{"ID":2825556,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2512.21300","arxiv_id":"2512.21300","title":"Closed-form empirical Bernstein confidence sequences for scalars and matrices","abstract":"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.","short_abstract":"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...","url_abs":"https://arxiv.org/abs/2512.21300","url_pdf":"https://arxiv.org/pdf/2512.21300v1","authors":"[\"Ben Chugg\",\"Aaditya Ramdas\"]","published":"2025-12-24T17:34:11Z","proceeding":"math.ST","tasks":"[\"math.ST\",\"math.PR\",\"stat.ME\"]","methods":"[]","has_code":false}