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犹疑之矩：先验与样本冲突下的后验方差AI 解读

For a suitably thick-tailed prior density and a sufficiently large observation, the sample information dominates the prior information. Similarly, a thick-tailed sampling distribution can be dominated by the prior if the observation is sufficiently large. We show, in the case where the dominating density is normal, that less extreme observations produce very large posterior variances.

矩阵变量分布理论：记号考量与贝叶斯应用AI 解读

We introduce and justify a convenient notation for certain matrix-variate distributions which, by its emphasis on the important underlying parameters, and the theory on which it is based, eases greatly the task of manipulating such distributions. Important examples include the matrix-variate normal, t, F and beta, and the Wishart and inverse Wishart distributions. The theory is applied to compound matrix distributions and to Bayesian prediction in the multivariate linear…

封闭种群多次重捕抽样的贝叶斯分析AI 解读

This paper considers from a Bayesian viewpoint inferences about the size of a closed animal population from data obtained by a multiple-recapture sampling scheme. The method developed enables prior information about the population size and the catch probabilities to be utilized to produce considerable improvements in certain cases on ordinary maximum likelihood methods. Several ways of expressing such prior information are explored and a practical example of the uses of…

摘要暂缺

Partial prior knowledge is quantified by an interval $I(L, U)$ of $\sigma$-finite prior measures $Q$ satisfying $L(A) \leq Q(A) \leq U(A)$ for all measurable sets $A$, and is interpreted as acceptance of a family of bets. The concept of conditional probability distributions is generalized to that of conditional measures, and Bayes theorem is extended to accommodate unbounded priors. According to Bayes theorem, the interval $I(L, U)$ of prior measures is transformed upon…

可靠性分析的贝叶斯非参数方法AI 解读

It is suggested that problems in a reliability context may be handled by a Bayesian nonparametric approach. A stochastic process is defined whose sample paths may be assumed to be increasing hazard rates by properly choosing the parameter functions of the process. The posterior distribution of the hazard rates is derived for both exact and censored data. Bayes estimates of hazard rates and $\operatorname{cdf's}$ are found under squared error type loss functions. Some…

未知均匀分布下的极小极大停止规则AI 解读

An invariance-based method of obtaining the minimax stopping rule when sampling from an unknown uniform distribution is presented and applied to two problems, maximizing the probability of selecting the smallest observation and minimizing the expected quantile of the observation selected. In the first problem, the minimax rules use only the relative ranks of the observations; in the second problem they are shown to achieve asymptotic risk (3 + 2 √2)1/√2/n ≈ 3.4780/n, which…

稳健的序贯近似贝叶斯估计AI 解读

SUMMARY An approximation to the sequential updating of the distribution of location parameters of a linear time series model is developed for non-normal observations. The behaviour of the resulting non-linear recursive filtering algorithm is examined and shown to have certain desirable properties for a variety of non-normal error distributions. Illustrative examples are given and relationships with previous work on robustness and sequential estimation are mentioned.

经验贝叶斯分布函数渐近最优性的一则注记AI 解读

This paper establishes the asymptotic optimality (in the sense of Robbins) of the empirical Bayes distribution function created from the Bayes rule relative to the Dirichlet process prior with unknown parameter $\alpha(.)$. It will follow that the same result applies to the estimation of the mean of a distribution function.

贝叶斯自助法AI 解读

The Bayesian bootstrap is the Bayesian analogue of the bootstrap. Instead of simulating the sampling distribution of a statistic estimating a parameter, the Bayesian bootstrap simulates the posterior distribution of the parameter; operationally and inferentially the methods are quite similar. Because both methods of drawing inferences are based on somewhat peculiar model assumptions and the resulting inferences are generally sensitive to these assumptions, neither method…

