Saturday, January 21, 2012

Prior, posterior, likelihood, MAP, ML

Let be a parameter and be the observation or data.


The posterior density is the probability density of given the observation, that is, .


The a priori (or prior) density is the probability density of before any observations are made, that is, .


The likelihood density is the probability of observed data, given the model parameters, that is, .


Using Bayes' rule, it is easy to see that




or, posterior prior likelihood.


When learning an unknown parameter from data, two commonly used estimation methods are as follows.


1. Maximum a posteriori (MAP) estimate




2. Maximum likelihood (ML) estimate



Notice that under a uniform prior, MAP and ML are identical.

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