1966, p. 319; Luce, 1968; Domotor, 1969; Krantz et al., 1971; and Suppes, 1973). The only one of these axiomatizations to address the problem of giving necessary and sufficient conditions is the work of Domotor, which approaches the subject in the finite case in a style similar to that of Scott (1964). By using indicator functions or, more generally, extended indicator functions, the difficulty of A I B not being an object is eliminated, for AZ IB is just the indicator function of the set A restricted to the set B, that is, AZ IB is a partial function whose domain is B. In similar fashion if X is an extended indicator function, X I A is that function restricted to the set A. The use of such partial functions requires care in formulating the algebra of functions in which we are interested, for functional addition X IA + Y [ B will not be well defined when A +B but A n B =t= 8. Thus, to be completely explicit we begin with a nonempty set Q, the probability space, and an algebra F of events, that is, subsets of Q, with it understood that 9 is closed under union and complementation. Next we extend this algebra to the algebra F* of extended indicator functions, that is, the smallest semigroup (under function addition) containing the indicator functions of all events in F. This latter algebra is now

Abstract: A Berry-Esseen bound is given for the rate of convergence to normality of the number of empty boxes when balls are distributed independently and at random to boxes with possibly unequal probabilities The method of proof uses the equivalence of this distribution to a certain conditional distribution based on independent Poisson random variables Then methods based on the characteristic function of this conditional distribution are used to obtain the result

Abstract: Two lists are given. The first is a list of properties of ancillary statistics; the second is a list of examples of ancillary statistics. It is then indicated which of the properties are satisfied by each example. Many of the models have the property that the parameter θ is a location parameter for the maximum likelihood estimator (MLE) in every conditional distribution determined by a fixed value of the ancillary statistic. In certain other models the same state of affairs is achieved by parameter transformation. In either of these cases we call the ancillary an “exact precision index.” There exist irregular models in which the precision of estimation depends not only on the ancillary but on θ as well. Some comments are offered on approximate ancillarity and on the conditionality principle.

Abstract: Suppose that for every independent sequence of random variables satis­ fying some hypothesis condition H, it follows that the partial sums converge almost surely. Then it is shown that for every arbitrarily-dependent sequence of random variables, the partial sums converge almost surely on the event where the conditional distributions (given the past) satisfy precisely the same condition H. Thus many strong laws for independent sequences may be immediately generalized into conditional results for arbitrarily-dependent sequences. 1. Introduction. If every sequence of independent random variables having property A has property B almost surely, does every arbitrarily-dependent sequence of random variables have property B almost surely on the set where the conditional distributions have property A? Not in general, but comparisons of the conditional Borel-Cantelli Lemmas, the condi­ tional three-series theorem, and many martingale results with their independent counter­ parts suggest that the answer is affirmative in fairly general situations. The purpose of this note is to prove Theorem 1, which states, in part, that if "property B" is "the partial sums converge," then the answer is always affIrmative, regardless of "property A." Thus many strong laws for independent sequences (even laws yet undiscovered) may be immediately generalized into conditional results for arbitrarily-dependent sequences.

Abstract: Stalnaker 's Thesis states that a conditional probabil i ty is the same as the probabil i ty of a conditional. At first sight, this s ta tement sounds like a harmless pun: \"Wha t is the probabil i ty that I throw a six if I throw an even number , if not the probabil i ty that: if I throw an even number , it will be a six?\" in van Fraassen ' s suggestive phrasing [7]. He goes on to discuss David Lewis ' s surprising demonstra t ion [3] that the thesis is wrong or trivial: assuming the Thesis, Lewis showed that no probabil i ty ass ignment can have more than four distinct values. In the sequel, van Fraassen revives Sta lnaker ' s Thesis on grounds which will not be discussed here. We shall instead give necessary and sufficient conditions for the Stalnaker thesis provided that: (a.1) the conditional probabil i ty of an event \"/3, given that the event A obtains, is defined by

Abstract: Application of the conditional probablity integral transformations for multivariate normal distribtions derived in Rincon-Gallardo,Quesenberry and O'Reality (1979) are considered assuming multiple samples. a component-wise analysis with pooling over smples is introduced and illustrated with three examples

Abstract: Data classified into several groups usually depend on structural and accessory parameters, the structural parameter 0 being a constant for all the data, while the accessory parameters 0, vary between groups. For the structural parameter, inferences that are free of the accessory parameters can be made when the 0, admit sufficient statistics or surrogates S,, by conditioning on the S, or by constructing statistics independent of the S,. When the Si are functions of the structural parameter 0, 0 is said to be endomorphic. Conditional likelihood methods are not unequivocal for endomorphic parameters; instead, inference is based on statistics independent of the S,, and so free of the 4,, derived from conditional distribution functions. With endomorphic parameters, since the surrogates and the statistics independent of them are functions of 0, tests of independence of these sets of statistics provide a means of making inferences about 0. STRUCTURAL AND ACCESSORY PARAMETERS; CONDITIONAL INFERENCE; CONDITIONAL DISTRIBUTION FUNCTION; STATISTICS INDEPENDENT OF A GIVEN STATISTIC; TESTS OF INDEPENDENCE

Abstract: Publisher Summary In a number of disciplines, data analysts are confronted with the problem of classifying an observation into one of the distinct groups when the number of variables is very large. The selection of variables is important, as there are situations where inclusion of unimportant variables may actually decrease the ability for discrimination. It is more feasible to analyze the data from cost and computational considerations if the number of variables is small. This chapter discusses various procedures for the selection of variables in discriminant analysis. It discusses procedures to find out whether variables associated with certain discriminant coefficients are important for discrimination between two populations. Generalizations of these procedures for several populations are discussed. These procedures are based upon using conditional distributions. Various procedures to determine the number of important discriminant functions are also discussed in the chapter.

