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*Ryan Martin*

## Inferential models

#### Abstract

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.

#### Outline

Part 1 |
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1. Setup of the statistical inference problem |

2. Probabilistic inference |

3. Valid probabilistic inference |

4. Can probabilities be valid? |

5. If not probabilities, then what can be valid? |

6. Characterisation of frequentist procedures via plausibility |

Part 2 |
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1. IM construction |

2. Validity theorem |

3. Examples |

4. Beyond validity: efficiency |

5. Dimension reduction, I: Conditioning |

6. Dimension reduction, II: Marginalisation |

7. Extensions: generalised IMs and prediction |

8. Back to the frequentist characterisation theorem |

Part 3 |
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1. Efficiency and optimality |

2. Computations |

3. Partial prior information |

4. False confidence phenomenon |

5. Weakening the validity requirement |

6. Fundamental role of imprecise probability |

7. Maybe more |