Empirical Bayes Selection for Value Maximization
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
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2022
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| _version_ | 1866918132195524608 |
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| author | Coey, Dominic Hung, Kenneth |
| author_facet | Coey, Dominic Hung, Kenneth |
| contents | We study the problem of selecting the best $m$ units from a set of $n$ as $m / n \to α\in (0, 1)$, where noisy, heteroskedastic measurements of the units' true values are available and the decision-maker wishes to maximize the aggregate true value of the units selected. Given a parametric prior distribution, the empirical Bayes decision rule incurs $O_p(n^{-1})$ regret relative to the Bayesian oracle that knows the true prior. More generally, if the error in the estimated prior is of order $O_p(r_n)$, regret is $O_p(r_n^2)$. In this sense \emph{selection} of the best units is fundamentally easier than \emph{estimation} of their values. We show this regret bound is sharp in the parametric case, by giving an example in which it is attained. Using priors calibrated from a dataset of over four thousand internet experiments, we confirm that empirical Bayes methods perform well in detecting the best treatments with only a modest number of experiments. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_03905 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Empirical Bayes Selection for Value Maximization Coey, Dominic Hung, Kenneth Methodology Econometrics 62C12, 62C25 (Primary) 62P20 (Secondary) We study the problem of selecting the best $m$ units from a set of $n$ as $m / n \to α\in (0, 1)$, where noisy, heteroskedastic measurements of the units' true values are available and the decision-maker wishes to maximize the aggregate true value of the units selected. Given a parametric prior distribution, the empirical Bayes decision rule incurs $O_p(n^{-1})$ regret relative to the Bayesian oracle that knows the true prior. More generally, if the error in the estimated prior is of order $O_p(r_n)$, regret is $O_p(r_n^2)$. In this sense \emph{selection} of the best units is fundamentally easier than \emph{estimation} of their values. We show this regret bound is sharp in the parametric case, by giving an example in which it is attained. Using priors calibrated from a dataset of over four thousand internet experiments, we confirm that empirical Bayes methods perform well in detecting the best treatments with only a modest number of experiments. |
| title | Empirical Bayes Selection for Value Maximization |
| topic | Methodology Econometrics 62C12, 62C25 (Primary) 62P20 (Secondary) |
| url | https://arxiv.org/abs/2210.03905 |