Computational Bayesian Approach to the Logistic Regression Model

2 COMPUTATIONAL BAYESIAN APPROACH TO THE LOGISTIC REGRESSION MODEL 

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. 

The Joint posterior is proportional to the Joint prior times the likelihood. 

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. 

Independent flat priors for O,, I3p. Sometimes we have no prior 

---