[This article was first published on R Works, and kindly contributed to R-bloggers]. (You can report issue about the content on this page here)Want to share your content on R-bloggers? click here if you have a blog, or here if you don't. In 1839, the gifted mathematician Peter Gustav Lejeune Dirichlet was attached to the Philosophy department at the University of Berlin working for less than full pay even though he had become a member of the Prussian Academy of Sciences in 1832. At that time, to become a “full professor” at the university it was required that a candidate deliver a Habilitationsschrift lecture in Latin. Apparently, Dirichlet’s facility with Latin wasn’t up to the task, so like many proficient “adjunct professors” today, Dirichlet took a side gig to support his family. He taught math at a military school. Anyway, I digress. It was about that time that Dirichlet began to work on a problem in celestial mechanics which involved this integral:Here which is attracted to a point where is the force of attraction and is the Euclidean norm.After a supernaturally insightful series manipulations detailed by Gupta and Richards, Dirichlet arrived at the following integral which you will recognize as the Beta function, the normalizing constant for the Dirichlet distribution:with mean and variance where and The Dirichlet distribution is a multivariate generalization of the Beta distribution that is often used in Bayesian statistics as a prior distribution for categorical and multinomial distributions. I illustrated this use of the Dirichlet in a previous post while constructing a Bayesian model for a three-state Markov chain. The Dirichlet distribution is remarkable in that it brings together 18th and 19th century work in analysis as exemplified by the Gamma, Beta and digamma functions with early 20th ideas from geometry and topology (the simplex) and modern Bayesian statistics.The (2)-SimplexA simplex is a generalization of the notion of a triangle to multiple dimensions. Informally in K dimensions, a simplex is the simplest polygon that is the convex hull of its K vertices. The vectors that comprise the simplex must be non-negative and sum to 1. So, a simplex is a natural way to represent probabilities that sum to 1 in multidimensional spaces.The support for the three dimensional Dirichlet distribution, the points on which the distribution is defined, is a (2)-simplex the triangular subset of a 2-dimensional plane intersecting the Euclidean axes at the points (1,0,0), (0,1,0), and (0,0,1). (Orient the triangle in the interactive plot below so that the reference plane is on top and the tip is pointing downward and you will see how the axes line up.)R packages used in this postlibrary(ggplot2)library(gganimate)library(dplyr)library(magick)library(MCMCpack) # for rdirichletlibrary(gtools) # for ddirichlet#library(patchwork) # for combining plotslibrary(threejs)library(extraDistr)Show the codeset.seed(42)# Sample from Dirichlet distribution over 3 categoriesn_samples