The generalized IFS Bayesian method and an associated variational principle covering the classical and dynamical cases

The paper’s Theorem 30 shows that, under the stated boundedness and positivity assumptions (bounded π_a, and l, ψ bounded and bounded away from zero), the variational problem sup over holonomic π̃ of ∫[log l + log π_a − log ϕ] dπ̃ + H_{dθ}(π̃) has value 0 attained by any posterior π; the argument proceeds via (i) the entropy identity H_ν(π) = −∫ log l̄ dπ and the change of base to H_{dθ}, (ii) the decomposition l̄ = (l ψ∘τ)/(ψ ϕ), and (iii) the holonomy identity with g = log ψ, which cancels ψ-terms and yields the announced value and maximizers (see (17), (18), (31) and the discussion leading to (32) and Theorem 30) . The candidate solution repeats these same steps: explicit change-of-base for entropy, bounding H_ν by −∫ log l̄, using log l − log ϕ − log l̄ = log ψ − log ψ∘τ, and constructing maximizers from stationary ρ, concluding J(π̃) ≤ 0 for all holonomic π̃ and J(π) = 0 for posteriors. Assumptions match those in the paper and ensure boundedness of the logarithms so holonomy applies; minor integrability technicalities around log π_a are handled implicitly in both treatments. Overall, both are correct and essentially the same proof .