随机删失数据下非参数生存曲线估计量的强一致收敛AI 解读

Let $X_1, \cdots, X_n$ be i.i.d. $P(X > u) = F(u)$ and $Y_1, \cdots, Y_n$ be i.i.d. $P(Y > u) = G(u)$, where both $F$ and $G$ are unknown continuous distributions. For $i = 1, \cdots, n$ set $\delta_i = 1$ if $X_i \leq Y_i$ and 0 if $X_i > Y_i$ and $Z_i = \min \{X_i, Y_i\}$. One way to estimate $F$ from the observations $(Z_i, \delta_i) i = 1, \cdots, n$ is by means of the product limit (PL) estimator, $F^\ast_n$ (Kaplan-Meier, [8]). In this paper it is shown that $F^\ast_n$…

利用删失生存数据序贯比较两个指数分布AI 解读

A sequential stopping procedure is proposed for a clinical trial which compares two exponential distributions using censored survival data. The procedure extends an approach first developed by Anscombe (1963) for comparing two treatments with normally distributed and instantaneously observed responses. Patients are assumed to arrive at random and are assigned alternately to one of the two treatments. An optimal stopping region for the ratio of failure rates of the two…

平均寿命贝叶斯估计量的预后验分析AI 解读

Abstract : In considering life test experiments from a behavioristic Bayes point of view, we may speculate on the effect of model misspecification on the posterior mean under alternative stopping rules. Of course, under the life distribution model, we expect the mean of the posterior to equal the mean of the prior. However, if we think the model may have been misspecified, we would no longer expect this equality. We show that for Bayes estimators of mean life under the…

线性纯死亡过程的初始规模估计AI 解读

Consider a linear pure death process in which the common death rate and the initial size n of the population are both unknown. An estimate of n is required, based on the observed times of the first j deaths, i.e. with censored sampling. It is shown that the only reliable non-Bayesian method is to use interval estimates, for all the standard point estimation techniques are liable to fail. Appropriate interval estimators are derived and discussed. The corresponding truncated…

利用秩计算变点的后验概率AI 解读

Simple approximate methods of analysis are given for change-point models using ranked observations. The models can be applied to continuous observations, providing a robust analysis in this case. For ranked observations, the models provide analyses for association. Numerical examples, involving continuous and ranked observations, are given.

犹疑的一刻：先验与数据冲突中的后验方差AI 解读

We consider inference from a single observation X about a location parameter 0. That is, given 0 = 0, the density function of X has the form f (x -0) for some known function f. The prior distribution for 0 has density g(6). It is well known that if f and g are both normal densities then the posterior density is normal with a variance independent of x and less than the variance of either f or g. Very different behaviour can occur if either component is not normal. Dawid…

正态化变换的贝叶斯方法AI 解读

The analysis of transformation of observations in the linear model with normal errors proposed by Box & Cox (1964) is considered. A different choice of noninformative unnormed prior is advocated, which is not outcome dependent. This new selection of prior leads to a formal identity between likelihood and Bayesian inference, both for the estimation of the best transformation to normality and for the presence of homoscedasticity and additivity under this transformation.…

We introduce and justify a convenient notation for certain matrix-variate distributions which, by its emphasis on the important underlying parameters, and the theory on which it is based, eases greatly the task of manipulating such distributions. Important examples include the matrix-variate normal, t, F and beta, and the Wishart and inverse Wishart distributions. The theory is applied to compound matrix distributions and to Bayesian prediction in the multivariate linear…

封闭总体多次重捕抽样的贝叶斯分析AI 解读

This paper considers from a Bayesian viewpoint inferences about the size of a closed animal population from data obtained by a multiple-recapture sampling scheme. The method developed enables prior information about the population size and the catch probabilities to be utilized to produce considerable improvements in certain cases on ordinary maximum likelihood methods. Several ways of expressing such prior information are explored and a practical example of the uses of…

单参数指数族的贝叶斯序贯估计AI 解读

The Bayesian sequential estimation problem for one-parameter exponential families is considered using loss related to the Fisher information and linear sampling cost. Tractible expressions for the Bayes estimator and the posterior expected loss are found, and the myopic, or one-step-ahead, stopping rule is defined. Sufficient conditions are given for optimality of the myopic procedure, and the myopic procedure is shown to be asymptotically optimal in all cases considered.

线性回归中线性后验期望对正态分布的刻画AI 解读

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.