## Lectures on Algebraic StatisticsHow does an algebraic geometer studying secant varieties further the understanding of hypothesis tests in statistics? Why would a statistician working on factor analysis raise open problems about determinantal varieties? Connections of this type are at the heart of the new field of "algebraic statistics". In this field, mathematicians and statisticians come together to solve statistical inference problems using concepts from algebraic geometry as well as related computational and combinatorial techniques. The goal of these lectures is to introduce newcomers from the different camps to algebraic statistics. The introduction will be centered around the following three observations: many important statistical models correspond to algebraic or semi-algebraic sets of parameters; the geometry of these parameter spaces determines the behaviour of widely used statistical inference procedures; computational algebraic geometry can be used to study parameter spaces and other features of statistical models. |

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### Contents

Markov Bases | 1 |

12 Markov Bases of Hierarchical Models | 11 |

13 The Many Bases of an Integer Lattice | 19 |

Likelihood Inference | 29 |

22 Likelihood Equations for Implicit Models | 40 |

23 Likelihood Ratio Tests | 48 |

Conditional Independence | 60 |

32 Graphical Models | 69 |

52 Exact Integration for Discrete Models | 114 |

Exercises | 123 |

62 Quasisymmetry and Cycles | 128 |

63 A Colored Gaussian Graphical Model | 131 |

64 Instrumental Variables and Tangent Cones | 135 |

65 Fisher Information for Multivariate Normals | 142 |

66 The Intersection Axiom and Its Failure | 144 |

67 Primary Decomposition for CI Inference | 147 |

33 Parametrizations of Graphical Models | 79 |

Hidden Variables | 89 |

42 Factor Analysis | 99 |

Bayesian Integrals | 105 |

68 An Independence Model and Its Mixture | 150 |

Open Problems | 157 |

164 | |