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Abstract:
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How can uncertainty quantification account for physical (and other) constraints on unknown parameters? Frequentist methods capture such constraints directly; Bayesian methods use a prior for which the probability that the constraints hold is 1. Practitioners often pick an approach for computational convenience, ignoring assumptions and their effect on the interpretation of the results. Bayesian and frequentist measures of uncertainty have similar names, but Bayesian uncertainties typically involve expectations with respect to the posterior distribution of the parameter holding the data (and prior) fixed, while frequentist uncertainties typically involve expectations with respect to the distribution of the data, holding the parameter fixed. Bayesian approaches require a prior distribution for the parameter; frequentist approaches do not even require the assumption that the parameter is random. This can cause frequentist and Bayesian estimates, uncertainties, and inferences to differ substantially, even for "uninformative" priors. Worst of all, perhaps, are (unfortunately common) hodgepodge analyses that combine both approaches with scant attention to what anything means.
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