We consider the general linear model Y = Aβ + ε 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 ε are normally distributed. For most of the practical situations in which linear models are used, only normal distributions have linear posterior expectations.
Regions containing all possible Bayes estimates under certain classes of bounded loss functions are given for many standard distributions. The properties of such regions are studied in depth for general classes of posterior distributions. Estimators are proposed that replace the posterior mean.
Michael Goldstein·刊期 1980-12 · 卷 67（3）·Biometrika已编入正刊
弱先验假设下的线性贝叶斯回归估计量AI 解读
A simple set of prior assumptions is given with respect to which the linear Bayes regression estimator, in the case of univariate regression, is determined explicitly. The risk, also determined explicitly, is shown to decrease,at rate 0 ( n-1 ), to the minimum risk attainable by any linear fitting procedure.
This paper considers the problem of sequentially estimating the mean of a normal distribution when the variance is unknown. A continuous time analogue of the discrete time problem is studied. For $L$ in a class of loss functions, properties of the value function and optimal continuation region of $L$ are presented. Asymptotic expansions are found for the value function and the optimal boundary function of the loss function $L$.
This paper is a continuation of a paper in the Annals of Statistics (1976), 4 1159-1189 where, among other things, a Bayesian approach to testing independence in contingency tables was developed. Our first purpose now, after allowing for an improvement in the previous theory (which also has repercussions on earlier work on the multinomial), is to give extensive numerical results for two-dimensional tables, both sparse and nonsparse. We deal with the statistics $X^2, \Lambda$…
Summary Many physiological characteristics such as, for example, blood pressure, level of serum cholesterol, vital capacity, may be considered as ageing factors contributing to the overall hazard rate function in an additive fashion (formula (3.2)). If so, the regression function of the characteristic on time, in a population of survivors to time t, depends on t even if the characteristic, itself, does not vary with t, for any one individual. Also, if there is a relationship…
The problem considered is that of testing sequentially between two separated composite hypotheses concerning the mean of a normal distribution with known variance. The parameter space is the real line, on which is assumed an a priori distribution, $W,$ with full support. A family $\{\delta(c)\}$ of sequential tests is defined and shown to be asymptotically Bayes, as the cost, $c$, per observation tends to zero, relative to a large class of fully supported a priori…
We discuss ridge regression estimators as a class, showing how six recently proposed estimators can thus be viewed, and give reasons for preferring one of these (empirical Bayes maximum likelihood estimator) to the other five. Results are based on analysis rather than simulations and on a frequentist rather than Bayesian viewpoint.
An example is given to show that there exist fixed sample size designs for any sample size greater than two, for which the Yates-Grundy estimator is inadmissible in the class of nonnegative unbiased quadratic estimators of the variance of the Horvitz-Thompson estimator. A necessary condition for the Horvitz-Thompson variance estimator to be nonnegative definite is pointed out. It is also pointed out that a posterior lower bound can be obtained for any nonnegative definite…
Let X be a real-valued random variable with a family of possible densities (with respect to σ-finite μ) belonging to a one-parameter exponential family with the natural parameter θ ∈ Ω = natural parameter space. Let C(μ) = convex hull of the support of μ. Fix x 0> in the interior of C(μ), then δ is generalized Bayes for θ (with squared error loss) if and only if g(t) = exp { f 0 t δ(y + x 0)dy} (< ∞ for t + x 0 ∈ C(μ)) is a moment-generating function for a probability…
Let X be a real-valued random variable with a family of possible densities (with respect to σ-finite μ) belonging to a one-parameter exponential family with the natural parameter θ ∈ Ω = natural parameter space. Let C(μ) = convex hull of the support of μ. Fix x 0> in the interior of C(μ), then δ is generalized Bayes for θ (with squared error loss) if and only if g(t) = exp { f 0 t δ(y + x 0)dy} (< ∞ for t + x 0 ∈ C(μ)) is a moment-generating function for a probability…
We discuss ridge regression estimators as a class, showing how six recently proposed estimators can thus be viewed, and give reasons for preferring one of these (empirical Bayes maximum likelihood estimator) to the other five. Results are based on analysis rather than simulations and on a frequentist rather than Bayesian viewpoint.
An example is given to show that there exist fixed sample size designs for any sample size greater than two, for which the Yates-Grundy estimator is inadmissible in the class of nonnegative unbiased quadratic estimators of the variance of the Horvitz-Thompson estimator. A necessary condition for the Horvitz-Thompson variance estimator to be nonnegative definite is pointed out. It is also pointed out that a posterior lower bound can be obtained for any nonnegative definite…
Summary In response to a suggestion by Bather, Gittins proposed the adoption of rules for sampling one at a time from several Bernoulli populations which were based on randomized versions of his dynamic allocation indices. Such rules, Gittins conjectured, might sometimes be both asymptotically optimal and also (almost) Bayes with respect to the usual discounted reward criterion. This conjecture is investigated and verified.