Abstract: A statistical theory of nonuniform systems is developed, based on the conditional distribution method. Junior correlation functions of a nonuniform system have been used to obtain an explicit expression for the Helmholtz free energy and the effective Hamiltonian as functionals of the particle number density field. The structure of a transient layer at the liquid-gas interface and a surface sorption effect have been studied. A method of reduced description in the fluctuation theory, using the conditional correlation functions of the particle number density distribution, is suggested. An infinite system of integro-differential equations is obtained for the correlation functions introduced. A method for its truncation is proposed. As a result, the grand statistical integral, which takes into account the density field fluctuations of the system in equilibrium with a thermostat, is calculated.

Abstract: : Time series models with autoregressive, moving average and mixed autoregressive-moving average correlation structure and with positive-valued non-normal marginal distribution are considered. First, a flexible mixed model GLARMA(p,q) with Gamma marginals is investigated. The correlation structure for several special cases is derived. For the first-order autoregressive case, GLAR(1), the conditional density of X sub n given X sub n-1 is derived. This leads to the formation of a likelihood function and a numerical approximation to and a simulation study of the maximum likelihood method of parameter estimation. Multivariate extensions of the model are considered briefly. Second, three methods for generating first-order moving average sequences with Exponential marginals are examined. These generalize the EMA (1) Exponential model. Negative correlation using antithetic variables is investigated in the moving average models. A preliminary analysis of wind speed data obtained over a 15-year period in the Gulf of Alaska is presented. A model with four harmonic deterministic mean multiplying random innovative factors modeled by a GLAR (1) process is developed. Correlograms and periodograms are used to determine the model for the mean and the structure of the innovation process. (Author)

Abstract: We solve a problem posed by J. P. Burgess and R. D. Mauldin, and thus show that a result of theirs on conditional distributions and orthogonal measures cannot be improved.

Abstract: Consider the half-line (0,∞] (0 excluded, ∞ included) equipped with the Borel σ-algebra β of subsets generated by the subintervals of (0,∞].

Abstract: Several models exist for the observed number Z of correct answers in a multiple choice test. One of these is the binomial error model which assumes that the conditional distribution of Z given the proportion-correct true score is binomial. The specification of a beta distribution to the proportion-correct true score has led to the development of the beta-binomial model. Recently, extensions of the beta-binomial model have been given that allow for guessing. Letting X be the number of known items in the test the regression of X on Z has received much attention. In the present paper it is argued that the number W of known items in the item pool is of interest rather than X, and the regression of W on Z is studied.

Abstract: Mathematical details of obtaining the likelihood functions for testing hypotheses under the mixed model for quantitative traits are given. Since families are assumed to be chosen by the value of the trait of a particular family member, conditional distributions are required. Algorithms for efficient computation of the likelihood are given.

Abstract: Formulas are presented for the calculation of the cosines of n-tet phase invariants and embedded seminvariants in all the space groups. They are shown to be of the form of a particular type of expected value formula that is derivable from the joint probability distribution. In the recent literature, formulas for phase invariants and seminvariants have been given in the form of conditional probability distributions. A detailed comparison of the relative merits of the two types of formulas, expected value and conditional distribution, has not yet been made. The variety of potential applications is quite vast and therefore it may require much effort to make evaluations of current theories. Should it seem worthwhile, the determinantal joint probability distributions employed in this paper could provide the basis for the derivation of additional conditional probability distributions. They are likely to be much more complex, however, than the expected value formulas. Some simple calculations with triplet and quartet invariants involving random structures in space group P1 show a considerable decrease in the reliability of the expected value formulas as the complexity of the structure increases. A comparable observation had been made in the past for conditional probability distributions for triplet phase invariants. Current theories present the possibility of obtaining information in special circumstances, for example, with respect to selected embedded seminvariants. How extensive and how useful such information might be, particularly with respect to the truly difficult structures that occur among the essentially equal-atom, noncentrosymmetric crystals with 100 or more nonhydrogen atoms in the asymmetric unit, remains to be seen.

Abstract: : Employing minimal sufficient statistics maximal statistical noise, several Kolmogorov-type statistics; and conditional distributions, optimal detection procedures are constructed for various one-and-two-sample problems involving Pareto Renewal processes. Cases with and without nuisance parameters are treated. Optimal parametric and distribution-free procedures are developed.

Abstract: : Because of the recent popularity of the three-parameter logistic model among the researchers who apply latent trait theory, it will be worthwhile to investigate the effect of noise accommodate in different models. In this paper, four types of models on the dichotomous response level, Types A, B, C and D, are considered. Type A does not include noise, and the other three types do. Observations are made as to how much total item information is lost because of the noise, how the item response information functions are affected, how the speed of convergence to the normality of the conditional distribution of the maximum likelihood estimate of ability, given a specific ability level, is affected, and so forth. (Author)

Abstract: This paper concerns a control problem in which a system represented by a stochastic differential equation is to be steered so as to follow a similar system the whereabouts of which is known only through noisy observations. It is shown that an optimal control exists and that this control depends in a Markovian fashion on the state of the controlled system and on the conditional distribution of the position of the target system.