Dirichlet distribution

In plain terms

Suppose you want to pick a random set of proportions, say the way a unit of probability splits across several possible outcomes. You can't pick each share independently, because the shares are chained together and must total one, so if you pick the first freely 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, as every legal way of dividing the whole is weighted equally. It's what a statistician writes down to express complete ignorance about a set of proportions, a starting assumption declared openly before any data arrive.

Usually, then, it enters an analysis as a modelling choice. What earns it a page here is that it enters CSD's story in a completely different way, as a computed consequence and not a choice.

In CSD

Take a quantum state and compress it down to its outcome probabilities for a given measurement, forgetting the phases, and then ask what the natural volume on the space of states looks like after the compression. The answer, by the Duistermaat- Heckman theorem, a result about how symmetric curved spaces project onto flat ones, is the flat Dirichlet distribution exactly, so the curved geometry of state space projects to perfect fairness on the set of all possible proportions.

This is the workhorse behind the Born result for systems with any number of levels, as it converts curved volume ratios into elementary volumes of flat regions. It also carries a small philosophical sting. The distribution statisticians must assume, the geometry here derives, and nobody chose fairness as it fell out of symmetry.

Mathematically

The Dirichlet family lives on the probability simplex, indexed by positive concentration parameters, and setting all parameters equal to one gives flat Lebesgue measure on the simplex.

The theorem is that the pushforward of the Fubini-Study measure on CP^(N-1) along the (normalised) moment map is this flat member, at general N. It's 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. He worked at Breslau, then for nearly thirty years in Berlin, and then took Gauss's own chair at Gottingen in 1855, which he held for only four years before his death.

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

Related entries

• Fubini-Study measureThe space of quantum states allows exactly one way of measuring size that respects its symmetry, and that one measure is what turns regions of the space into Born-rule probabilities.
• Born volume ratioIn every finite dimension the Born weight of an outcome equals a ratio of volumes in the space of quantum states, for every outcome and every state, with no genericity condition attached.
• Duistermaat-Heckman formulaProject the curved space of quantum states down to its outcome probabilities and the curvature disappears exactly, which is what makes the Born rule computable in every dimension.
• Typicality volumeAlmost every way the experiment could have started gives the quantum statistics, and that isn't an assumption about probability, as it's a measured volume.
• Haar measureA closed and bounded family of symmetries has exactly one even-handed notion of volume, and every measure CSD calls forced inherits its uniqueness from that one fact.