Closed-form empirical Bernstein confidence sequences for scalars and matrices
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909975493738496 |
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| author | Chugg, Ben Ramdas, Aaditya |
| author_facet | Chugg, Ben Ramdas, Aaditya |
| contents | We derive a new closed-form variance-adaptive confidence sequence (CS) for estimating the average conditional mean of a sequence of bounded random variables. Empirically, it yields the tightest closed-form CS we have found for tracking time-varying means, across sample sizes up to $\approx 10^6$. When the observations happen to have the same conditional mean, our CS is asymptotically tighter than the recent closed-form CS of Waudby-Smith and Ramdas [38]. It also has other desirable properties: it is centered at the unweighted sample mean and has limiting width (multiplied by $\sqrt{t/\log t}$) independent of the significance level. We extend our results to provide a CS with the same properties for random matrices with bounded eigenvalues. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_21300 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Closed-form empirical Bernstein confidence sequences for scalars and matrices Chugg, Ben Ramdas, Aaditya Statistics Theory Probability Methodology We derive a new closed-form variance-adaptive confidence sequence (CS) for estimating the average conditional mean of a sequence of bounded random variables. Empirically, it yields the tightest closed-form CS we have found for tracking time-varying means, across sample sizes up to $\approx 10^6$. When the observations happen to have the same conditional mean, our CS is asymptotically tighter than the recent closed-form CS of Waudby-Smith and Ramdas [38]. It also has other desirable properties: it is centered at the unweighted sample mean and has limiting width (multiplied by $\sqrt{t/\log t}$) independent of the significance level. We extend our results to provide a CS with the same properties for random matrices with bounded eigenvalues. |
| title | Closed-form empirical Bernstein confidence sequences for scalars and matrices |
| topic | Statistics Theory Probability Methodology |
| url | https://arxiv.org/abs/2512.21300 |