A guest article by Nic Lewis
In a recent article I discussed Bayesian parameter inference in the context of radiocarbon dating. I compared Subjective Bayesian methodology based on a known probability distribution, from which one or more values were drawn at random, with an Objective Bayesian approach using a noninformative prior that produced results depending only on the data and the assumed statistical model. Here, I explain my proposals for incorporating, using an Objective Bayesian approach, evidence-based probabilistic prior information about of a fixed but unknown parameter taking continuous values. I am talking here about information pertaining to the particular parameter value involved, derived from observational evidence pertaining to that value. I am not concerned with the case where the parameter value has been drawn at random from a known actual probability distribution, that being an unusual case in most areas of physics. Even when evidence-based probabilistic prior information about a parameter being estimated does exist and is to be used, results of an experiment should be reported without as well as with that information incorporated. It is normal practice to report the results of a scientific experiment on a stand-alone basis, so that the new evidence it provides may be evaluated.
In principle the situation I am interested in may involve a vector of uncertain parameters, and multi-dimensional data, but for simplicity I will concentrate on the univariate case. Difficult inferential complications can arise where there are multiple parameters and only one or a subset of them are of interest. The best noninformative prior to use (usually Bernardo and Berger’s reference prior) may then differ from Jeffreys’ prior.
Where there is an existing parameter estimate in the form of a posterior PDF, the standard Bayesian method for incorporating (conditionally) independent new observational information about the parameter is “Bayesian updating”. This involves treating the existing estimated posterior PDF for the parameter as the prior in a further application of Bayes’ theorem, and multiplying it by the data likelihood function pertaining to the new observational data. Where the parameter was drawn at random from a known probability distribution, the validity of this procedure follows from rigorous probability calculus. Where it was not so drawn, Bayesian updating may nevertheless satisfy the weaker Subjective Bayesian coherency requirements. But is standard Bayesian updating justified under an Objective Bayesian framework, involving noninformative priors?
A noninformative prior varies depending on the specific relationships the data values have with the parameters and on the data-error characteristics, and thus on the form of the likelihood function. Noninformative priors for parameters therefore vary with the experiment involved; in some cases they may also vary with the data. Two studies estimating the same parameter using data from experiments involving different likelihood functions will normally give rise to different noninformative priors. On the face of it, this leads to a difficulty in using objective Bayesian methods to combine evidence in such cases. Using the appropriate, individually noninformative, prior, standard Bayesian updating would produce a different result according to the order in which Bayes’ theorem was applied to data from the two experiments. In both cases, the updated posterior PDF would be the product of the likelihood functions from each experiment, multiplied by the noninformative prior applicable to the first of the experiments to be analysed. That noninformative priors and standard Bayesian updating may conflict, producing inconsistency, is a well known problem (Kass and Wasserman, 1996).
Modifying standard Bayesian updating
My proposal is to overcome this problem by applying Bayes theorem once only, to the joint likelihood function for the experiments in combination, with a single noninformative prior being computed for inference from the combined experiments. This is equivalent to the modification of Bayesian updating proposed in Lewis (2013a). It involves rejecting the validity of standard Bayesian updating for objective inference about fixed but unknown continuously-valued parameters, save in special cases. Such special cases include where the new data is obtained from the same experimental setup as the original data, or where the experiments involved are different but the same form of prior in noninformative in both cases. Continue reading