Multivariate normal distribution posterior

In the case that objects of all classes obey a multivariate normal distribution, an optimal classification rule can be based on Bayes theorem. The assignment of a sample, x, characterized by p features to a class j of all classes g is based on maximizing the posterior probability ... 

We suggest that when the posterior has multiple nodes a similar strategy be used. At each mode, we find the multivariate normal distribution that matches the curvature of the target at that mode. The candidate density should be a mixture of the multivariate Student s t distributions with low degrees of fieedom where each component of the mixture corresponds to the multivariate normal found for that node. Using this mixture density as the independent candidate density will give fast convergence to the posterior. 

In other words, the product of the likelihood of the observed data [e.g., Eq. (3.12)] and the prior distribution is proportional to the probability of the posterior distribution, or the distribution of the model parameters taking into account the observed data and prior knowledge. Suppose the prior distribution is p-dimensional multivariate normal with mean 0 and variance O, then the prior can be written as... 

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... 

However, our multivariate normal prior does not have to be made up from independent components. In Chapter 4 we saw that the posterior distribution for this case will be multivariate rtorma/(bi, Vi) where the mean vector and covariance matrix are found by the simple updating rules we developed in Chapter 4. In this case they give... 

The prior density of fij is normal T, ), and given the parent node g. and the coparent node the observations yij,..., yujj are a random sample from a normal nj + Xj/3,a ). Thus zj = yj - Xj/3 is a multivariate normal random vector with mean vector Hj and covariance matrix o l. The posterior distribution of Hj is normally distributed with variance Wj and mean Uj where... 

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... 

The true posterior is given by the binomial likelihood times the prior. We can find a random sample from the true posterior by using the Metropolis-Hastings algorithm using a multivariate Student s t with low degrees of freedom that corresponds to the normal approximation to the posterior as the independent candidate distribution. 

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