贝叶斯统计-多变量模型_Multiparameter Models

Bayesian Methods Computation Lecture 3 Multi Parameter Models Dr Ke Deng Center for statistical Science Tsinghua University Beijing 邓 柯 清华 学统计学研究中 kdeng 1 Multi Parameter Models 2 v Multivariate normal with unknown mean vector and covariance matrix with unknown mean vector and known covariance matrix with known mean vector and unknown covariance matrix v Multinomial v Univariate normal with unknown mean variance Nuisance Parameters Marginal Posterior Distribution 3 Univariate normal with unknown mean variance Parameter of interest Nuisance parameter Joint posterior Marginal posterior Prior General framework of Bayesian inference for multiparameter models Averaging over the nuisance parameter Conditional posteriorMarginal posterior Univariate Normal with a Noninformative Prior 4 Univariate normal with unknown mean variance Non informative prior prior independence of location and scale parameters Jeffreys s principle Joint posterior Univariate Normal with a Noninformative Prior 5 Univariate normal with unknown mean variance Non informative prior prior independence of location and scale parameters Jeffreys s principle Joint posterior Conditional posterior Univariate Normal with a Noninformative Prior 6 Marginal posterior Joint posterior Conditional posterior Univariate Normal with a Noninformative Prior 7 Joint posterior Marginal posterior Conditional posterior Univariate Normal with a Conjugate Prior Conjugate prior Univariate normal with unknown mean variance Marginal conditional posterior Joint posterior 8 Random samples via simulation Univariate Normal with a Conjugate Prior Conjugate prior Univariate normal with unknown mean variance Joint posterior 9 Marginal posterior Multinomial Model with a Conjugate Prior Conjugate prior Multinomial model for categorical data Joint posterior Dirichletdistribution with as hyper parameter 10 Dirichletdistribution with y as hyper parameter Multivariate Normal with Known Variance Conjugate Prior Multivariate normal Joint posterior 9 Conjugate prior Likelihood Multivariate Normal with Unknown Mean Variance Joint posterior 9 Conjugate prior Likelihood Normal Inverse Wishart Normal Inverse Wishart 13 Multivariate Normal with Unknown Mean Variance Conjugate 14 Multivariate Normal with Unknown Mean Variance Conjugate Multivariate Normal with Unknown Mean Variance Conjugate Joint posterior 15 Conjugate prior Likelihood Normal Inverse Wishart Normal Inverse Wishart Meaning of the 4 hyper parameters Prior mean Prior covariance matrix of additional date for prior mean of additional date for prior covariance matrix Multivariate Normal with Unknown Mean Variance Non Informative 16 multivariate Jeffreysprior Each ofthe correlations in has marginally a uniformpriordistribution The joint distribution is not uniform however because of the constraint that the correlation matrixb