Maximum a Posteriori Estimator

Introduction

Given random variable \(X\), such as poses and/or landmarks, and the measurements \(Z\), we are interested in finding \(X\) given \(Z\). The maximum a posteriori (MAP) estimator is defined as:

\[ \begin{align} \hat{X}^{MAP} &= \arg \max_{X} p(X | Z) \\ &= \arg \max_{X} \frac{p(Z | X) p(X)}{p (Z)} \\ &= \arg \max_{X} L(X; Z) p (X), \end{align} \]

where \(L(X; Z)\) is the likelihood of the states \(X\) given the measurements \(Z\) and is defined as any function proportional to \(p(Z | X)\):

\[ L(X; Z) \propto p(Z | X). \]

Bayes' rule tells us that solving the maximum posterior probability is equivalent to the esitmate of the product of maximum likelihood and a priori. The notation \(L(X; Z)\) emphasizes the fact that the likelihood is a function of \(X\) and not \(Z\), which acts merely as a paremeter in this context.

If we have no information about the prior, the estimator converts to MLE.