多重线性回归模型的贝叶斯预报分析

Bayesian Predictive Analysis of the Multiple Linear Regression Model

多重线性回归模型的贝叶斯预报分析是贝叶斯线性模型理论的重要组成部分。通过模型系统的统计结构,证明了矩阵正态—Wishart分布为模型参数的共轭先验分布;利用贝叶斯定理,根据模型的样本似然函数和参数的先验分布推导了参数的后验分布;然后,从数学上严格推断了模型的预报分布密度函数,证明了模型预报分布为矩阵t分布。研究结果表明:由于参数先验分布的作用,样本的预报分布与其原统计分布有着本质性的差异,前者服从矩阵正态分布,而后者为矩阵t分布。

The Bayesian analysis of the multiple equation linear system is an important part of Bayesian inference theory about linear model. According to the statistical structure of the model, we first prove that the matrix Normal-Wishart distribution is its parameters' conjugate prior. Then, based on the Bayesian theorem, prior distribution and likelihood function, we infer their joint posterior distribution, which belongs to the fa-(mily) of Normal-Wishart distributions. Finally, we compute the predictive density of a future sample, whose distribution is a matrix t distribution. The result in this paper shows that, due to the effect of the parameters' prior, there is difference between the predictive distribution of the future sample and its original statistical distribution, the former being the matrix t distribution and the latter Matrix normal distribution.