Electronic Structure Reading Group: Quantifying and propagating model-form uncertainty via misspecification-aware Bayesian regression

Essentially all models are misspecified, i.e. no one parameter choice is able to exactly reproduce observations. As a result, model parameters are fundamentally uncertain, i.e. have model-form error, as there is no unique “best choice”. In addition, finite capacity models such as polynomials or partially frozen neural networks are often underparametrized, i.e. the number of training data is much greater than the number of parameters, meaning epistemic uncertainties are minimal.

I will discuss recent work[1] which treats the true generalisation error, a misspecification-aware error measure for which the misspecification-blind log likelihood of Bayesian inference is only an (Hoeffding/Jensen) upper bound, closely analogous to the Gibbs-Bogoliubov bound (F<U). Whilst direct minimisation of the generalisation error is not tractable, we derive a novel condition any valid posterior must obey, then design an ansatz valid for any model, which we efficiently evaluate for linear (or linearised) models. Our method, POPS[1,2] provides a new form of posterior distribution on model parameters which enables robust *bounding* of test errors. Fitting and inference has minimal (< x2) overhead compared to standard Bayesian regression schemes. Importantly, by assigning error to parameters rather than simply the model output we have an excellent starting point for propagation of model-form uncertainty through multi scale simulations[3,4,5].

[1]T. Swinburne, D. Perez, Mach. Learn.: Sci. Technol., 6, 015008 (2025)
[2] https://github.com/tomswinburne/POPS-Regression.git <https://github.com/tomswinburne/POPSRegression.git>
[3] I. Maliyov, P. Grigorev, T. Swinburne, npj. Comput. Mater., 11, 22 (2025)
[4] D. Perez, A. P. A. Subramanyam, I. Maliyov, T. D. Swinburne, arXiv:2502.07104 (2025)
[5] T. D. Swinburne, C. Lapointe, M-C. Marinica, arXiv:2502.18191 (2025)

This event is jointly organized with the COSMO seminar.
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The electronic structure reading group brings together researchers and students interested in mathematical aspects of electronic structure problems and adjacent topics, including:

• Density Functional Theory
• Many-body Schrödinger equation for electrons
• Born-Oppenheimer Molecular Dynamics
• Numerical analysis and error control