Multivariate normal conjugate prior

This gives the shape of the approximate posterior for any prior. However, for most priors, numerical integration would be required to obtain the scale factor needed to make it a density. However, there are two types of priors where we can find the approximate posterior without integration. These are where we use independent flat priors, and when we use multivariate normal conjugate priors. We will restrict ourselves to choosing our prior from one of these two types. 

Multivariate normal conjugate priors. Suppose we use a multivariate nor-ma/(bo, Vo) prior density for the parameter vector )3. For instance, we could let the prior for the component be normal bi, s ) and let the prior for each component be independent of each other. Then... 

When we have eensored survival times data, and we relate the linear predictor to the hazard function we have the proportional hazards model. The function BayesCPH draws a random sample from the posterior distribution for the proportional hazards model. First, the function finds an approximate normal likelihood function for the proportional hazards model. The (multivariate) normal likelihood matches the mean to the maximum likelihood estimator found using iteratively reweighted least squares. Details of this are found in Myers et al. (2002) and Jennrich (1995). The covariance matrix is found that matches the curvature of the likelihood function at its maximum. The approximate normal posterior by applying the usual normal updating formulas with a normal conjugate prior. If we used this as the candidate distribution, it may be that the tails of true posterior are heavier than the candidate distribution. This would mean that the accepted values would not be a sample from the true posterior because the tails would not be adequately represented. Assuming that y is the Poisson censored response vector, time is time, and x is a vector of covariates then... 

When we have a random sample of multivariate normal fx) observations with known covariance matrix S, the conjugate prior for /lx is multivariate nor-ma/(mo, Vo). The posterior is multivariate norma/(mi, Vi) where... 

When we have n independent observations from the normal linear regression model where the observations all have the same known variance, the conjugate prior distribution for the regression coefficient vector /3 is multivariate normal(bo, Vq). The posterior distribution of /3 will be multivariate nor-mal y>i, Vi), where... 

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