In a MAP estimate:

$\pi_{MAP} = \underset{\pi}{\operatorname{argmax}} P(\pi\mid\chi)$

$=\underset{\pi}{\operatorname{argmax}} \frac{P(\chi\mid\pi)P(\pi)}{P(\chi)}$

$=\underset{\pi}{\operatorname{argmax}} P(\chi\mid\pi)P(\pi)$

How can the distribution $P(\chi)$ be discarded in the second step?


The reason we can discard $P(\chi)$ is that it is not a function of $\pi$, so it doesn't matter in terms of finding the value of $\pi$ that maximizes the posterior probability, just as the constant $c$ doesn't matter in terms of finding the $\arg \min_x$ of $c(x-\theta)^2$.

It would matter if what we cared about was what the value of the maximum posterior probability was, but we don't, we only care that it is the maximum posterior probability, and what the value of $\pi$ is that corresponds to it.


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