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Ryan Martin

Statistical inference is concerned with the quantification of uncertainty about unknowns via data-dependent degree of belief measures. An inferential model (IM) formalizes this as a mapping from the data, posited statistical model, etc., to a general degree of belief measure. Important questions include:

• what properties should an IM satisfy?
• what do these properties imply concerning the mathematical form of the IM output?
• and how to construct an IM that satisfies these properties?

In Part 1 of the course (about 1 lecture), I introduce a validity condition designed to ensure that the belief measure output is reliable in a specific sense, and then I investigate the implications this has on the mathematical form of the IM's output. First, I will show that an IM whose output is a probability distribution cannot be valid and, second, I will demonstrate that valid IMs can take the form of consonant belief/plausibility functions. Furthermore, I will give a characterization of frequentist procedures having error rate control guarantees in terms of the same consonant belief/plausibility functions, suggesting that valid IMs can only take this form.

Having an understanding of what a valid IM looks like, the next question is how to construct such a thing. In Part 2 of course (about 1.5 lectures), I will focus on the construction presented in the Inferential Models monograph, co-authored with Chuanhai Liu, which is based on the use of random sets. With validity being guaranteed by construction, I turn to questions about efficiency. In particular, I will provide details about two fundamental dimension reduction strategies, namely, conditioning and marginalization, that lead to significant efficiency gains. Several non-trivial examples will be presented to illustrate the practical utility of this theory.

Finally, in Part 3 of the course (about 0.5 lectures), I will consider a number of open problems and unanswered questions, including, the construction of optimal/most-efficient IMs, the incorporation of partial prior information, the consequences of relaxing the validity condition, and the potential impacts of imprecise probability on the foundations of statistical inference.