多重线性回归模型系统的贝叶斯预报分析

Bayesian predictive analysis of multiple linear regression model system

指出多重线性模型系统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,前者为从矩阵正态分布,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 for linear model. According to the statistical structure of the model, it is proved that the matrix normal Wishart distribution is conjugate priori distribution of model parameters. Then, based on the Bayesian theorem, sample likelihood function and priori distribution of the model, the posteriori distribution of parameters was derived. Finally, the density function of predictive distribution of the model was rigorously inferred mathematically and the predictive distribution was proved to be a matrix t distribution. The result shows that, due to the effect of the parameters＇ priori distribution, there is essential difference between the sample predictive distribution and its original statistical distribution, the former being the matrix t distribution and the latter matrix normal distribution.