# MAP and one solved question

Is there anyone can describe me, why this sentence is false? I couldn't get the point from this answer.

$$p(\theta | \mathcal{D}) = \frac{p(\mathcal{D}|\theta)p(\theta)}{p(\mathcal{D})}$$
where $$\mathcal{D}$$ is your set of data points, $$p(\mathcal{D}|\theta)$$ is the likelihood and $$p(\theta)$$ is the prior for the parameter(s) $$\theta$$. Imagine a degenerate prior, which assigns probability $$1$$ to a specific values $$\tilde{\theta}$$ of the parameters. This prior is always equal to $$0$$, except at $$\theta = \tilde{\theta}$$ (in other words, it is a Dirac located at $$\tilde{\theta}$$). Then your posterior is also going to be $$0$$ for any $$\theta$$ different from $$\tilde{\theta}$$, and will be independent of $$\mathcal{D}$$.
• The Bernstein-von Mises theorem (en.wikipedia.org/wiki/Bernstein%E2%80%93von_Mises_theorem) states (roughly) that, under some conditions, as the number of data points $s$ increases, the MAP converges to the MLE. But this is only true under some conditions (which are detailed in the article), including the smoothness (ie the continuity) of the prior density. A degenerate prior, as illustrated in your first question, does not validate this assumption, and the theorem is thus not applicable. – Camille Gontier Dec 15 '20 at 19:14