I am wondering if Maximum-a-Posteriori (MAP) estimates are consistent in the frequentist sense.
When I am searching for this, usually what pops up is posterior consistency, for example in the sense of Doobs' consistency theorem. I am rather referring to pointwise consistency in the sense of frequentist estimates, as specified below. Maybe the latter follows from the former, but I could not quite figure out why, and I did not find any reference on this either.
Formally, consider a model $\{ \mathbb{P}_\theta | \theta \in \Theta \}$ admitting densities $p(x|\theta)$ and a prior measure $\Pi$ on $\Theta$ with density $p(\theta)$.
Let $X_1,...,X_n$ $\overset{\text{i.i.d}}{\sim} \mathbb{P_{\theta_0}}$.
Assume a MAP-estimate $ \hat{\theta}_n = \underset{\theta}{\arg \max} \ p(\theta|X_1,...,X_n) = \underset{\theta}{\arg \max} \ p(X_1,...,X_n|\theta) p(\theta)$
When is $\hat{\theta}_n $ consistent, i.e. $\hat{\theta}_n \overset{\mathbb{P}_{\theta_0}}{\rightarrow} \theta_0$ ?
Especially I am wondering if under conditions where the Bernstein-von-Mises theorem holds, this form is given. Under some conditions, which may be found e.g. in here, one has that the posterior measure approximates a Gaussian in total variation distance.
That is, one has $$ ||\mathbb{P}_{\sqrt{n}(\overline{\Theta}_n-\theta_0) |X_1,...,X_n} - \mathcal{N} (\Delta_{n,\theta_0},I_{\theta_0}^{-1})||_{TV} \overset{\mathbb{P}_{\theta_0}}{\rightarrow} 0$$
where $\overline{\Theta}_n$ is marginally distributed as $\Pi$, $I_{\theta_0}$ is the fisher-information-matrix and $$\Delta_{n,\theta_0} = I_{\theta_0}^{-1} \frac{1}{\sqrt{n}} \sum_i \frac{\partial}{\partial \theta} \ln p(X_i|\theta)$$
Typically one has $\Delta_{n,\theta_0}$ converging in distribution to $\mathcal{N}(0,I_{\theta_0}^{-1})$. Hence, intuitively, one could assume that the MAP-estimate, considering the rescaling that took place, should be consistent for the true parameter. And indeed, as can be seen in the reference, a reasoning like this works for typical Bayes-Estimators like the posterior mean. However, the conditions that are given there, seem to not hold for the posterior mode.
