Prem K. Goel

Information about Hyperparameters in Hierarchical Models

Abstract We consider situations in which the prior distribution of a parameter vector θ1 in the distribution of an observable random vector X contains a hyperparameter vector θ2. The experimenter specifies another distribution for θ2 that contains hyperparameters θ3, and so forth. One wants to learn about the hyperparameters at each level of this hierarchical model. We show that for many measures of information, the gain in information decreases as one moves to higher levels of hyperparameters. These results are illustrated for univariate normal models and a general linear hierarchical model. Examples of measures of information are given for which this property does not hold.

Information Measures and Bayesian Hierarchical Models

Abstract We consider the Bayesian statistical models in which the prior distribution of the parameter vector θ1 in the distribution of an observable random vector X is to be specified in a hierarchical fashion and one wants to learn about the hyperparameters at each level of this prior distribution. It is shown that for a wide class of information measures, based on the so-called f divergence, the information decreases as one moves to higher levels of hyperparameters. This result unifies all the theorems in Goel and DeGroot (1981) and provides several other information measures for which the above desirable property holds.

Only Normal Distributions Have Linear Posterior Expectations in Linear Regression

Abstract We consider the general linear model Y = Aβ + e in the Bayesian framework and examine the implications of the statement that the posterior expectation of β, given Y, is a linear function of Y. Under various conditions on the model, it is shown that this linear posterior expectation implies that both β and e are normally distributed. For most of the practical situations in which linear models are used, only normal distributions have linear posterior expectations.

On implications of credible means being exact bayesian

Abstract In the literature on credibility theory, Jewell showed that the credible means are exact Bayesian for exponential families. In this paper we examine the implications of the statement that credible means are exact Bayesian for certain special forms of distributions. It is conjectured that this statement is valid only for exponential families.

Consistency and asymptotic normality of maximum likelyhood estimators

Abstract Let the random variable X denote the time taken in completion of a process. For a fixed a, if the observed value of X is less than a, the X is observable, but if X is greater than a, the process is tampered with and is accelerated or decelerated at time a by some unknown factor α, and Y=a+α(X-a) is observed. If the experimenter has only partial control over the experiment, it may be difficult to get several observations on Y corresponding to the same a value. Thus we have a set of independent but not identically distributed observations. The large sample behavior of m.l.e. of the unknown parameters based on tampered random variables Y b1 , ..., Y bn is studied. If X follows an exponential distribution with mean (1/--), ... the consistency and asymptotic normality of the m.l.e. of α and -- is established under mild conditions on a b1, a b2, ... The conditions needed for establishing the consistency of m.l.e. of lX are given when X follows a uniform distribution U(O, --) or when X has any known dis...

A note on the consistency of maximum likelihood estimators

Abstract A sequence of maximum likelihood estimators based on a sequence of independent but not necessarily identically distributed random variables is shown to be consistent under certain assumptions. Some examples are given to show that these assumptions are easy to verify and not very restrictive.