Dirichlet distribution

In plain terms

Here is the problem. You want to pick a random set of proportions -- say, how a unit of probability splits across N outcomes. You cannot pick each share independently, because they are chained together: they must total one. Pick the first freely and the others feel it.

The Dirichlet family is the standard machinery for randomness under exactly that constraint, and its flat member is the fair one: every legal way of dividing the whole is weighted equally. It is what a Bayesian writes down to mean I know nothing about these proportions.

Usually it enters an analysis as a modelling choice -- a confessed prior. What earns it a page here is that it enters CSD's story a completely different way: as a computed consequence, not a choice.

In CSD

Compress a quantum state down to its outcome probabilities -- forget the phases -- and ask what the natural measure on states looks like after the compression. Answer, by the Duistermaat-Heckman theorem: the flat Dirichlet. Exactly. The curved geometry of state space projects to perfect fairness on the simplex.

This is the workhorse behind the general-N Born result: it converts curved volume ratios into elementary simplex volumes. And it carries a small philosophical sting -- the distribution that statisticians must assume, the geometry here derives. Nobody chose fairness; it fell out of symmetry.

Mathematically

The Dirichlet family lives on the probability simplex, indexed by positive concentration parameters; all parameters equal to one gives flat Lebesgue measure. The theorem: the pushforward of the Fubini-Study measure along the (normalised) moment map is this flat member, at general N -- a computed identity, with the M-factorial normalisation explicit.

The name

Peter Gustav Lejeune Dirichlet (1805-1859) was born at Duren, between Aachen and Cologne, to a family of Belgian descent, and studied in Paris -- where he reportedly kept Gauss's Disquisitiones under his pillow, having found the one book too expensive to buy and too essential to return. Breslau, then nearly thirty years in Berlin, then Gauss's own chair at Gottingen in 1855, which he enjoyed for only four years before his death.

The modern concept of a function -- any rule, not necessarily a formula -- is substantially his, as is the pigeonhole principle and the first honest convergence proof for Fourier series. The distribution bearing his name arose from an integral he evaluated; the statistics came much later and kept the name.