Bayesian Methods & Computation
Lecture 3
Multi-Parameter Models
Dr. Ke Deng
Center for statistical Science
Tsinghua University, Beijing
邓柯
清华⼤学统计学研究中⼼
kdeng@
Multi-Parameter Models
v
Multivariate normal:
Ø
with unknown mean vector and covariance matrix Øwith unknown mean vector and known covariance matrix
Ø
with known mean vector and unknown covariance matrix
v Multinomial:
v Univariate normal with unknown mean & variance:
prior independence of location and scale parameters Joint posterior:
prior independence of location and scale parameters Joint posterior:
Conditional posterior:
Joint posterior: Conditional posterior:
Joint posterior: Conditional posterior:
Joint posterior: Marginal & conditional posterior:
Univariate Joint posterior:
Multinomial Model with a Conjugate Prior Conjugate prior:
Multinomial model for categorical data:Joint posterior:Dirichlet distribution with αas hyper-parameter
Dirichlet distribution with α+y as hyper-parameter
Likelihood:
Conjugate prior: Joint posterior:
Multivariate Normal with Unknown Mean & Variance
Joint posterior:Conjugate prior:Likelihood:
Normal-Inverse-Wishart Normal-Inverse-Wishart
Multivariate Normal with Unknown Mean & Variance (Conjugate)
Multivariate Normal with Unknown Mean & Variance (Conjugate)
Normal-Inverse-Wishart
Multivariate Normal with Unknown Mean & Variance (Non-Informative)multivariate Jeffreys prior
ØEach of the correlations in Σhas,marginally,a uniform prior distribution.ØThe joint distribution is not uniform,however,because of the constraint that the correlation matrix be positive definite.
Reference
•Gelman, A., Carlin, J.B., Stern, H.S. and Rubin, D.B. (2003). Bayesian Data Analysis (3rd ed), Chapman & Hall: London. (Textbook) –Chapter 3。