Maximum Likelihood, Maximum A Posteriori and Bayesian

Definitions

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   : random variables, usually state and observations respectively
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   : probability distribution function for a random variable 
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  .

Bayes Rule

Combining the rules for conditional probabilities 

Examples:

1. Likelihood - coin flipping

Parameter Estimation

Bayes rule is often used (in robotics) to do parameter estimation. There are two ways you can do this:

1. Maximum Likelihood : find 
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    for which the likelihood is maximum.
2. Maximum a Posteriori : find 
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    for which the posterior is maximum.

Maximising the likelihood generates a result purely based on observations whereas maximising the posterior involves working with the prior belief, which is usually embedded via information from experts or machine learning. This MIL/MAP Tutorial Presentation has a couple of great simple examples which highlight the situation. MAP estimation pulls the estimate to the prior. The more focused the prior belief, the larger the pull towards the prior.