Integrated organic inference (IOI): A reconciliation of statistical paradigms

Abstract

It is recognised that the Bayesian approach to inference can not adequately cope with all the types of pre-data beliefs about population quantities of interest that are commonly held in practice. In particular, it generally encounters difficulty when there is a lack of such beliefs over some or all the parameters of a model, or within certain partitions of the parameter space concerned. To address this issue, a fairly comprehensive theory of inference is put forward called integrated organic inference that is based on a fusion of Fisherian and Bayesian reasoning.Depending on the pre-data knowledge that is held about any given model parameter, inferences are made about the parameter conditional on all other parameters using one of three methods of inference, namely organic fiducial inference, bispatial inference and Bayesian inference. The full conditional post-data densities that result from doing this are then combined using a framework that allows a joint post-data density for all the parameters to be sensibly formed without requiring these full conditional densities to be compatible. Various examples of the application of this theory are presented. Finally, the theory is defended against possible criticisms partially in terms of what was previously defined as generalised subjective probability.

A revision to the theory of organic fiducial inference

“…An example of a statistic that may often be regarded as being an approximately sufficient statistic for θ j is any one-to-one function of a unique maximum likelihood estimator of θ j . The use of a maximum likelihood estimator as a fiducial statistic was illustrated in Section 5.7 of Bowater (2018) and Section 3.5 of Bowater (2020).…”

“…In the context of the above algorithm, the variable Γ will be referred to as the primary random variable (primary r.v. ), which is the way that this term was used in Bowater (2018Bowater ( , 2019Bowater ( , 2020. To clarify, if it is possible, which it is in many cases, to rewrite this algorithm so that, after the data set x is generated from the sampling density or mass function g(x | θ) by using some black-box procedure, the value γ of the variable Γ is generated by setting it equal to a deterministic function of the data x and the parameter θ j , then Γ would not be the primary r.v.…”