# Seven Lectures in Algebraic Statistics

The goal of the Seven Lectures in Algebraic Statistics is to give a practical introduction to several topics in Algebraic Statistics. No previous knowledge of statistics will be assumed. Familiarity with algebra at the level of Cox, Little, O’Shea’s book “Ideals, varieties, and algorithms” is a plus. The lecture will be 60 minutes long and be followed by a 30 minute practical session in R or Macaulay2.

## 21.10.2019 - Lecture 1 - Introduction

Key words: Parametric statistical model, exponential families, toric varieties. This lecture is mainly based on Chapter 6, Sections 1 and 2 from Sullivant’s book “Algebraic Statistics”. Lecture notes available here.R code. Macaulay2 code.

## 28.10.2019 - Lecture 2 - Fisher’s exact test for discrete exponential families

Key words: Hypothesis testing, Fisher’s exact test, Markov basis, contingency tables. This lecture is based on Chapter 1 Section 1 from “Lectures in Algebraic Statistics” and Chapter 9 Section 1 from Sullivant’s book “Algebraic Statistics”. Lecture notes available here.

## 04.11.2019 - Lecture 3 - Conditional independence models

We will finish the overview of Fisher’s exact test and connection to toric ideals. Key words: Conditional independence statements, CI models, primary decomposition, axioms for conditional independence. Macaulay2 code.. Lecture notes available here.. Exercise Sheet.

## 11.11.2019 - Lecture 4 - Maximum Likelihood Estimation and Geometry

Key words: maximum likelihood estimation, maximum likelihood degree, iterative proportional scaling. Lecture Notes.

## Causality

How to infer causal relations between observed variables? One path we can take is to try to find a directed acyclic graph (DAG) that describes the causal relations between the variables in our data. In this three lectures we will see an example of this statistical approach for vectors of Gaussian random variables. We will study Gaussian graphical models, the PC-algorithm and highlight the connections to algebra throughout this study.

## 18.11.2019 - Lecture 5 - Graphical Models

## 02.12.2019 - Lecture 6 - Causality I

Key words: local Markov property, global Markov property, d-separation, Markov equivalence Lecture Notes.

## 09.12.2019 - Lecture 7 - Causality II

Key words: Learning a DAG, PC-algorithm, faithfulness assumption, algebraic hypersurface, singularities.