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I was wondering if anyone could spare a moment to help with the answers to the following questions.

Suppose we have an estimator $\hat{\theta}:\mathbb{R}^{d}\rightarrow\mathbb{R}$ such that the number of parameters $p\gg d$. For a squared objective $(\hat{\theta}-\theta)^{2}$ one can perform a bias-variance decomposition of the estimator $\hat{\theta}$.

  1. Does the bias-variance analysis hold for different objectives (e.g. cross-entropy)?
  2. Is there a Bayesian analysis of bias-variance decomposition of an estimator (or something equivalent)?
  3. If we wanted to evaluate the implications of the choice of our prior in terms of the functions that it allows to be induced in the function space of the estimator $\hat{\theta}$ how would we do that? (I'm looking for any sources, math theorems or maybe there's a specific field of study whose name I might be missing at the moment?)
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