Category Theory and Statistics

A statistical model is a mathematical representation of a random phenomenon, and it can be viewed as a functor from the category of possible outcomes to the category of probability distributions.

A category is a collection of objects and morphisms, and it can be viewed as a generalization of a directed graph. In a Bayesian network, the objects are the random variables, and the morphisms are the conditional dependencies.

A Bayesian update is a way of updating a probability distribution in light of new evidence, and it can be viewed as a natural transformation between functors from the category of possible outcomes to the category of probability distributions.

A terminal object in a category is an object that has a unique morphism from every other object in the category. In statistics, a sufficient statistic is a statistic that has a unique distribution for each value of the parameter of interest.

A probability distribution is a way of combining a collection of possible outcomes into a single mathematical object. In category theory, a limit is a way of combining a collection of objects and morphisms into a single object.

A morphism in a category is a way of mapping one object to another. In statistics, a statistical test is a way of mapping a sample to a decision about a parameter of interest.

A statistical model is a way of combining a collection of possible outcomes into a single mathematical object. In category theory, a colimit is a way of combining a collection of objects and morphisms into a single object.

A directed graph is a collection of vertices and edges, where each edge has a direction. In a Markov chain, the vertices are the states of the system, and the edges are the transitions between states.

A random variable is a function from a sample space to a set of possible outcomes. In category theory, a monoid is a category with a single object and a single morphism from that object to itself. This can be viewed as a generalization of a random variable, where the sample space is the object and the set of possible outcomes is the morphism.

A natural transformation is a morphism between functors that preserves the structure of the functors. In a chi-squared test, the two functors are the functors from the category of possible outcomes to the category of probability distributions for the two categorical variables. The chi-squared test statistic is a natural transformation between these two functors.

A probability distribution is a function from a sample space to a set of possible outcomes. In category theory, a group is a category with a single object and a single morphism from that object to itself that is invertible. This can be viewed as a generalization of a probability distribution, where the sample space is the object and the set of possible outcomes is the morphism.

A limit is a way of combining a collection of objects and morphisms into a single object. In a t-test, the collection of objects is the set of possible outcomes of the experiment, and the morphisms are the functions that map each outcome to the corresponding t-statistic. The t-test statistic is the limit of this collection of objects and morphisms.

A probability distribution is a function from a sample space to a set of possible outcomes. In category theory, a ring is a category with two objects and two morphisms between those objects that satisfy certain axioms. This can be viewed as a generalization of a probability distribution, where the sample space is the first object, the set of possible outcomes is the second object, and the two morphisms are the addition and multiplication operations.

A colimit is a way of combining a collection of objects and morphisms into a single object. In an ANOVA, the collection of objects is the set of possible outcomes of the experiment, and the morphisms are the functions that map each outcome to the corresponding ANOVA statistic. The ANOVA statistic is the colimit of this collection of objects and morphisms.

A statistical model is a mathematical representation of a random phenomenon. In category theory, a topos is a category that satisfies certain axioms. This can be viewed as a generalization of a statistical model, where the category of possible outcomes is the topos.