On Rates of Convergence for Bayesian Density Estimation

Abstract.  We consider the problem of estimating a compactly supported density taking a Bayesian nonparametric approach. We define a Dirichlet mixture prior that, while selecting piecewise constant densities, has full support on the Hellinger metric space of all commonly dominated probability measures on a known bounded interval. We derive pointwise rates of convergence for the posterior expected density by studying the speed at which the posterior mass accumulates on shrinking Hellinger neighbourhoods of the sampling density. If the data are sampled from a strictly positive, α -Hölderian density, with α  ∈ ( 0,1] , then the optimal convergence rate n^{− α / (2 α +1)} is obtained up to a logarithmic factor. Smoothing histograms by polygons, a continuous piecewise linear estimator is obtained that for twice continuously differentiable, strictly positive densities satisfying boundary conditions attains a rate comparable up to a logarithmic factor to the convergence rate n ^−4/5 for integrated mean squared error of kernel type density estimators.     

我国建筑消防现状

近年来 ,中国城市建筑火灾发生率呈上升趋势 ,仅 1999年 ,全国共发生建筑火灾 18万起 ,比前一年增长 4 1% ;死亡 2 70 0多人 ,比前一年增长 15 % ,直接经济损失多达 14亿元。     

Subjective probabilities and utility with event-dependent preferences

This article generalizes Savage's theory to include event-dependent preferences. The state space is partitioned into finitely many events. The induced preferences over consequences are assumed independent of the underlying states within, but not across, these events. This results in an additively separable representation of preferences over acts. The dependence of the preference relation over consequences on the events is represented by event-dependent mappings of the set of consequences onto itself. Given these mappings, the preferences on acts are represented by the expectation of event-dependent utilities on the consequences with respect to unique subjective probabilities on the states.Helpful discussions with David Schmeidler are gratefully acknowledged.     

动机与第二语言习得

动机是学习第二语言成就大小的重要因素之一。很多研究实验已经证明动机在人类认知过程中起非常关键性的作用。本文打算对动机，融合性动机和功用性动机及其与第二语言习得的关系做一些探讨     

Improving on truncated linear estimates of exponential and gamma scale parameters

We consider the problem of estimating the scale parameter of an exponential or a gamma distribution under squared error loss when the scale parameter θ is known to be greater than some fixed value θ₀. Natural estimators in this setting include truncated linear functions of the sufficient statistic. Such estimators are typically inadmissible, but explicit improvements seem difficult to find. Some are presented here. A particularly interesting finding is that estimators which are admissible in the untruncated problem which take values only in the interior of the truncated parameter space are found to be inadmissible for the truncated problem.     

存在战略购买需求的定价、供给、机制选择和配给研究

本文研究存在战略购买需求的易逝资产销售策略问题。垄断厂商基于利润最大化目标确定易逝资产定价、供给、机制选择和配给策略,战略消费者通过锚定预期价格安排战略购买时机。不同于通常基于效用理论研究定价的思路,本文首先基于锚定效应和跨期价格均衡思想探寻不同战略等待购买规模的市场预期需求曲线和动静态定价区域;其次在众多预期需求曲线中寻找市场有效定价前沿（即有效预期需求曲线）;再次在利润曲面上找出与有效定价前沿对应的容量扩展线（即最大利润曲线）;最后沿容量扩展线和有效定价前沿搜寻最大期望利润及相应策略。研究表明,消费者保留价异质和需求不确定性是动态定价和战略购买存在的根本原因;市场在不同战略等待购买规模状态拥有不同预期需求曲线,最大战略等待购买规模状态预期需求曲线是市场有效定价前沿。动静态定价机制各有其所适用的容量和价格空间,消费者保留价水平和战略消费者规模决定动态定价空间大小,随机需求分布差异只影响动态定价空间形状（即影响需求弹性）。在跨期价格均衡区域内,提价和扩容都会加剧消费者战略购买程度,供给越大定价往往越低。战略购买不仅会降低厂商供给、定价和利润水平,改变不同类型消费者之间高低价购买机会,甚至还可能影响定价机制选择和配给策略。压缩过度供给和虚高价格空间可降低战略购买导致的利润损失。本文研究结果可为考虑消费者行为的需求价格理论研究和运营管理实践提供参考。     

The calibration of P-values,posterior Bayes factors and the AIC from the posterior distribution of the likelihood

The posterior distribution of the likelihood is used to interpret the evidential meaning of P-values, posterior Bayes factors and Akaike's information criterion when comparing point null hypotheses with composite alternatives. Asymptotic arguments lead to simple re-calibrations of these criteria in terms of posterior tail probabilities of the likelihood ratio. (Prior) Bayes factors cannot be calibrated in this way as they are model-specific.     

Three pseudo-paradoxes in ‘quantum’ decision theory: Apparent effects of observation on probability and utility

In quantum domains, the measurement (or observation) of one of a pair of complementary variables introduces an unavoidable uncertainty in the value of that variable's complement. Such uncertainties are negligible in Newtonian worlds, where observations can be made without appreciably disturbing the observed system. Hence, one would not expect that an observation of a non-quantum probabilistic outcome could affect a probability distribution over subsequently possible states, in a way that would conflict with classical probability calculations. This paper examines three problems in which observations appear to affect the probabilities and expected utilities of subsequent outcomes, in ways which may appear paradoxical. Deeper analysis of these problems reveals that the anomalies arise, not from paradox, but rather from faulty inferences drawn from the observations themselves. Thus the notion of quantum decision theory is disparaged.     

试析《隆中对》兼论关羽之失

诸葛亮未出茅庐便知三分天下,在刘备三顾茅庐时,便提出<隆中对>,为刘备指出发展方向和治国方略.<隆中对>前后可分两大部分.前半部分劝告刘备不要与已拥有百万之众,挟天子以令诸侯的曹操争锋;要与占据江东已历三世,国险而民附,贤能为之用的孙权结盟,不要为敌.指出刘备的发展方向是占有荆州和益州.后半部分为刘备指出五点治国方略和战略部署:(一)联吴抗曹;(二)跨有荆、益;(三)西和西戎,南抚夷越;(四)内修政理;(五)两路出军,一向宛、洛,一出秦川.这五大治国方略和战略部署,由于关羽大意失荆州而受到严重打击,使"跨有荆、益"和"两路出军"的计划落空,吴蜀联盟中断,但诸葛亮在"联吴抗曹"、"西和诸戎,南抚夷越"等方面仍取得了辉煌的成就,对开发四川、云南、贵州作出了突出的贡献.诸葛亮是三国时期一流的政治家、军事家和战略家.     

Regression models using the LINEX loss to predict lower bounds for the number of points for approximating planar contour shapes

Researchers in statistical shape analysis often analyze outlines of objects. Even though these contours are infinite-dimensional in theory, they must be discretized in practice. When discretizing, it is important to reduce the number of sampling points considerably to reduce computational costs, but to not use too few points so as to result in too much approximation error. Unfortunately, determining the minimum number of points needed to achieve sufficiently approximate the contours is computationally expensive. In this paper, we fit regression models to predict these lower bounds using characteristics of the contours that are computationally cheap as predictor variables. However, least squares regression is inadequate for this task because it treats overestimation and underestimation equally, but underestimation of lower bounds is far more serious. Instead, to fit the models, we use the LINEX loss function, which allows us to penalize underestimation at an exponential rate while penalizing overestimation only linearly. We present a novel approach to select the shape parameter of the loss function and tools for analyzing how well the model fits the data. Through validation methods, we show that the LINEX models work well for reducing the underestimation for the lower bounds.