Bayesian Inference from the Numerical Posterior

In some cases we have a formula for the exact posterior. In other cases we only know the shape of the posterior using Bayes theorem. The posterior is proportional to prior times likelihood. In those cases we can find the posterior density numerically by dividing through by the scale factor needed to make the integral of the posterior over 

Understanding Computational Bayesian Statistics. By William M. Bolsiad Copyright (c) 2010 John Wiley Sons, Inc. 

The first type of inference is where a single statistic is calculated from the sample data and is used to estimate the unknown parameter. From the Bayesian perspective, point estimation is choosing a value to summarize the posterior distribution. The most important summary number of a distribution is its location. The posterior mean and the posterior median are good measures of location and hence would be good Bayesian estimators of the parameter. Generally we will use the posterior mean as our Bayesian estimator since it minimizes the posterior mean squared error 

The posterior mean is the first moment (or balance point) of the posterior distribution. We find it by 

The posterior median could also be used as a Bayesian estimator since it minimizes the posterior mean absolute deviation 

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