Matched curvature covariance matrix

Most statistical packages will find the maximum likelihood estimator 0ml and also the matched curvature covariance matrix Vml- However they do not note that this covariance matrix does not represent the spread of the likelihood function. 

The vector of maximum likelihood estimators and their matched curvature covariance matrix gives us a place to start. We approximate the joint likelihood function by the multivariate normal 0ML, Vml) likelihood function, which has mean vector equal to 0ML, the mode of the likelihood function, and covariance matrix Vml, which matches the curvature of the likelihood function at its mode. 

Actually it is the matched curvature covariance matrix, which is the covariance matrix of a multivariate normal distribution, that matches the curvature of the likelihood function at its maximum. This matched curvature covariance matrix has no relation to the spread of the likelihood function. 

In the computational Bayesian approach, we want to draw a sample from the actual posterior, not its approximation. As we noted before, we know its shape. Our approach will be to use the Metropolis-Hastings algorithm with an independent candidate density. We want a candidate density that is as close as possible to the posterior so many candidates will be accepted. We want the candidate density to have heavier tails than the posterior, so we move around the parameter space quickly. That will let us have shorter burn-in and use less thinning. We use the maximum likelihood vector 0ml matched curvature covariance matrix Vj fz, as the... 

We approximate the likelihood function by a multivariate normal 0ML where 0ml the MLE and is the matched curvature covariance matrix that is output by the iteratively reweighted least squares. We use a multivariate nor-ma/[bo, Vo] prior for or we can use a "flat" prior if we have no prior information. The approximate posterior will be... 

The maximum likelihood and the matched curvature covariance matrix can be found by iteratively reweighted least squares. 

The Poisson regression model is an example of the generalized linear model. The maximum likelihood estimates of the coefficients of the predictors can be found by iteratively reweighted least squares. This also finds the covariance matrix of the normal distribution that matches the curvature of the likelihood... 

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... 

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