There is no formal requirement for the choice of prior to be evidence-based, although in scientific inference one might hope that it often would be.In an objective approach, the prior is instead normally selected to be noninformative, in the sense of letting inference for the parameter(s) of interest be determined, to the maximum extent possible, solely by the data.Fisher information – the expected value of the negative second derivative of the log-likelihood function with respect to the parameters, in regular cases – is a measure of the amount of information that the data, on average, carries about the parameter values.In simple univariate cases involving a fixed, symmetrical measurement error distribution, Jeffreys’ prior will generally be proportional to the derivative of the data variable being measured with respect to the parameter.A noninformative prior primarily reflects (at least in straightforward cases) how informative, at differing values of the parameter of interest, the data are expected to be about that parameter.In the univariate parameter continuous case, Jeffreys’ prior is known to be the best noninformative prior, in the sense that, asymptotically, Bayesian posterior distributions generated using it provide closer probability matching than those resulting from any other prior. Jeffreys’ prior is the square root of the (expected) Fisher information.I argued that this did not, as claimed, equate to not including anything but the radiocarbon dating information, and was not a scientifically sound method for inference about isolated examples of artefacts. My article attracted many comments, not all agreeing with my arguments.This article follows up and expands on points in my original article, and discusses objections raised. Radiocarbon dating involves determining the radiocarbon age of (a sample from) an artefact and then converting that determination to an estimate of the true calendar age , using a highly nonlinear calibration curve.
That is to say, how good is the “probability matching” (frequentist coverage) of the method.In April 2014 I published a guest article about statistical methods applicable to radiocarbon dating, which criticised existing Bayesian approaches to the problem.A standard – subjective Bayesian – method of inference about the true calendar age of a single artefact from a radiocarbon date determination (measurement) involved using a uniform-in-calendar-age prior.The dotted green line shows the noninformative Jeffreys’ prior used in the objective Bayesian method, which reflects the derivative of the calibration curve.The posterior PDF using Jeffreys’ prior is shown as the solid green line.Both variants of the subjective Bayesian method using a uniform prior are unreliable.The HPD regions that Ox Cal provides give less poor coverage than two-sided credible intervals derived from percentage points of the uniform prior posterior CDF, but at the expense of not giving any information as to how the missing probability is divided between the regions above and below the HPD region.A calibration program is used to derive estimated calendar age probability density functions (PDFs) and uncertainty ranges from a radiocarbon determination.The standard calibration program Ox Cal that I concentrated on uses a subjective Bayesian method with a prior that is uniform over the entire calibration period, where a single artefact is involved.However, even if that body of knowledge is common to two people, their probability evaluations are not required to agree, and may for neither of them properly reflect the knowledge on which they are based.I do not regard this as a satisfactory paradigm for scientific inference.