Bayesian Linear Inference Theory And Classification Identification Method For Multiple Populations In The Modern Economics And Management

This paper mainly deals with the multivariate Bayesian inference theory used in the modern economical and management science. This includes the Bayesian inference theory about three important kinds of linear models, including the single equation model, multiple equation model system and VAR(p) predictive model, and their application in economic forecasting and quality control, and also the design for the Bayesian classification identification method among multiple populations.First, the two kinds of the parameters' prior distributions are considered. One is the diffusion prior distribution of the location parameter, scale parameter and location-scale parameter established in terms of Radon-Nikodym theorem and Harr invariant measure. The other is the conjugate prior ones of the parameter σ in the normal distribution N(u0,σ2), (u,σ2) in the normal distribution N(u,σ2), the precision matrix ∑-1 in the multivariate normal distribution Nm(u0,∑) and the parameter (u,∑) in the multivariate normal distribution Nm(u,∑).On the strength of the square loss function, this part also defines the vector loss function and matrix loss function, and discusses the Bayesian risk decision solutions about random vector parameter and random matrix parameter under these loss functions respectively.Secondly, the Bayesian inference theory about single equation model is explored. In this part, five main problems are discussed. (1) The Bayesian estimation theory about the model' parameters, including their posterior distribution deduction. (2) The Bayesian analysis about the coefficient vector when the design matrix is singular. (3) How to design the Bayesian test method about the parameter's linear hypothesis according to the relationship between the multivariate t distribution and F distribution. (4) The Bayesian diagnosis and unit root test method about the random error series. (5) The Bayesian mean value quality control chart when the variance is known and the mean value -standard error control chart when the variance is unknown.At the same time, the multiple equation model system is also examined. This part mainly solves three problems. (1) The posterior distribution of the coefficient matrix, the precision matrix and covariance matrix, and their Bayesian estimation under the matrix normal - Wishart conjugate prior distribution. (2) The deduction of the predictive distribution, proved to be Matrix t distribution. (3) The designs of Bayesianmultivariate mean value control charts in terms of the relationship between the multivariate Wishart distribution and x2 distribution , the Bayesian process capability index and its confidence lower limi.Furthermore, the Bayesian inference theory about unrestricted and restricted VAR(p) model under the parameter's prior distributions is explored. The structure of Minnesota conjugate prior distribution, its hyper-parameters and determination, and the Bayesian theory about VAR(p) model under the special conjugate prior distribution are all analyzed in detail.Finally, a new kind of methods on how to classify a sample into one of the several known populations in terms of posterior probability ratio established by the sample's predictive density functions when the unknown parameters' prior distributions are diffuse prior and Minnesota prior or normal-inverted Wishart distribution. The method doesn't require the consistency of each population's covariance.