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Then the unconstrained logistic MLE is consistent for the corresponding parameter values: ˆaZ p − →−logK0, ˆbZ p − →1 (on the unstandardisedlogwi scale) as the number of accountsN→∞. Proof sketch.For any observation with true expected rater0 = E[θi|¯fi,wi], Poisson devianceℓ(r) =r−Clogr satisfies E[ℓ(r)] =r−r0 logr +const, which is uniquely minimised atr =r0 (proper scoring rule property). Under the B-S data-generating process (DGP), the Bayesian posterior mean isr0 = (1−Z∗ i )µ+Z∗ i ¯fi with Z∗ i = wi/(wi +K0). For the logistic model to achieveˆri = r0 for every account simultaneously (i.e. for every value ofwi) requiresσ(aZ +bZ logwi) =wi/(wi +K0)to hold identically inwi. 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