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242 Statistics’ two theorieswhere the subscripts to F denote partial differentiation with respect to the indicatedargument. Each of the integrals records an F (y, θ) valueasanintegralof its derivative — the fundamental theorem of calculus — one with respect toθ and the other with respect to y. Thisispurecomputation,entirelywithoutBayes! And then, quite separately, the Bayes survivor value using a profferedprior π(θ) is∫s 0 (θ) = π(θ)F y (y 0 ; θ)dθ.θ(vi) Validity of Bayes posterior: Simple scalar model. Thesecondintegralfor p 0 (θ) and the integral for s 0 (θ) are equal if and only if the integrands areequal. In other words if and only ifπ(θ) =− F θ(y 0 ; θ)F y (y 0 ; θ)=∂y(θ; u)∂θ∣∣fixed F (y;θ);y 0with an appropriate norming constant included. The second equality comesfrom the total derivative of u = F (y; θ) set equal to 0, thus determininghow a θ-change affects y for fixed probability position. We can also viewv(θ) =∂y(θ; u)/∂θ for fixed u as being the change in y caused by a change inθ, thus giving at y 0 a differential version of the y, θ analysis in the precedingsubsection.Again, with this simple scalar model analysis, there is no frequency-Bayescontradiction; it is just a matter of getting the prior right. The correct priordoes depend on the data point y 0 but this should cause no concern. If theobjective of Bayesian analysis is to extract all accessible information from anobserved likelihood and if this then requires the tailoring of the prior to theparticular data, then this is in accord with that objective. Data dependentpriors have been around for a long time; see, e.g., Box and Cox (1964). But ofcourse this data dependence does conflict with a conventional Bayes view thata prior should be available for each model type. The realities of data analysismay not be as simple as Bayes might wish.(vii) What’s the conclusion? With a location model, Bayes and frequencyapproaches are in full agreement: Bayes gets it right because the Bayes calculationis just a frequency confidence calculation in mild disguise. However,with a non-location model, the Bayes claim with a percentage attached toan interval does require a data-dependent prior. But to reference the conditionalprobability lemma, relabeled as Bayes lemma, requires that a missingingredient for the lemma be created, that a density not from the reality beinginvestigated be given objective status in order to nominally validate the termprobability: this violates mathematics and science.