Maximum Likelihood (ML) vs Maximum a Posteriori Estimation (MAP), When to Use Which?

Question

ML = Maximum Liklihood

MAP = Maximum a-posteriori

ML is intuitive/naive in that it starts only with the probability of observation given the parameter (i.e. the likelihood function) and tries to find the parameter best accords with the observation. But it take into no consideration the prior knowledge.

MAP seems more reasonable because it does take into consideration the prior knowledge through the Bayes rule.

So, I think MAP is much better. Is that right? And when should I use which?

Here is a related question, but the answer is not thorough - Differences Using Maximum Likelihood or Maximum a Posteriori for Deconvolution / Deblur?

Answer 1

You can think of MAP as a regularization of the ML.
Just like you have regularization for Least Squares Problem (They can be built, mostly, as MAP problem).

The nice thing is that, as always, the best regularization is more data, namely, in most case when there is a lot of data they collide (Namely, low sensitivity fir the Posterior PDF).

So they differ mainly when there is (Relatively) low amount of data.
Now, when you have low amount of data and you know nothing about the parameters you're trying to estimate, ML is the way to go.
If you have low amount of data yet some prior knowledge about it, or reasonable assumption, make those assumption as regularization.
It even good to make the effort and describe this knowledge in Posterior PDF form.

Update

Found really nice tutorial about the topic - The Truth About Priors and Overfitting.