MAP vs. ML Decoding

Likelihood explains the data; posterior probability picks the best overall guess.

The idea

Maximum likelihood (ML) and maximum a posteriori (MAP) decoding are the two standard rules for guessing which message $x$ produced an observed output $y$. They differ in whether the decoder uses the prior probabilities of the messages.

The ML rule chooses the $x$ that maximizes the likelihood $\mathbb{P}(Y = y \mid X = x)$ — the message under which the observed output is most probable. It uses no prior.

The MAP rule chooses the $x$ that maximizes the posterior $\mathbb{P}(X = x \mid Y = y)$. By Bayes' rule the posterior equals $\mathbb{P}(X = x)\,\mathbb{P}(Y = y \mid X = x)$ divided by a normalizer that does not depend on $x$, so MAP maximizes the product of prior and likelihood.

When the prior is uniform, it multiplies every candidate by the same constant, and the two rules always agree. When the prior is uneven they can disagree: the message with the larger likelihood can lose to one that was much more probable a priori. The disagreement can be drawn: give two candidate messages $A$ and $B$ each a column whose width is its prior $p(x)$ and whose height is its likelihood $p(y \mid x)$, so that ML compares the heights while MAP compares the areas $p(x)\,p(y \mid x)$. Because MAP selects the most probable message given the data, it is the rule that minimizes the probability of a decoding error.

Ways to work on it

• Walkthrough. Separate likelihood from posterior probability and see why MAP is the natural error-minimizing decoder.
• Practice. Compute ML and MAP decisions from priors and likelihoods.
• Hardest. Compute a MAP decision and compare it with the ML decision.

Not sure where to start? Take the ten-question placement test.