Harmonic-prior predictive dominance with unknown variance
Let $m \ge k \ge 3$ and let $N$ be a positive integer with $n = mN > k$. Fix a real $m \times k$ matrix $\widetilde X$ of full column rank, put $X = \mathbf{1}_N \otimes \widetilde X$, and observe $Y \sim N_n(X\beta, sI_n)$, where $\beta \in \mathbb{R}^k$ and $s > 0$ are both unknown. The target is the density $p_{\beta,s}$ of an independent future vector $Z \sim N_m(\widetilde X\beta, sI_m)$. Consider the improper priors, with densities relative to $d\beta\, ds$, $$\pi_H(\beta, s) = (\beta^T \widetilde X^T \widetilde X \beta)^{1 - k/2} s^{-2}, \qquad \pi_R(\beta, s) = s^{-1}.$$ For $-1 < \alpha < 1$ the generalized Bayes predictive density $q_{\pi,\alpha}(\cdot \mid y)$, defined for almost every $y$, is the normalization in $z$ of $$\left[\mathbb{E}_{\pi(\beta, s \mid y)}\, p_{\beta,s}(z)^{(1 - \alpha)/2}\right]^{2/(1 - \alpha)},$$ the loss between densities $q$ and $p$ is $$D_\alpha(q, p) = \frac{4}{1 - \alpha^2}\left(1 - \int_{\mathbb{R}^m} q(z)^{(1 + \alpha)/2} p(z)^{(1 - \alpha)/2}\, dz\right),$$ and the risk is $R_\alpha(\beta, s; q) = \mathbb{E}_{Y \mid \beta, s}\, D_\alpha(q(\cdot \mid Y), p_{\beta,s})$. Komaki proved the corresponding dominance for known variance under Kullback–Leibler loss, and Maruyama, Matsuda and Ohnishi under $\alpha$-divergence loss with known variances when the ratio of the past to the future variance is below an explicit bound depending on $\alpha$ and the dimension. For unknown variance, Kato proved it at the endpoint $\alpha = -1$, the loss $\int p \log(p/q)$, and Maruyama and Strawderman at $\alpha = 1$, the loss $\int q \log(q/p)$.
Problem. For every allowed design, $N$ and $\alpha$, does $q_{\pi_H,\alpha}$ dominate $q_{\pi_R,\alpha}$, that is, is its risk no larger for every $(\beta, s)$ and strictly smaller for at least one parameter?
Source: Maruyama and Strawderman (2012), Problem 1.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- ST — Statistics Theory
- Keywords
- AIM Open Applied Problems, alpha-divergences, harmonic priors, predictive densities
- Literature checked
- Nov. 5, 2026 by clemens-admin
References
@incollection{maruyama2012,
author = {Maruyama, Yuzo and Strawderman, William E.},
title = {Bayesian predictive densities for linear regression models under $\alpha$-divergence loss: some results and open problems},
booktitle = {Contemporary Developments in {B}ayesian Analysis and Statistical Decision Theory: A Festschrift for {W}illiam {E}. {S}trawderman},
series = {IMS Collections},
volume = {8},
publisher = {Institute of Mathematical Statistics},
year = {2012},
pages = {42--56},
doi = {10.1214/11-IMSCOLL803},
eprint = {1002.3786},
archivePrefix = {arXiv},
}
@article{komaki2001,
author = {Komaki, Fumiyasu},
title = {A shrinkage predictive distribution for multivariate {N}ormal observables},
journal = {Biometrika},
volume = {88},
number = {3},
year = {2001},
pages = {859--864},
doi = {10.1093/biomet/88.3.859},
}
@article{kato2009,
author = {Kato, Kengo},
title = {Improved prediction for a multivariate normal distribution with unknown mean and variance},
journal = {Ann. Inst. Statist. Math.},
volume = {61},
number = {3},
year = {2009},
pages = {531--542},
doi = {10.1007/s10463-007-0163-z},
}
@article{maruyama2019,
author = {Maruyama, Yuzo and Matsuda, Takeru and Ohnishi, Toshio},
title = {Harmonic {B}ayesian prediction under $\alpha$-divergence},
journal = {IEEE Trans. Inform. Theory},
volume = {65},
number = {9},
year = {2019},
pages = {5352--5366},
doi = {10.1109/TIT.2019.2915245},
eprint = {1605.05899},
archivePrefix = {arXiv},
}
@misc{aim-harmonic-prior-predictive-dominance-with-unknown-variance,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Harmonic-prior predictive dominance with unknown variance}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/4f5b1659355cb61d194da94ef14615214e1363a8/problems/668-harmonic-prior-predictive-dominance-with-unknown-variance.md},
note = {Entry 668 at commit 4f5b165, 2026-11-04},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-01117, "Harmonic-prior predictive dominance with unknown variance", clopen.solutions.