An Entropy-Energy Identity for Predictive Kullback-Leibler Regret in Infinitely Divisible Location Models
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866917536656785408 |
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| author | Takanashi, Kōsaku McAlinn, Kenichiro |
| author_facet | Takanashi, Kōsaku McAlinn, Kenichiro |
| contents | We consider predictive density estimation under logarithmic score for $d$-dimensional infinitely divisible location models. Taking the formal Bayes predictive density under the Lebesgue prior as a benchmark, we study the Kullback-Leibler regret of competing Bayes predictive densities. Our main contribution is an exact entropy-energy identity: the integrated regret of a Bayes predictive density $\hat{p}^π$ under prior $π$ relative to the benchmark admits an exact representation as the Dirichlet-form energy of the square-rooted marginal distribution $\sqrt{M^π}$ for the symmetric Markov semigroup induced by the benchmark kernel. This converts regret comparisons into a potential-theoretic problem and yields a sharp recurrence/transience characterization of when the benchmark predictive density can or cannot be uniformly improved. We introduce an $\mathcal{A}$-harmonic class of improper priors -- defined through the generator $\mathcal{A}$ of the induced process -- and give explicit tail conditions -- an integral test on the induced marginal, equivalent to power-law prior decay in heavy-tailed models -- that guarantee admissibility of the resulting Bayes predictive density. We illustrate the theory with new results for several distributions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_27253 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | An Entropy-Energy Identity for Predictive Kullback-Leibler Regret in Infinitely Divisible Location Models Takanashi, Kōsaku McAlinn, Kenichiro Statistics Theory Probability We consider predictive density estimation under logarithmic score for $d$-dimensional infinitely divisible location models. Taking the formal Bayes predictive density under the Lebesgue prior as a benchmark, we study the Kullback-Leibler regret of competing Bayes predictive densities. Our main contribution is an exact entropy-energy identity: the integrated regret of a Bayes predictive density $\hat{p}^π$ under prior $π$ relative to the benchmark admits an exact representation as the Dirichlet-form energy of the square-rooted marginal distribution $\sqrt{M^π}$ for the symmetric Markov semigroup induced by the benchmark kernel. This converts regret comparisons into a potential-theoretic problem and yields a sharp recurrence/transience characterization of when the benchmark predictive density can or cannot be uniformly improved. We introduce an $\mathcal{A}$-harmonic class of improper priors -- defined through the generator $\mathcal{A}$ of the induced process -- and give explicit tail conditions -- an integral test on the induced marginal, equivalent to power-law prior decay in heavy-tailed models -- that guarantee admissibility of the resulting Bayes predictive density. We illustrate the theory with new results for several distributions. |
| title | An Entropy-Energy Identity for Predictive Kullback-Leibler Regret in Infinitely Divisible Location Models |
| topic | Statistics Theory Probability |
| url | https://arxiv.org/abs/2605.27253 |