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docs(05): scale-normalize the ACNet generator plot - #8
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Archimedean generators are identified only up to a scaling of their argument (phi(t) and phi(ct) give the same copula), so the raw overlay looked far off even though the fitted copula matched (LL near the Clayton ceiling). Normalize both generators to phi'(0) = -1 before plotting; the learned ACNet now visibly tracks the true Clayton, with a small residual = genuine approximation error. Added a markdown note explaining the identifiability.
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The ACNet notebook's generator overlay looked badly off from Clayton even though the copula was well-recovered (fitted LL ≈ Clayton ceiling). Cause: Archimedean generators are identified only up to a scaling of the argument —$\varphi(t)$ and $\varphi(ct)$ give the same copula, so the likelihood can't fix the scale.
Fix: normalize both generators to$\varphi'(0) = -1$ before plotting. The learned ACNet now clearly tracks the true Clayton, with a small residual = genuine approximation error (finite mixture of exponentials vs Clayton's Gamma-frailty transform). Added a markdown note on the identifiability. Notebook re-executed and committed.