M-open, M-complete, M-closed

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I came across the M-open concept and found it useful to mention, though I haven’t had time to really delve into it. M-open, M-complete, and M-closed describe different relations between our hypothesis class and reality. Basically, do we have the true model in our hypothesis class? (Spoiler: no, we usually don’t, except with synthetic data.) If not, what does our estimation procedure provide us?

Fancy folks write this as \(\mathcal{M}\)-open etc., but life is too short for indulgent typography.

Le and Clarke (2017) summarises:

For the sake of completeness, we recall that Bernardo and Smith (Bernardo and Smith 2000) define M-closed problems as those for which a true model can be identified and written down but is one amongst finitely many models from which an analyst has to choose. By contrast, M-complete problems are those in which a true model (sometimes called a belief model) exists but is inaccessible in the sense that even though it can be conceptualised it cannot be written down or at least cannot be used directly. Effectively this means that other surrogate models must be identified and used for inferential purposes. M-open problems according to Bernardo and Smith (2000) are those problems where a true model exists but cannot be specified at all.

They also mention Clyde and Iversen (2013) as a useful resource.

You’ll notice that many references find this concept interesting because of its application to Bayesian model stacking.

Related: likelihood principle, decision-theory, black swans, misspecified models, Robust bayes …