MAP estimator

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In statistics, MAP estimates come from maximizing the likelihood function multiplied by a prior probability distribution.

For example, if the density of the data is given by $f$ , then the likelihood function is given by

$L(\theta) = f(x_1,\dots,x_n \mid \theta).\,$

When θ is unknown, the method of maximum likelihood uses the value of θ that maximizes L(θ) as an estimate of θ. This is the maximum likelihood estimator (MLE) $\widehat{\theta}_{MLE}$ of θ

In contrast, the MAP estimator $\widehat{\theta}_{MAP}$ of θ postulates the existence of an a priori distribution $π(θ)$ , and the MAP estimator is given by

$\widehat{\theta}_{MAP} \in \arg\max \pi(\theta) f(x_1, \dots, x_n \mid \theta).$

[edit] Example

Suppose that we are given a sequence $(x_1, \dots, x_n)$ of IID $N(\mu,\sigma_v^2 )$ random variables and an a prior distribution of $μ$ is given by $N(0,\sigma_m^2 )$ . We wish to find the MAP estimate of $μ$ .

The function to be maximized is then given by

$\pi(\mu) L(\mu) = \frac{1}{\sqrt{2 \pi \sigma_m}} \exp\left(-\frac{1}{2} \left(\frac{\mu}{\sigma_v}\right)^2\right) \prod_{j=1}^n \frac{1}{\sqrt{2 \pi \sigma_v}} \exp\left(-\frac{1}{2} \left(\frac{x_j - \mu}{\sigma_v}\right)^2\right),$