Consider an estimate $\theta^*$ of a parameter $\theta$ based on repeated observations from a family of densities $f_\theta$ evaluated by the Kullback–Leibler loss function $K(\theta, \theta^*) = \int \log(f_\theta/f_{\theta^*})f_\theta$. The maximum likelihood prior density, if it exists, is the density for which the corresponding Bayes estimate is asymptotically negligibly different from the maximum likelihood estimate. The Bayes estimate corresponding to the maximum…
Suppose that $X_{\sigma} | \mathbf{\theta} \sim N(\mathbf{\theta}, \sigma^2)$ and that $\sigma \to 0$. For which prior distributions on $\mathbf{\theta}$ is the posterior distribution of $\mathbf{\theta}$ given $X_{\sigma}$ asymptotically $N(X_{\sigma}, \sigma^2)$ when in fact $X_{\sigma} \sim N(\theta_0, \sigma^2)$? It is well known that the stated convergence occurs when $\mathbf{\theta}$ has a prior density that is positive and continuous at $\theta_0$. It turns out that…
Observations $X_i$ are uncorrelated with means $\theta_i, i = 1,\cdots, n$, and variances 1. The linear estimators $\widehat{\theta} = T\mathbf{X}$, for some $n \times n$ matrix $T$, are widely used in smoothing problems, where it is assumed that neighbouring parameter values are similar. The smoothness assumption is violated in change point problems, where neighbouring parameter values are equal, except at some unspecified change points where there are jumps of unknown size…
A sequence of observations undergoes sudden changes at unknown times. We model the process by supposing that there is an underlying sequence of parameters partitioned into contiguous blocks of equal parameter values; the beginning of each block is said to be a change point. Observations are then assumed to be independent in different blocks given the sequence of parameters. In a Bayesian analysis it is necessary to give probability distributions to both the change points and…
Product partition models assume that observations in different components of a random partition of the data are independent. If the probability distribution of random partitions is in a certain product form prior to making the observations, it is also in product form given the observations. The product model thus provides a convenient machinery for allowing the data to weight the partitions likely to hold; and inference about particular future observations may then be made…
Observations $y_i$ are made at points $x_i$ according to the model $y_i = F(x_i) + e_i$, where the $e_i$ are independent normals with constant variance. In order to decide whether or not $F(x)$ is constant, a likelihood ratio test is constructed, comparing $F(x) \equiv \mu$ with $F(x) = \mu + Z(x)$, where $Z(x)$ is a Brownian motion. The ratio of error variance to Brownian motion variance is chosen to maximize the likelihood, and the resulting maximum likelihood statistic $B$…
If a density in two dimensions has a convex contour containing probability α, the contour may be estimated from a sample by finding the convex polygon of smallest area containing a proportion α of the sample points. An algorithm for finding a particular contour is given that takes O(n 2) space and O(n 3) time for n sample points.
If a density in two dimensions has a convex contour containing probability α, the contour may be estimated from a sample by finding the convex polygon of smallest area containing a proportion α of the sample points. An algorithm for finding a particular contour is given that takes O(n 2) space and O(n 3) time for n sample points.
The dip test measures multimodality in a sample by the maximum difference, over all sample points, between the empirical distribution function, and the unimodal distribution function that minimizes that maximum difference. The uniform distribution is the asymptotically least favorable unimodal distribution, and the distribution of the test statistic is determined asymptotically and empirically when sampling from the uniform.
High-density clusters are defined on a population with density f in r dimensions to be the maximal connected sets of form {x | f(x) ≥ c}. Single-linkage clustering is evaluated for consistency in detecting such high-density clusters—other standard hierarchical techniques, such as average and complete linkage, are hopelessly inconsistent for these clusters. The asymptotic consistency of single linkage closely depends on the percolation problem of Broadbent and Hammersley—if…
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…
A set of observations is partitioned into $k$ clusters by optimizing a clustering criterion $W$. The asymptotic distribution of this clustering criterion may be determined simply in certain cases where the optimal sample partition differs negligibly from the optimal population partition. Detailed proofs are given in the one-dimensional case when the clustering criterion to be minimized is within cluster sum of squares. The asymptotic distributions are used to compute…
The empirical distribution function $P_n$ converges with probability 1 to a true distribution $P$ in $R^k$, uniformly over measurable convex sets, if and only if $P$ is a countable mixture of distributions, each of which is carried by a flat and gives zero probability to the relative boundaries of convex sets included in the flat.
