Showing papers on "Posterior probability published in 1971"

A Bayesian Approach to Short-term Forecasting

Abstract: A new approach to short-term forecasting is described, based on Bayesian principles. The performance of conventional systems is often upset by the occurrence of changes in trend and slope, or transients. In this approach events of this nature are modelled explicitly, and successive data points are used to calculate the posterior probabilities of such events at each instant of time. The system produces not only single-figure forecasts but distributions of trend and slope values which are relevant to subsequent decisions based on forecasts.

Estimation of the Probability that Y<X

Abstract: The problem of estimating θ = Pr[Y < X] has been considered in the literature in both distribution-free and parametric frameworks. In this article, using a Bayesian approach, we consider the estimation of θ from two approaches. The first, analogous to the classical procedure, is concerned with the problem of parametric estimation. The second, peculiar to the Bayesian approach, is directed to the query, “For two future observations, × and Y, what is the probability (given only the available sample data) that Y is less than X” This probability, termed the predictive probability, is not an estimate but is, in fact, a probability. These two views are related in that this predictive probability is the mean of the posterior distribution of θ. In the following sections, these Bayesian procedures are applied to the case of independent exponentially distributed random variables and to various cases of the normal distribution. The Bayesian estimates thus obtained are compared, whenever possible, with their...

A note on the selection of data transformations

Abstract: SUMMARY Recently Box & Cox (1964) and Fraser (1967) have proposed likelihood functions as a basis for choosing a transformation of data to yield a simple linear model. Here a simple, exact, test of significance is proposed for the consistency of the data with a postulated transformation within a given family. Confidence sets can be derived from this test. The power of the test may be estimated and used to predict the sharpness of the inferences to be derived from such an analysis. The methods are illustrated with examples from the paper by Box & Cox (1964). Box & Cox (1964) considered the choice of a transformation among a parametric family of data transformations to yield a simple, normal, linear model. They investigated two approaches to this problem and derived a likelihood function and a posterior distribution for the parameters of the transformation. Draper & Cox (1969) have found approximations for the precision of the maximum likelihood estimate. Fraser (1967) derived a different likelihood function which yields quite different inferences from those of Box & Cox (1964) in extreme cases where the number of parameters is close to the number of observations. Likelihood methods require repeated computations using a number of transforms of the original data. This can be troublesome if there is a multiparameter family of transformations. A further defect of the likelihood methods is that confidence limits and tests based on them have only asymptotic validity; the number of parameters must be small compared with the number of observations. This will not be the case for small data sets, paired comparison experiments and extreme cases. In the present paper, a method is proposed which has three possible advantages over direct calculation of likelihoods. Its main disadvantage is that it does not lead to such a clear graphical summary of conclusions as is given by a plot of a likelihood. The advantages are: (i) an 'exact' test of significance is obtained from which 'exact' confidence limits can be calculated; (ii) the amount of calculation is reduced if only one or a few transforms are to be tested; (iii) the precision with which the transformation can be estimated is capable of theoretical calculation.

Jaynes's maximum entropy prescription and probability theory

Abstract: Jaynes's prescription of maximizing the information-theoretic entropy is applied in a special situation to determine a certain set of posterior probabilities (when evidence fixing the expected value of a dynamical variable is given) and also the corresponding set of prior probabilities (when this evidence is not given). It is shown that the resulting values of these probabilities are inconsistent with the principles of probability theory. Three possible ways of avoiding this inconsistency are briefly discussed.

Estimating the Conditional Probability of Misclassification

Abstract: Let x be an observation assumed to be from one of two p-dimensional normal populations II, II2 with unknown mean vectors t , t2 and common known covariance matrix M. The observation is to be classified as coming from one of the populations. Samples S1 , S2 consisting of N1 , N2 observations known to come from HI , II2 respectively are also available, and a classification rule is based on these samples. This paper is concerned with the problem of estimating the conditional probability of misclassifying x given a fixed classification rule. Let xi , x2i (i = 1, ... , N ; j = 1, *. , N2) denote the observations in S1, S2, and let i , 22 denote the sample means. The classification rule will be the commonly used one (see for example Anderson, (1958)) that classifies x as HII if

Interval Estimates for Linear Combinations of Means

Abstract: If x Q(i = 1,..., k; i = 1, ..., ni) are independent normal random variables with mean ,ti and variance gr2, then interval estimates for linear combinations of the ,t4's based on a simple normal approximation to Student's t-distribution, are shown to be adequate approximations to the available alternative intervals for values of ox near 0 05. The approximate intervals can be regarded either as conservative confidence intervals or as posterior probability intervals and do not require the use of special tables such as those for the Behrens-Fisher or Welch solutions of the two means problems. The approximation holds good for k>2 for which no tables of the alternative solutions exist.

Modelling Conditional Probability

Abstract: Many studies of the joint frequency of the initial and final conditions of weather elements such as cloud cover, visibility, rainfall or temperature attest to the importance of the initial event as a predictor of the later event. Most efforts have involved the actual collection of the data in contingency tables but there is a strong need for an analytical tool to estimate the conditional probabilities from more readily available climatic frequencies. By assuming the Markov process, and with the help of published tables detailing the bivariate normal distribution, a succinct two-parameter model, using the climatic frequencies in a single equation, has been developed to estimate conditional probabilities, of both frequent and rare events, within a few percentage points. The two parameters have been charted as direct functions of the probability of the initial event and the temporal persistence of the element.

