Cristiano Bocci - An Introduction to Algebraic Statistics with Tensors

At the beginning of a book on Algebraic Statistics, it is undoubtedly a good idea to give the reader some idea of the goals of the discipline.

A reader who is already familiar with the basics of Statistics and Probability is probably curious about what the prefix Algebraic might mean. As we will see, Algebraic Statistics has its own way of approaching statistical problems, exploiting algebraic, geometric, or combinatorial properties. These problems are somewhat different from the one studied by Classical Statistics.

We will illustrate this point of view with some examples, which consider well-known statistical models and problems. At the same time, we will point out the difference between the two approaches to these examples.

The initial concern of Classical Statistics is the behavior of one random variable . Usually is identified with a function with values in the real numbers. This is clearly an approximation. For example, if one records the height of the members of a population, it is unlikely that the measure goes much further than the second decimal digit (assume that the unit is m). So, the corresponding graph is a histogram, with a basic interval of m. This is translated to a continuous variable, by sending the length of the basic interval to zero (and the size of the population increases).

For random variables of this type, the first natural distribution that one expects is the celebrated Gaussian distribution, which corresponds to the function

where and are parameters which describe the shape of the curve (of course, other types of distributions are possible, in connection with special behaviors of the random variable ).

The first goal of Classical Statistics is the study of the shape of the function , together with the determination of its numerical parameters.

When two or more variables are considered in the framework of Classical Statistics, their interplay can be studied with several techniques. For instance, if we consider both the heights and the weights of the members of a population and our goal is a proof of the (obvious) fact that the two variables are deeply connected, then we can consider the distribution over pairs (height, weight), which is represented by a bivariate Gaussian, in order to detect the existence of the connection.

The starting point of Algebraic Statistics is quite different. Instead of considering variables as continuous functions, Algebraic Statistics prefers to deal with a finite (and possibly small) range of values for the variable