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The Bayesian Cramér-Rao Bound (BCRB) is generally attributed to Van Trees who published it in 1968. According to Stigler’s law of eponymy, no scientific discovery is named after its first discoverer. This is the case not only for the Cramér-Rao bound itself—due in particular to the French mathematicians Fréchet and Darmois—but also for the van Trees inequality: The French physician, geneticist, epidemiologist and mathematician Marcel-Paul (Marco) Schützenberger, in a paper of just fifteen lines written in 1956—more than a decade before van Trees—had not only derived the BCRB but, as a close examination of his proof shows, used a very original approach based on the Weyl-Heisenberg uncertainty principle on the square root of the posterior distribution. This work reviews and extends Schützenberger’s approach to Fisher information matrices, which opens up new perspectives.

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A Historical Perspective on the Schützenberger-van Trees Inequality: A Posterior Uncertainty Principle

  • 会议论文

  • 发表日期：

    2026年

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  • Olivier Rioul

摘要

The Bayesian Cramér-Rao Bound (BCRB) is generally attributed to Van Trees who published it in 1968. According to Stigler’s law of eponymy, no scientific discovery is named after its first discoverer. This is the case not only for the Cramér-Rao bound itself—due in particular to the French mathematicians Fréchet and Darmois—but also for the van Trees inequality: The French physician, geneticist, epidemiologist and mathematician Marcel-Paul (Marco) Schützenberger, in a paper of just fifteen lines written in 1956—more than a decade before van Trees—had not only derived the BCRB but, as a close examination of his proof shows, used a very original approach based on the Weyl-Heisenberg uncertainty principle on the square root of the posterior distribution. This work reviews and extends Schützenberger’s approach to Fisher information matrices, which opens up new perspectives.