Optimum sequential search with discrete locations and random acceptance errors

Abstract: : Much work has been done in search theory. However, very little effort has occurred where an object's presence at a location can be accepted when no object is present there. The case analyzed is of this type. The number of locations is finite, a single object is stationary at one location, and only one location is observed each step of the search. The object's location has a known prior probability distribution. Also known are the conditional probability of acceptance given the object's absence (small) and the conditional probability of rejection given the object's presence (not too large); these probabilities remain fixed for all searching and locations. The optimum sequential search policy specifies that the next location observed is one with the largest posterior probability of the object's presence (evaluated after each step from Bayes Rule) and that the object is at the first location where acceptance occurs. Placement at the first acceptance seems appropriate when the conditional probability of acceptance given the object's absence is sufficiently small. The policy is optimum in that, for any number of steps, it minimizes the probability of no acceptances and, simultaneously, maximizes the probability that an acceptance occurs and the object is accurately located. Search always terminates (with probability one). Optimum truncated sequential policies are also considered. Methods are given for evaluating some pertinent properties and for investigating the possibility that no object occurs at any location. (Author)

Sensor-array Processing With Channel-recursive Bayes Techniques

Abstract: Arrays of seismometers, hydrophones, and electromagnetic receivers have several signal processing problems in common. This paper is concerned primarily with source location and secondarily with signal extraction. The basic problem can be described as follows: A transient signal from an event is detected in the outputs of the sensor array. We determine the location of the source from the temporal positions of the signal in the array outputs. Further, if the signal is unknown, we estimate it. The approach taken here differs from previous investigations in three ways: (i) a Bayes estimation approach is used, (ii) the estimates are evaluated recursively with respect to channels, and (iii) a time‐domain approach is used, as opposed to a frequency‐domain approach. The proposed estimation technique is optimum with respect to a large class of loss functions, since it is based on the expectation of the posterior distribution. Recursive evaluation of the posterior expectation has several advantages. At each step we...

Some Upper Bounds on Error Probability for Multiclass Pattern Recognition

Abstract: An upper bound on the probability of error for the general pattern recognition problem is obtained as a functional of the pairwise Kolmogorov variational distances. Evaluation of the bound requires knowledge of a priori probabilities and of the class-conditional probability density functions. A tighter bound is obtained for the case of equal a priori probabilities, and a further bound is obtained that is independent of the a priori probabilities.

Bayesian Modelling of a Non-Stationary Poisson Process1

Abstract: In this paper we consider a Poisson process where there is uncertainty as to the exactvalue of the parameter (λ) of the process. The usual Bayesian procedure is to select a gamma prior distribution on the λ values. When the process is observed (i.e. a certain number of Poisson events are observed in a particular time period), use of Bayes's Rule gives a posterior distribution which is still a member of the gamma family. However, this updating procedure assumes that the true λ value, although unknown, does not change with time. In practical applications, such as the modelling of consumer purchasing behaviour, the assumption of a stationary value of λ is inappropriate. In this paper we investigate the consequences (in terms of an implied updating procedure) of allowing λ to change at specified points in time, the changes being probabilistic in nature. Included is a discussion of how the various parameters might be estimated.

Probability and statistics

Abstract: Surely, there should be something useful and tangible that comes out of the experiment. This is usually in the form of probability. Assuming the sample size was large enough and represented the entire population [1] of applicability, the statistics should be able to predict [2] what the probability is of the drug being effective against a pathogen if a person takes it. Thus the experimenter should be able to tell a patient “If you take this drug, the probability that you will be cured is x%”. This shows the interrelation between probability and statistics.

Randomized Models and the Dilution and Bioassay Problems

Abstract: Randomization has been proposed [1] for the dilution-series problem with the purpose of producing a continuous random variable; the randomization is performed by selecting a value from the uniform distribution (0, 1]. This paper develops a structural model appropriate to a range of problems involving such randomization as part of the design. The model applies to the dilution series problem and to the bioassay problem, and produces a posterior distribution for the primary parameter and a marginal likelihood for other parameters.

A Comment on “Probability of Survival as an Investment Criterion” By Fred Hanssmann

Abstract: In a recent article of this journal, Hanssmann [Hanssmann, Fred. 1968. Probability of survival as an investment criterion. Management Sci. 15(1, September) 33–48.] proposed probability of survival as an investment planning criterion. An earlier paper by Roy [Roy, A. D. 1952. Safety first and the holding of assets. Econometrica (July) 431–439.] has shown that the normality assumption is not necessary in order to utilize probability of survival in investment planning and that a result identical and Hanssmann's is possible in the absence of any specific assumption concerning the form of the probability distributions involved. Hence, use of probability of survival as an investment planning criterion requires only a relatively weak assumption concerning probability distributions of portfolio returns, i.e., finite means and variances. There is no need to assume normality or, for that matter, any specific form for portfolio probability distributions.

Sum of event probabilities and subjective estimates of pi in probability learning

Abstract: Students had 300 prediction trials on two-choice contingent-probability event ratios with sums less than, equal to, or greater than one; then they estimated the probability of the most frequent event. Their asymptotic prediction varied with their subjective estimates of the event's probability, rather than with its actual probability of occurrence. Specifically, they underestimated when the ratio summed to less than one and overestimated when it summed to more than one, apparently because they assumed that the events were mutually exclusive.

BRIEF REPORT: THE EQUIVALENCE OF TWO MULTIVARIATE CLASSlFICATION SCHEMES.

Abstract: The equivalence of two multivariate classification schemes is shown when the sizes of the samples drawn from the populations to which assignment is required are identical. One scheme is based on posterior probabilities determined from a Bayesian density function; the second scheme is based on likelihood ratio discriminated scores. Both of these procedures involve prior probabilities; if estimates of these priors are obtained from the identical sample sizes, the equivalence follows.