Is MAP a point estimator? Why? In Bishop's Pattern Recognition and Machine Learning book, page 30, section 1.2.6, it says that the MAP estimator for the curve fitting model derived in section 1.2.5 is a point estimator of the model parameters $\textbf{w}$:

My question is: why are we calling MAP a point estimator? The posterior distribution can give probabilities for a particular choice of $\textbf{w}$, and it is possible to evaluate the confidence intervals around a particular choice of $\textbf{w}$, say $\textbf{w}_{MAP} \pm \epsilon$.

The maximum a posteriori estimate is
$$ \theta_{MAP} = \underset{\theta \in \mathbb{R}^p}{\arg \min} \ \ \ -\log\big( p(\theta \vert y) \big) $$
Hence, the MAP estimate is the mode of the posterior; the point where the posterior achieves largest density.  It is not a distribution, it is an element in $\mathbb{R}^P$, hence it is a point esitmate.
