Vladimir Vovk, Alexander Gammerman, and Glenn Shafer.Algorithmic Learning in a Ran- dom World

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Aki Vehtari, Andrew Gelman, and Jonah Gabry. Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC . Statistics and Computing, 27 0 (5): 0 1413--1432, September 2017. ISSN 0960-3174, 1573-1375. doi:10.1007/s11222-016-9696-4

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Vehtari, A., Gelman, A., Gabry, J.: Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC. Statistics and Computing27(5), 1413–1432 (2017) https://doi.org/10.1007/s11222-016-9696-4

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author Vehtari, A. , author Gelman, A. , author Gabry, J. , year 2017 . title Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC . journal Statistics and Computing volume 27 , pages 1413--1432 . :10.1007/s11222-016-9696-4

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author Vehtari, A., author Gelman, A., author Gabry, J., year 2017 . title Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC . journal Statistics and Computing volume 27, pages 1413--1432 . :10.1007/s11222-016-9696-4

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Vehtari, A., Gelman, A., Gabry, J.: Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC. Statistics and Computing27(5), 1413– 1432 (2017) https://doi.org/10.1007/s11222-016-9696-4

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Aki Vehtari, Andrew Gelman, and Jonah Gabry. Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC . Statistics and Computing, 27 0 (5): 0 1413--1432, 2017. doi:10.1007/s11222-016-9696-4

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author Vehtari, A. , author Gelman, A. , author Gabry, J. , year 2017 . title Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC . journal Statistics and computing volume 27 , pages 1413--1432 . :10.1007/s11222-016-9696-4

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author Vehtari, A., author Gelman, A., author Gabry, J., year 2017 . title Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC . journal Statistics and computing volume 27, pages 1413--1432 . :10.1007/s11222-016-9696-4

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Practical. Statistics and Computing , author =. 2017 , keywords =. doi:10.1007/s11222-016-9696-4 , abstract =

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Practical. Statistics and Computing, author =. 2017, keywords =. doi:10.1007/s11222-016-9696-4, abstract =

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Vehtari, A. and Gelman, A. and Gabry, J. , year =. Statistics and Computing , volume =. doi:10.1007/s11222-016-9696-4 , title =

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Vehtari, A. and Gelman, A. and Gabry, J., year =. Statistics and Computing, volume =. doi:10.1007/s11222-016-9696-4, title =

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doi: 10.1007/s11222-016-9696-4. Vladimir Vovk, Alexander Gammerman, and Glenn Shafer.Algorithmic Learning in a Ran- dom World. Springer, New York, 2005. Vladimir Vovk, Jieli Shen, Valery Manokhin, and Min-ge Xie. Nonparametric predictive distributions based on conformal prediction.Machine Learning, 108(3):445–474, 2019. doi: 10.1007/s10994-018-5755-8. Conference version inProceedings of COPA 2017,PMLR 60:82–102 (2017). A Proofs A.1 Proof of Proposition 2 We follow the argument of Foutz [1977], applying the Inverse Function Theorem to establish existence, consistency, and uniqueness ofban before deriving the asymptotic distribution. 27 LetI(a) =E[−∂ aψ(Y;a)]. The domination condition in (A2) and dominated convergence implyI(a) is continuous neara ∗ 0, withI(a ∗

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Vehtari, Aki and Gelman, Andrew and Gabry, Jonah , year=. Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC , volume=. Statistics and Computing , publisher=. doi:10.1007/s11222-016-9696-4 , number=

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Vehtari, Aki and Gelman, Andrew and Gabry, Jonah, year=. Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC, volume=. Statistics and Computing, publisher=. doi:10.1007/s11222-016-9696-4, number=

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doi: 10.1007/s11222-016-9696-4. 50 §A — Nesting Proof and Rolling Bühlmann–Straub Exposition MLE Consistency Under the B-S Data-Generating Process Proposition (MLE recovery under a B-S data-generating process).Suppose the data are generated by the B-S mechanism with true structural parameterK0: that is, the true credibility weight isZi =wi/(wi+K0), the complement is flat, and there is no temporal decay. Then the unconstrained logistic MLE is consistent for the corresponding parameter values: ˆaZ p − →−logK0, ˆbZ p − →1 (on the unstandardisedlogwi scale) as the number of accountsN→∞. Proof sketch.For any observation with true expected rater0 = E[θi|¯fi,wi], Poisson devianceℓ(r) =r−Clogr satisfies E[ℓ(r)] =r−r0 logr +const, which is uniquely minimised atr =r0 (proper scoring rule property). Under the B-S data-generating process (DGP), the Bayesian posterior mean isr0 = (1−Z∗ i )µ+Z∗ i ¯fi with Z∗ i = wi/(wi +K0). For the logistic model to achieveˆri = r0 for every account simultaneously (i.e. for every value ofwi) requiresσ(aZ +bZ logwi) =wi/(wi +K0)to hold identically inwi. Using the identity σ(log(x/K)) = x/(x +K), this is satisfied if and only ifaZ =−logK0 and bZ =

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Practical. Statistics and Computing , author =. 2017 , keywords =. doi:10.1007/s11222-016-9696-4 , abstract =

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Practical. Statistics and Computing, author =. 2017, keywords =. doi:10.1007/s11222-016-9696-4, abstract =

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Aki Vehtari, Andrew Gelman, and Jonah Gabry. 2017. Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC.Statistics and com- puting27 (2017), 1413–1432. doi:10.1007/s11222-016-9696-4