Both likelihood inference and Bayesian inference arise from a surface defined on the inference universe which is the Cartesian product of the parameter space and the sample space. Likelihood inference uses the sampling surface which is a probability distribution in the sampling dimension only. Bayesian inference uses the joint probability distribution defined on the inference universe The likelihood function and the Bayesian posterior distribution come from cutting the respective surfaces with a (hyper)plane parallel to the parameter space and through the observed sample values. Unlike the likelihood function, the posterior distribution always will be a probability distribution. This is responsible for the different choices of estimators, and the different way the two approaches have of dealing with nuisance parameters. In this paper we present a graphical approach for teaching the difference between the two approaches.
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