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Q: On the MSE of MAP estimators and model mismatch

GoulifetWe consider an estimation problem where the parameter $\theta$ is assigned with the prior $g_\alpha$ depending on some parameter $\alpha$ (e.g. the variance of a Gaussian prior) and the observation $x$ has likelihood $f(x|\theta)$. Then, the MAP estimator is given by: $$\theta_\alpha^* (x) = \a...

In statistics, the mean squared error (MSE) or mean squared deviation (MSD) of an estimator (of a procedure for estimating an unobserved quantity) measures the average of the squares of the errors—that is, the average squared difference between the estimated values and the true value. MSE is a risk function, corresponding to the expected value of the squared error loss. The fact that MSE is almost always strictly positive (and not zero) is because of randomness or because the estimator does not account for information that could produce a more accurate estimate. In machine learning, specifically...

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