What's the difference between a Bayes Estimator and a Maximum a posteriori (MAP) Estimator?

Question

I have been practicing with two textbooks and one has a chapter on Bayes Estimator (Sheldon Ross) and another has a Chapter on MAP Estimation (Pishro Nik). I seem to get the same answers (to the simple Example Problems) using either and it looks like the process is the same, but I feel like there is some difference, I just can't pin it down.

Could someone explain it to me?

Thank you very much

Answer 1

In Bayesian Statistic there are different kind of estimators for a parameter. Different estimators come out because of different hypothesis about the loss function.

Here are some examples:

1. an estimator which minimize Mean Square Error, that is posterior expectation

2. MAP, Maximum a Posteriori, the estimator which maximize posterior density

3. Posterior Median

4. Posterior Mode

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Consider the following example:

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Throwing a coin, we are intersted in estimating the parameter $\theta$: probability to get Head.

We have no information "a Priori" about the parameter's value thus the prior is uniform

$$\pi(\theta)=1$$

Suppose we flip the coin 10 times getting 6 H and 4T thus the posterior is

$$\pi(\theta|\mathbf{x})\propto \theta^6(1-\theta)^4$$

that is a $\text{Beta}(7;5)$, thus

$$\pi(\theta|\mathbf{x})=2310\cdot\theta^6(1-\theta)^4$$

Depending on which estimator we want to use we get

$$\hat{\theta}_{\text{MMSE}}=\mathbb{E}\left[\theta|\mathbf{x}\right]=\frac{7}{12}$$

$$\hat{\theta}_{\text{MAP}}=\frac{6}{10}$$