This paper provides explicit solutions to the problem of estimating the arrival rate $\lambda$ of a Poisson process using a Bayes sequential approach. The loss associated with estimating $\lambda$ by $d$ is assumed to be of the form $(\lambda - d)^2\lambda^{-p}$ and the cost of observation includes both a time cost and an event cost. A discrete time approach is taken in which decisions are made at the end of time intervals having length $t$. Limits of the procedures as $t$ ap…
It is observed that in selecting an alternative to the usual maximum likelihood estimator, $\delta^0$, of a multivariate normal mean, it is important to take into account prior information. Prior information in the form of a prior mean and a prior covariance matrix is considered. A generalized Bayes estimator is developed which is significantly better than $\delta^0$ if this prior information is correct and yet is very robust with respect to misspecification of the prior…
In this note, we give a lemma which shows that the expected squared difference between the Bayes estimator with a Dirichlet process prior and the Kaplan-Meier product limit (PL) estimator for a survival function based on censored data is $O(n^{-2})$. This lemma, together with already proven pointwise consistency properties of the Bayes estimator, is used to establish two properties of the PL estimator; namely, the mean square consistency of the PL estimator with rate $O(n^{-1})$…
Let $\tau$ be a prior distribution over the parameter space $\Theta$ for a given parametric model $P_\theta, \theta \in \Theta$. For the sample space $\mathscr{X}$ (over which $P_\theta$'s are probability measures) belonging to a general class of topological spaces, which include the usual Euclidean spaces, it is shown that this parametric Bayes model can be approximated by a nonparametric Bayes model of the form of a mixture of Dirichlet processes prior, so that (i) the…
A general calculus of conditional independence is developed, suitable for application to a wide range of statistical concepts such as sufficiency, parameter-identification, adequacy and ancillarity. A vehicle for this theory is the statistical operation, a structure-preserving map between statistical spaces. Concepts such as completeness and identifiability of mixtures arise naturally and play an important part. Some general theorems are exemplified by applications to…
Let $\mathbf{S}_{p \times p}$ have a Wishart distribution with scale matrix $\Sigma$ and $k$ degrees of freedom. Estimators of $\Sigma$ are given for each of the loss functions $L_1(\hat{\Sigma}, \Sigma) = \operatorname{tr} (\hat{\Sigma}\Sigma^{-1}) - \log \det (\hat{\Sigma}\Sigma^{-1}) - p$ and $L_2(\hat{\Sigma}, \Sigma) = \operatorname{tr} (\hat{\Sigma}\Sigma^{-1} - I)^2$. The obvious estimators of $\Sigma$ are the scalar multiples of $\mathbf{S}$, i.e., $a\mathbf{S}$ where…
Forensic scientists often have to interpret refractive index measurements made on fragments of window glass, some taken from the scene of a crime and some found on a suspect. Adopting a model used recently by Lindley (1977) and Seheult (1978) this contribution proposes a non-Bayesian solution based on a likelihood ratio. The problem of deciding whether the fragments have come from a common source is distinguished from the problem of deciding the guilt or innocence of the…
This paper considers the problem of sequentially estimating the mean of a normal distribution when the variance is unknown. A continuous time analogue of the discrete time problem is studied. For L in a class of loss functions, properties of the value function and optimal continuation region of L are presented. Asymptotic expansions are found for the value function and the optimal boundary function of the loss function L.
This paper is a continuation of a paper in the Annals of Statistics (1976), 4 1159-1189 where, among other things, a Bayesian approach to testing independence in contingency tables was developed. Our first purpose now, after allowing for an improvement in the previous theory (which also has repercussions on earlier work on the multinomial), is to give extensive numerical results for two-dimensional tables, both sparse and nonsparse. We deal with the statistics X2,Λ…
The problem considered is that of testing sequentially between two separated composite hypotheses concerning the mean of a normal distribution with known variance. The parameter space is the real line, on which is assumed an a priori distribution, W, with full support. A family {δ(c)} of sequential tests is defined and shown to be asymptotically Bayes, as the cost, c, per observation tends to zero, relative to a large class of fully supported a priori…
This paper provides explicit solutions to the problem of estimating the arrival rate λ of a Poisson process using a Bayes sequential approach. The loss associated with estimating λ by d is assumed to be of the form (λ−d)2λ−p and the cost of observation includes both a time cost and an event cost. A discrete time approach is taken in which decisions are made at the end of time intervals having length t. Limits of the procedures as t ap…
It is observed that in selecting an alternative to the usual maximum likelihood estimator, δ0, of a multivariate normal mean, it is important to take into account prior information. Prior information in the form of a prior mean and a prior covariance matrix is considered. A generalized Bayes estimator is developed which is significantly better than δ0 if this prior information is correct and yet is very robust with respect to misspecification of the prior…
In this note, we give a lemma which shows that the expected squared difference between the Bayes estimator with a Dirichlet process prior and the Kaplan-Meier product limit (PL) estimator for a survival function based on censored data is O(n−2). This lemma, together with already proven pointwise consistency properties of the Bayes estimator, is used to establish two properties of the PL estimator; namely, the mean square consistency of the PL estimator with rate O(n−1)…
Let τ be a prior distribution over the parameter space Θ for a given parametric model Pθ,θ∈Θ. For the sample space X (over which Pθ's are probability measures) belonging to a general class of topological spaces, which include the usual Euclidean spaces, it is shown that this parametric Bayes model can be approximated by a nonparametric Bayes model of the form of a mixture of Dirichlet processes prior, so that (i) the…
Let Sp×p have a Wishart distribution with scale matrix Σ and k degrees of freedom. Estimators of Σ are given for each of the loss functions L1(Σ^,Σ)=tr(Σ^Σ−1)−logdet(Σ^Σ−1)−p and L2(Σ^,Σ)=tr(Σ^Σ−1−I)2. The obvious estimators of Σ are the scalar multiples of S, i.e., aS where…