Discovering Governing Equations in the Presence of Uncertainty

The paper adopts the same posterior reweighting formula as the model, namely π_Λ(λ|Y) = π_Λ(λ) · π_Y(Q̂(λ)|Y)/π_Q(Q̂(λ)) (its Equation (6)) and claims a “Consistency of push-forward” theorem (Equations (11)–(12)) stating that push-forwarded posterior matches the observed data distribution . However, the paper’s statement is incomplete and internally inconsistent: it (i) omits the essential hypothesis μ_Y ≪ μ_Q (existence of the Radon–Nikodym derivative), (ii) presents a malformed display (11) that conflates densities/measures and introduces ad hoc “volume measures,” and (iii) asserts matching “for some measurable set A” rather than the required “for all A,” while also deferring proof to a mis-cited reference. In contrast, the model’s solution gives the standard Radon–Nikodym construction assuming μ_Y ≪ μ_Q and correctly proves normalization and the identity Q̂_# μ_{Λ|Y} = μ_Y for all measurable sets via the push-forward (LOTUS) identity, which is the known consistent solution in the stochastic inverse/Bayesian setting (the paper’s methodology context also aligns with this push-forward/Bayes construction) .