Conditionality principle

The conditionality principle is a Fisherian principle of statistical inference that Allan Birnbaum formally defined and studied in his 1962 JASA article. Informally, the conditionality principle can be taken as the claim that experiments which were not actually performed are statistically irrelevant. Together with the sufficiency principle, Birnbaum's version of the principle implies the famous likelihood principle. Although the relevance of the proof to data analysis remains controversial among statisticians, many Bayesians and likelihoodists consider the likelihood principle foundational for statistical inference.

Formulation

The conditionality principle makes an assertion about an experiment E that can be described as a mixture of several component experiments E_h where h is an ancillary statistic (i.e. a statistic whose probability distribution does not depend on unknown parameter values). This means that observing a specific outcome x of experiment E is equivalent to observing the value of h and taking an observation x_h from the component experiment E_h, for example, rolling a dice (whose value is h = 1 ... 6) to determine which of six experiments to conduct (experiment E₁ ... E₆).

The conditionality principle can be formally stated thus:

Conditionality Principle: If E is any experiment having the form of a mixture of component experiments E_h, then for each outcome [math]\displaystyle{ (E_h, x_h) }[/math] of E, the evidential meaning of any outcome x of any mixture experiment E is the same as that of the corresponding outcome x_h of the corresponding component experiment E_h actually conducted, ignoring the overall structure of the mixed experiment (see Birnbaum 1962).

An illustration of the conditionality principle, in a bioinformatics context, is given by (Barker 2014).

References

• Barker, D. (2014). "Seeing the wood for the trees: Philosophical aspects of classical, Bayesian and likelihood approaches in statistical inference and some implications for phylogenetic analysis". Biology and Philosophy 30 (4): 505–525. doi:10.1007/s10539-014-9455-x. 
• Berger, J.O.; Wolpert, R.L. (1988). The Likelihood Principle (2nd ed.). Haywood, CA: The Institute of Mathematical Statistics. ISBN 978-0-940600-13-3. 
• Birnbaum, Allan (1962). "On the foundations of statistical inference". Journal of the American Statistical Association 57 (298): 269–326. doi:10.2307/2281640.  (With discussion.)

Further reading

• Kalbfleisch, J. D. (1975). "Sufficiency and conditionality". Biometrika 62 (2): 251–259. doi:10.1093/biomet/62.2.251.