Minimum Message Length and Statistically Consistent Invariant (Objective?) Bayesian Probabilistic Inference—From (Medical) “Evidence”

Abstract

“Evidence” in the form of data collected and analysis thereof is fundamental to medicine, health and science. In this paper, we discuss the “evidence‐based” aspect of evidence‐based medicine in terms of statistical inference, acknowledging that this latter field of statistical inference often also goes by various near‐synonymous names—such as inductive inference (amongst philosophers), econometrics (amongst economists), machine learning (amongst computer scientists) and, in more recent times, data mining (in some circles).

Three central issues to this discussion of “evidence‐based” are (i) whether or not the statistical analysis can and/or should be objective and/or whether or not (subjective) prior knowledge can and/or should be incorporated, (ii) whether or not the analysis should be invariant to the framing of the problem (e.g. does it matter whether we analyse the ratio of proportions of morbidity to non‐morbidity rather than simply the proportion of morbidity?), and (iii) whether or not, as we get more and more data, our analysis should be able to converge arbitrarily closely to the process which is generating our observed data.

For many problems of data analysis, it would appear that desiderata (ii) and (iii) above require us to invoke at least some form of subjective (Bayesian) prior knowledge. This sits uncomfortably with the understandable but perhaps impossible desire of many medical publications that at least all the statistical hypothesis testing has to be classical non‐Bayesian—i.e. it is not permitted to use any (subjective) prior knowledge.

Keywords:

• Minimum Message Length
• MML
• Inference
• Bayesianism
• Statistical Invariance
• Statistical Consistency
• Evidence
• Evidence‐Based Medicine

Acknowledgements

I thank Chris Wallace (1933–2004) (Dowe 2008) for giving me a proper appreciation of Bayesianism and for introducing me to—and training me in—Minimum Message Length (Wallace and Boulton 1968; Wallace 2005). I thank Dr Jason Grossman for alerting me to the (subtle) distinction in the section “Statistical Consistency” between statistical consistency (or even efficiency) and small‐sample performance, and also for discussion motivating the section “Experimental Design, Data Collection Protocol and Likelihood Principle”. As in Dowe (2008, footnote 116), I am grateful to Claire Leslie for telling me about the bus number problem from the section “Maximum Likelihood”. And I thank those who anonymously reviewed the paper for useful guidance in helping me make the paper clearer and better.

Notes

[1] If tan⁻¹ ranges from −π/2 to π/2, then we take the negative square root for x < 0. We can write this more properly as (κ, θ) = (sign(x) · tan⁻¹(y/x)).