Beyond the Kullback--Leibler divergence: the likelihood-ratio distribution as a detectability measure
Abstract
The Kullback--Leibler (KL) divergence is the expectation value of the log-likelihood ratio between two specified models, evaluated with respect to one of the two models. It provides an average measure of the distinguishability of the models. We examine its connection to discovery potential and the accuracy of parameter estimates when one model represents a null hypothesis, and the alternative is a family of models characterized by one or more continuous parameters. For data drawn from a multivariate Gaussian distribution, we derive the log-likelihood-ratio distribution and its cumulants under two such hypotheses. We then search over the alternative model's parameter space and use peak prominence, the height of a likelihood-profile peak above its surrounding baseline, as an operational detection statistic. Repeating this search on simulated data drawn from the null model allows us to calibrate the prominence threshold for a specified false-positive rate. We study a toy Gaussian model as well as three cosmological examples involving cosmic topology, the tensor-to-scalar ratio, and dark energy. In all these cases, detectability of a particular model depends not only on KL divergence, but also on the size of the model parameter space, the likelihood-ratio distributions, and the distance of the model from any parameter space boundaries. The relationship between KL divergence, discovery potential, and parameter estimation must therefore be calibrated for each family of models and detection statistic under consideration.
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BibTeX
@misc{samandar2026beyond,
title = {{Beyond the Kullback--Leibler divergence: the likelihood-ratio distribution as a detectability measure}},
author = {Amirhossein Samandar and Mikel Martin Barandiaran and Linn Htat Lu and Andrew H Jaffe and Deyan P Mihaylov and Glenn D Starkman and Cynthia Trendafilova and Yashar Akrami and Stefano Anselmi and Javier Carrón Duque and Craig J Copi and Fernando Cornet-Gomez and Arthur Kosowsky and Anna Negro and Ricardo G Rodrigues and Andrius Tamosiunas},
howpublished = {HAL},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.17186},
eprint = {2610.17186},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05843958v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Beyond the Kullback--Leibler divergence: the likelihood-ratio distribution as a detectability measure
AU - Amirhossein Samandar
AU - Mikel Martin Barandiaran
AU - Linn Htat Lu
AU - Andrew H Jaffe
AU - Deyan P Mihaylov
AU - Glenn D Starkman
AU - Cynthia Trendafilova
AU - Yashar Akrami
AU - Stefano Anselmi
AU - Javier Carrón Duque
AU - Craig J Copi
AU - Fernando Cornet-Gomez
AU - Arthur Kosowsky
AU - Anna Negro
AU - Ricardo G Rodrigues
AU - Andrius Tamosiunas
PY - 2026
DA - 2026/10/12
PB - HAL
DO - 10.48550/arxiv.2610.17186
UR - https://hal.science/hal-05843958v1
AB - The Kullback--Leibler (KL) divergence is the expectation value of the log-likelihood ratio between two specified models, evaluated with respect to one of the two models. It provides an average measure of the distinguishability of the models. We examine its connection to discovery potential and the accuracy of parameter estimates when one model represents a null hypothesis, and the alternative is a family of models characterized by one or more continuous parameters. For data drawn from a multivariate Gaussian distribution, we derive the log-likelihood-ratio distribution and its cumulants under two such hypotheses. We then search over the alternative model's parameter space and use peak prominence, the height of a likelihood-profile peak above its surrounding baseline, as an operational detection statistic. Repeating this search on simulated data drawn from the null model allows us to calibrate the prominence threshold for a specified false-positive rate. We study a toy Gaussian model as well as three cosmological examples involving cosmic topology, the tensor-to-scalar ratio, and dark energy. In all these cases, detectability of a particular model depends not only on KL divergence, but also on the size of the model parameter space, the likelihood-ratio distributions, and the distance of the model from any parameter space boundaries. The relationship between KL divergence, discovery potential, and parameter estimation must therefore be calibrated for each family of models and detection statistic under consideration.
ER -