基于线性回归模型的贝叶斯方法的研究

Research on Bayesian Method Based on Linear Regression Model

线性回归模型是回归分析中比较简单且应用非常广泛的一种统计模型,它可以很好的确定变量之间的一种定量关系。但是在一些实际问题中,很多问题并不是严格的线性关系,这就需要结合历史经验来对模型做出相应的改进。针对这一问题,结合了贝叶斯理论,以多元线性回归模型和Logistic回归模型为例进行研究,使得这些没有严格线性关系的问题依然可以按照线性回归模型的思想来求解。在贝叶斯方法中,平方损失函数下,分别以无信息先验分布和联合正态分布作为参数的先验分布,得到多元线性回归模型和Logistic回归模型中参数的贝叶斯估计,以及求取了等式约束下的Logistic回归的贝叶斯参数估计,估计结果良好有效。

The linear regression model is a statistical model that is relatively simple and widely used in regression analysis.It can well determine a quantitative relationship between variables.However,in some practical problems,many problems are not strictly linear,which requires a combination of historical experience to make corresponding improvements to the model.In this paper,the Bayesian theory is combined with this problem.The multiple linear regression model and the logistic regression model are taken as examples to make these problems without strict linear relations still be solved according to the idea of linear regression model.In the Bayesian method,under the squared loss function,the prior distributions with no information prior distribution and joint normal distribution are selected as the parameters,and the Bayesian estimation of the parameters in the multiple linear regression model and the logistic regression model are obtained.The Bayesian parameter estimation of logistic regression under equality constraints is obtained,and the estimation results are effective.