Only Normal Distributions Have Linear Posterior Expectations in Linear Regression
研究了贝叶斯线性回归中，后验期望为Y的线性函数这一条件，证明这要求β和ε都服从正态分布，说明在大多数实际应用中只有正态分布具有此性质。
Abstract 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. Key Words: Linear modelLinear posterior expectationPosterior expectationConjugate priorNormal distributionCharacterization