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Learning from noisy data is a fundamental problem in many machine learning applications.

We provide the definitions of important terms used throughout the paper. Assumption 2.3 when the demand distribution is exponential. Note that Lemma B.1 implies that In the following result, we show that there exist appropriate constants such that prior distribution satisfies Assumption 2.3 when the demand distribution is a multivariate Gaussian with unknown The proof is a direct consequence of Theorem 3.2, Lemmas B.6, B.7, B.8, B.9, and Proposition 3.2. Theorem 6.19] the prior induced by Assumption 2.2 is a direct consequence of Assumption 2.4 and 2.5 are straightforward to satisfy since the model risk function Lemma B.13. F or a given Using the result above together with Proposition 3.2 implies that the RSVB posterior converges at C.1 Alternative derivation of LCVB We present the alternative derivation of LCVB. We prove our main result after a series of important lemmas.

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But theguarantees have strong conditions that donot often hold inpractice, resulting indegenerate component optimization problems; and weshowthat the ad-hoc regularization used to prevent degeneracyin practice can cause BVI to fail in unintuitiveways.

In what follows, we review the Decision-Estimation Coefficient and Estimation-to-Decisions meta-algorithm (Section 1.2), highlighting opportunities for improvement.