In the Gaussian case, it is well-known that the MSE, is minimizer by the mean value. However, in general, if $X \in L^2(\mathcal{F};\mathbb{P})$, is a random-variable in $\mathbb{R}$, then is the does $$ \mathbb{E}_{\mathbb{P}}[X] = \inf_{Z \in L^2(\mathcal{F};\mathbb{P})} \;\mathbb{E}_{\mathbb{P}}\left[(X-Z)^2\right]? $$

The argmin, is the conditional expectation, but what about the min itself?

  • $\begingroup$ ..... variance? $\endgroup$ – pointguard0 Feb 5 at 15:29
  • $\begingroup$ Probably not, since that isn't true for a 1-dimensional Gaussian. $\endgroup$ – N00ber Feb 5 at 15:36

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