A Bayesian-Geometric Reading of Classical Model-Based Optimal Design
Abstract
The classical model-based optimal design criteria (D-, T-, L-, C-, and A-optimality) are usually motivated via the Cramér--Rao bound as optimizing scalar functions of the Fisher information matrix (FIM). This paper offers a complementary reading in which each criterion appears asymptotically as N → ∞ as the leading-order term of an expected dissimilarity between prior and posterior, with the FIM entering through the Bernstein--von Mises limit. The Kullback--Leibler (KL) divergence of the posterior relative to the prior recovers D-optimality (classically known); a reverse KL of the prior relative to the posterior recovers T-optimality (here defined as maximization of the trace of the FIM, following one common convention) and its weighted-trace generalization; and a first-order approximation of the squared 2-Wasserstein distance recovers L-, C- and A-optimality. The five criteria emerge as instances of a common Bayesian-geometric template, indexed by three choices: the dissimilarity, the prior parametrization, and the approximation scheme used to evaluate it (e.g.\ Taylor expansions of the matrix square root). For the weighted criteria, the Gaussian prior covariance encodes the criterion's weight matrix, so any choice of weights corresponds to a specific prior in the family. The T-, L-, C-, and A-optimality derivations appear novel to the best of my knowledge. A sixth criterion, E-optimality, is recovered not as an independent expected-dissimilarity limit but as a min-max robustification of C-optimality against uncertainty in the modeler's prior direction. The framework is extended to distinguish between a designer's prior over unknown parameters at design time and a modeler's prior used for posterior inference, recovering the classical expected information gain objective as a special case.
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BibTeX
@misc{tarek2026bayesiangeometric,
title = {{A Bayesian-Geometric Reading of Classical Model-Based Optimal Design}},
author = {Mohamed Tarek},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.04450},
eprint = {2611.04450},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.04450},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Bayesian-Geometric Reading of Classical Model-Based Optimal Design
AU - Mohamed Tarek
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.04450
UR - https://arxiv.org/abs/2611.04450
AB - The classical model-based optimal design criteria (D-, T-, L-, C-, and A-optimality) are usually motivated via the Cramér--Rao bound as optimizing scalar functions of the Fisher information matrix (FIM). This paper offers a complementary reading in which each criterion appears asymptotically as N → ∞ as the leading-order term of an expected dissimilarity between prior and posterior, with the FIM entering through the Bernstein--von Mises limit. The Kullback--Leibler (KL) divergence of the posterior relative to the prior recovers D-optimality (classically known); a reverse KL of the prior relative to the posterior recovers T-optimality (here defined as maximization of the trace of the FIM, following one common convention) and its weighted-trace generalization; and a first-order approximation of the squared 2-Wasserstein distance recovers L-, C- and A-optimality. The five criteria emerge as instances of a common Bayesian-geometric template, indexed by three choices: the dissimilarity, the prior parametrization, and the approximation scheme used to evaluate it (e.g.\ Taylor expansions of the matrix square root). For the weighted criteria, the Gaussian prior covariance encodes the criterion's weight matrix, so any choice of weights corresponds to a specific prior in the family. The T-, L-, C-, and A-optimality derivations appear novel to the best of my knowledge. A sixth criterion, E-optimality, is recovered not as an independent expected-dissimilarity limit but as a min-max robustification of C-optimality against uncertainty in the modeler's prior direction. The framework is extended to distinguish between a designer's prior over unknown parameters at design time and a modeler's prior used for posterior inference, recovering the classical expected information gain objective as a special case.
ER -