nLab de Finetti's theorem

Idea

De Finetti’s theorem is one of the central results of probability theory, and it usually refers to any of the following related statements:

• Every exchangeable probability measure on an infinite product is a unique mixture of iid ones.

• Every infinite sequence of exchangeable random variables exhibits conditional independence given the empirical distribution.

• Every infinite sequence of exchangeable random variables exhibits conditional independence given the exchangeable sigma-algebra.

The statement involves mixtures of probability measures, and so probability measures over probability measures. This recursive construction is one motivation for introducing monads in probability theory. Indeed, categorically, the theorem can equivalently be expressed as follows:

• The Giry monad is the limit of an exchangeable process in the category of Markov kernels.

Motivation

There are several ways of motivating de Finetti’s theorem. Here is one, which is very close to de Finetti’s original motivation, as explained in his original paper (de Finetti’29).

Consider a repeated experiment where we are uncertain about the probability distribution of the outcomes. For the simplest possible example, imagine repeatedly flipping a coin, but without knowing in advance the bias of the coin. One may be tempted to assume that the coin flips are independent, since “a coin has no memory”. However, this is not quite the case:

• What we can say is that the coin flips are exchangeable, since the coin is always the same, and so every flip, and each subsequence of flips, follows the same distribution.
• However, in general independence fails: there is hidden information in past flips about future flips: by flipping the coin repeatedly, we can get more and more certain about the coin’s bias. (For example, if we keep getting “heads”, we may start suspecting that the coin is not fair.)
• If we instead know exactly the bias of the coin, there is no more hidden information that past flips have about future ones. Therefore, given the bias of the coin, the flips are conditionally independent.

De Finetti’s theorem says precisely that for every exchangeable process we have conditional independence of the variables given their distribution.

In measure-theoretic probability

Let $X$ be a standard Borel space (for example, the real line), and form the infinite product $X^\mathbb{N}$.

Notice first of all that given the space $P X$ of probability measures over $X$ (the Giry monad), the iid sampling map $samp_\mathbb{N}:P X\to X^\mathbb{N}$ is exchangeable. In other words, given any finite permutation of the coordinates $\sigma: X^\mathbb{N}\to X^\mathbb{N}$, and denoting the kernel induced by $\sigma$ again by $\sigma$, the following diagram of Markov kernels commutes:

Consider now any (other) exchangeable probability measure $p$ on $X^\mathbb{N}$. If we write it as a Markov kernel $1\to X^\mathbb{N}$, exchangeability means by definition that, once again, the following diagram commutes for every $\sigma$:

A way of stating de Finetti’s theorem is now to say that $P X$ is the categorical limit of the diagram formed by the $\sigma$ (the group action of finite permutations). In other words, given any exchangeable measure $p$ (or more generally any exchangeable kernel), there exists a unique measure (or kernel) $u$ making the following diagram commute:

Explicitly, this means that we can write $p$ as a convex mixture (with measure $u$) of iid measures: for every measurable $A_1,\dots,A_n\subseteq X$,

$$p(A_1\times\dots\times A_n) \;=\; \int_{P X} samp_\mathbb{N}(A_1\times\dots\times A_n|q) \, u(d q) \;=\; \int_{P X} \big(q(A_1)\cdots q(A_n)\big)\,u(d q) .$$

This allows to define the space $P X$, i.e. the Giry monad, purely from its universal property. In terms of representable presheaves, one can say that the space of probability measure represents exchangeable processes.

Moreover, since the iid sampling map denotes conditional independence of its outputs given its input, the theorem implies that the various copies of $X$ are conditionally independent given the distribution ($P X$).

In Markov categories

Just as it happens for standard Borel spaces, in any observationally representable Markov category with conditionals and Kolmogorov products, a version of de Finetti’s theorem holds, and the proof is completely analogous to the measure-theoretic one (FGP’21).

See also (CFGKL) for a similar statement under different assumptions.

Conversely, one can define de Finetti limits in a Markov category as particular limits which are compatible with the Markov structure, and show that such a limit always gives a probability monad, making the Markov category representable (FFL’23, section 3.6).

In dagger categories

(For now, see here.)

Quantum versions

(For now, see the references.)

See also

• probability theory, categorical probability

• infinitary tensor product, Kolmogorov extension theorem

• Giry monad, probability monad, Markov category

• Hewitt-Savage zero-one law, law of large numbers

• exchangeability, limit, permutation

• ergodicity, ergodic decomposition theorem

• Wikipedia, de Finetti’s theorem

References

The original paper by de Finetti:

• Bruno de Finetti, Funzione caratteristica di un fenomeno aleatorio (The characteristic function of a random phenomenon), Atti del Congresso Internazionale dei Matematici: Bologna, 3-10 settembre 1928 (1929), pages 179-290. (Full text in Italian, English translation)

Category-theoretic accounts:

• Bart Jacobs, Sam Staton, De Finetti’s construction as a categorical limit, Proceedings of CMCS, 2021. (arXiv:2003.01964)

• Tobias Fritz, Tomáš Gonda, Paolo Perrone, De Finetti’s theorem in categorical probability. Journal of Stochastic Analysis, 2021. (arXiv:2105.02639)

• Sean Moss, Paolo Perrone, Probability monads with submonads of deterministic states, LICS 2022. (arXiv:2204.07003)

• Tobias Fritz, Tomáš Gonda, Paolo Perrone, Eigil Fjeldgren Rischel, Representable Markov categories and comparison of statistical experiments in categorical probability, Theoretical Computer Science 961, 2023. (arXiv:2010.07416)

• Noé Ensarguet, Paolo Perrone, Categorical probability spaces, ergodic decompositions, and transitions to equilibrium, arXiv:2310.04267

• Leihao Chen, Tobias Fritz, Tomáš Gonda, Andreas Klingler, Antonio Lorenzin, The Aldous-Hoover Theorem in Categorical Probability, (arXiv:2411.12840)

For the quantum case:

• Sam Staton and Ned Summers, Quantum de Finetti Theorems as Categorical Limits, and Limits of State Spaces of $C^\ast$-algebras, Proceedings of Quantum Physics and Logic, 2022. (arXiv)

• Tobias Fritz and Antonio Lorenzin, Involutive Markov categories and the quantum de Finetti theorem, 2023. (arXiv)

Introductory videos:

• Tobias Fritz, Categorical Probability and the de Finetti theorem, New York Category Seminar, 2021. (YouTube)

• Paolo Perrone, Markov categories: a tutorial. Applied Category Theory (ACT) 2023, tutorial video. De Finetti’s theorem starting at time 1:16:30. (YouTube)

• Paolo Perrone, Universal Properties in Probability Theory. Keynote talk, Category Theory (CT) 2023. De Finetti theorem starting at time 39:25 (YouTube)

• Antonio Lorenzin, Involutive Markov categories and the quantum de Finetti theorem, The Oxford Advanced Seminar on Informatic Structures (OASIS), 2023. (YouTube)

category: probability