Clustering algorithms are now in widespread use for sorting heterogeneous data into homogeneous blocks. If the data consist of a number of variables taking values over a number of cases, these algorithms may be used either to construct clusters of variables (using, say, correlation as a measure of distance between variables) or clusters of cases. This article presents a model, and a technique, for clustering cases and variables simultaneously. The principal advantage in this…
Alan Forsythe · J. A. Hartigan·刊期 1970-12 · 卷 57（3）·Biometrika已编入正刊
重复子样本计算生成的置信区间效率AI 解读
Confidence intervals for a mean may be obtained from n observations continuously and symmetrically distributed about the mean, by computing the means of all 2n − 1 subsets, or subsamples, of the observations. The true mean lies in each of the intervals between the ordered subsample means with probability 2−n. In order to reduce the 2n − 1 computations of means, a random subset of subsample means may be used, or a ‘ balanced’ subset may be selected satisfying certain group…
Summary A method of linear prediction is proposed, which follows the Bayesian scheme in combining prior and present information, but which uses only the first two moments of the distribution of parameters and observations, rather than the complete probability model required of Bayesian techniques.
Summary The likelihood and invariance principles, when combined, imply the probability principle, which allows only very restricted inferences about a parameter, based on an observation. It is suggested, in explanation, that the likelihood and invariance principles are valid under two different viewpoints of the model, and that these viewpoints are incompatible.
J. A. HARTIGAN·刊期 1967 · 卷 54（1-2）·Biometrika已编入正刊
不等式约束拟合中残差平方和的分布AI 解读
SUMMARY Consider a model for n observations, Yi = ,ti + o'6i (i = 1, ..., n), where the 6i are n independent unit normal variables and the ,ui are restrained by p linear inequalities. The maximum-likelihood estimate of {1tij is {Yoi} minimizing (Y - Y0)2 subject to the linear inequalities; the computation of {Y0i} requires quadratic programming. This paper is concerned with the distribution of the residual sum of squares (Y - Y0)2, which it is natural to use for inference…
Summary Welch and Peers have shown that it is possible to determine a prior distribution for a one-dimensional parameter by requiring that the associated one-sided Bayes intervals are asymptotically confidence intervals. It is shown in this note that their procedure does not apply if certain natural two-sided Bayes intervals are used.
Consider an estimate θ∗ of a parameter θ based on repeated observations from a family of densities fθ evaluated by the Kullback–Leibler loss function K(θ,θ∗)=∫log(fθ/fθ∗)fθ. The maximum likelihood prior density, if it exists, is the density for which the corresponding Bayes estimate is asymptotically negligibly different from the maximum likelihood estimate. The Bayes estimate corresponding to the maximum…
Suppose that Xσ∣θ∼N(θ,σ2) and that σ→0. For which prior distributions on θ is the posterior distribution of θ given Xσ asymptotically N(Xσ,σ2) when in fact Xσ∼N(θ0,σ2)? It is well known that the stated convergence occurs when θ has a prior density that is positive and continuous at θ0. It turns out that…
Observations Xi are uncorrelated with means θi,i=1,⋯,n, and variances 1. The linear estimators θ=TX, for some n×n matrix T, are widely used in smoothing problems, where it is assumed that neighbouring parameter values are similar. The smoothness assumption is violated in change point problems, where neighbouring parameter values are equal, except at some unspecified change points where there are jumps of unknown size…
Observations yi are made at points xi according to the model yi=F(xi)+ei, where the ei are independent normals with constant variance. In order to decide whether or not F(x) is constant, a likelihood ratio test is constructed, comparing F(x)≡μ with F(x)=μ+Z(x), where Z(x) is a Brownian motion. The ratio of error variance to Brownian motion variance is chosen to maximize the likelihood, and the resulting maximum likelihood statistic B…
Partial prior knowledge is quantified by an interval I(L,U) of σ-finite prior measures Q satisfying L(A)≤Q(A)≤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…
A set of observations is partitioned into k clusters by optimizing a clustering criterion W. The asymptotic distribution of this clustering criterion may be determined simply in certain cases where the optimal sample partition differs negligibly from the optimal population partition. Detailed proofs are given in the one-dimensional case when the clustering criterion to be minimized is within cluster sum of squares. The asymptotic distributions are used to compute…
The empirical distribution function Pn converges with probability 1 to a true distribution P in Rk, uniformly over measurable convex sets, if and only if P is a countable mixture of distributions, each of which is carried by a flat and gives zero probability to the relative boundaries of convex sets included in the flat.