APTS 2020-21: Statistical Inference |

Lecturer: | Simon Shaw; s.shaw at bath.ac.uk |

Lecture Notes: pdf.

Slides covering all material: pdf.

**Material covered:**

Lecture 1: |
Introduction: reasoning about inferences, idea of an evidence function, equivalence of evidence, Weak Indifference Principle (WIP), Distribution Principle (DP), Transformation Principle (TP), DP /\ TP -> WIP. |

Lecture slides: pdf. Additional iPad notes: pdf. Lecture notes: p18 (start of Chapter 2) - p20 (end of Section 2.3). | |

Lecture 2: |
Weak Conditionality Principle (WCP), Strong Likelihood Principle (SLP), Birnbaum's Theorem: WIP /\ WCP < - > SLP, Strong Sufficiency Principle (SSP), Weak Sufficiency Principle (WSP), SLP -> SSP -> WSP -> WIP, Stopping Rules, Stopping Rule Principle (SRP), SLP -> SRP. |

Lecture slides: pdf. Additional iPad notes: pdf. Lecture notes: p21 (start of Section 2.4) - p25 (end of Section 2.6). | |

Lecture 3: |
Ancillarity, Strong Conditionality Principle (SCP), SLP -> SCP, Review of the principles and their logical framework, Key relation of the SLP, Bayesian statistics satisfies the SLP, MLE satisfies the SLP, Classical statistics typically violates the SLP, Reflections. |

Lecture slides: pdf. Additional iPad notes: pdf. Lecture notes: p25 (start of Section 2.7) - p30 (end of Chapter 2). |

Lecture 4: |
Definition of a loss function, Bayesian statistical decision problem, Definition of risk, Definition of Bayes rule and Bayes risk, Example of quadratic loss function, Example of quadratic loss with a Poisson likelihood and Gamma prior, Decision rule, Bayes (decision) rule and risk of the sampling procedure, Bayes rule theorem, Bayes rule for the posterior decision respects the strong likelihood principle. |

Lecture slides: pdf. Additional iPad notes: pdf. Lecture notes: p31 (start of Chapter 3) - p34 (end of Section 3.2). | |

Lecture 5: |
The Classical Risk, Example calculations of classical risks, Admissible decision rules, Theorem: the Bayes rule is admissible, Wald’s Complete Class Theorem. |

Lecture slides: pdf. Additional iPad notes: pdf. Lecture notes: p34 (start of Section 3.3) - p37 (end of Section 3.3). | |

Lecture 6: |
Point estimation, Quadratic loss, Bilinear loss, Absolute loss, Quadratic loss in higher dimensions, Set estimation, Level set, Level set property, Hypothesis tests. |

Lecture slides: pdf. Additional iPad notes: pdf. Lecture notes: p37 (start of Section 3.4) - p40 (end of Chapter 3). |

Lecture 7: |
Confidence procedure, Confidence set, Family of confidence procedures, Constructing confidence procedure using a pivot, Likelihood ratio test, Duality of acceptance regions and confidence sets. |

Lecture slides: pdf. Additional iPad notes: pdf. Lecture notes: p41 (start of Chapter 4) - p45 (end of Section 4.2). | |

Lecture 8: |
Level set property (LSP), Theorem: constructing a family of confidence procedures with the LSP, The linear model and exact confidence procedures with the LSP, Wilks’ Theorem, Wilks confidence procedures, Definition of a p-value, Definition of super-uniform. |

Lecture slides: pdf. Additional iPad notes: pdf. Lecture notes: p45 (start of Section 4.3) - p49 (end of Example 22). | |

Lecture 9: |
Definition of a significance procedure, Duality of significance procedures and confidence procedures, Proof of how to derive confidence procedures from significance procedures, Proof of how to derive significance procedures from confidence procedures, Theorem of how to construct significance procedures from ant test statistic, How to use simulation to compute a p-value. |

Lecture slides: pdf. Additional iPad notes: pdf. Lecture notes: p50 (top of page) - p56 (end of Section 4.7). |