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Statistical Models are Algebraic Varieties

My goal in this chapter of lecture notes is to introduce the basic notions of a statistical model and an algebraic variety and show how they are related to one another. The title “Statistical Models are Algebraic Varieties” could be considered as a mantra of algebraic statistics. It is the ubiquity of the polynomial structures that arise when describing statistical models that makes some statistical problems amenable to algebraic techniques. The algebraic perspective on statistical models provides a number of interesting applications, which I will discuss in more detail in later chapters. The goal of this chapter is only to show that it is not so far fetched to think that algebra could be useful in statistics by illustrating that a number of elementary statistical models have the structure of an algebraic variety. On a more pragmatic note, this chapter will also serve as a primary place to introduce much of the terminology and notation that will be used throughout the rest of the lecture notes. But what is algebraic statistics? I am sure that the answer will vary greatly depending on who is asked. In the broadest possible sense, algebraic statistics is merely the study of the algebraic structures underlying statistical inference. Which algebraic structures are emphasized will vary greatly from person to person. The focus of my research in this area has been on the ways that algebraic geometry and the ensuing commutative algebra and combinatorics arise in statistical inference, and so I will tend to focus on these aspects in these lecture notes. Of course, one cannot talk about everything so I will have to leave out lots of interesting results. Why study algebraic statistics? Again, the answer will vary greatly depending on who is asked. Algebraic statistics is certainly not a magic bullet to solve all statistical problems. For me, the approach has always been to merely try to understand a statistical problem by looking through algebraic glasses. At the end of the day, this new point of view may lead to new algebraic solutions to statistical problems. It might explain unusual phenomena that statisticians observe. It might lead to purely mathematical problems, interesting in their own right, that are not informative for statistics. It might lead to a unifying framework for discussing and exploring many related statistical problems.