Random Exchangeable Measures

• Random exchangeable measures are probability laws invariant under group actions, generalizing exchangeable sequences to complex structures.
• They underpin models in Bayesian nonparametrics, random graphs, and Pólya urn schemes by uniquely linking probabilistic symmetries with structural representations.
• Representation theorems such as de Finetti’s and Kallenberg’s offer practical methodologies to analyze stochastic processes in network models and invariant measures.

A random exchangeable measure is a probability law on a measurable space of random (possibly infinite) measures, or, more broadly, stochastic processes of measures, that is invariant under suitable group actions—typically permutations. Such measures generalize the classical notion of exchangeable sequences and arrays to the field of random measures, encompassing random distributions, processes, and random objects such as graphs, partitions, and Polya urns. The deep mathematical structure of random exchangeable measures encompasses de Finetti, Aldous–Hoover, and Kallenberg representation theorems, with major implications for Bayesian nonparametrics, random graph theory, and the study of probabilistic symmetries.

1. Foundations and Definitions

Let $(S, \mathcal{S})$ be a standard Borel space. A random measure %%%%1%%%% is a measurable map from a probability space into the space of (locally finite) measures on $S$. An exchangeable random measure (ERM) is defined by invariance under a group $G$ of symmetries, typically the infinite symmetric group $S_\infty$ acting by relabeling: for any $g \in G$ and Borel set $A \in \mathcal{S}$,

$\Law\bigl(\xi(A)\bigr) = \Law\bigl(\xi(gA)\bigr).$

For arrays or spaces such as $A^{\mathbb{N}^{(k)}}$ (arrays indexed by $k$-tuples), exchangeability refers to invariance of the law under all finitely supported permutations of indices:

$\Law\left(\left(\xi_{i_1,\ldots,i_k}\right)_{(i_1,\ldots,i_k)\in\mathbb{N}^k}\right) = \Law\left(\left(\xi_{\pi(i_1),\ldots,\pi(i_k)}\right)_{(i_1,\ldots,i_k)\in\mathbb{N}^k}\right) \quad \forall \pi\in S_\infty.$

In the context of random graphs or partitions, random exchangeable measures model structures whose distributions are invariant under relabeling of nodes or partition elements (Austin, 2013, Veitch et al., 2015).

2. Representation Theorems

Multiple classical results provide deep structural characterizations of random exchangeable measures:

• De Finetti’s Theorem: For exchangeable sequences $(X_n)$, there exists a random probability measure (the "directing measure") such that the $X_n$ are i.i.d. conditionally on this measure.
• Aldous–Hoover–Kallenberg Theorem: For exchangeable arrays (e.g., random graphs, hypergraphs), the law is represented as a mixture over suitable random measurable functions of i.i.d. random variables dependent on subsets of indices.
• Kallenberg’s Theorem for Random Measures: For random measures on $\mathbb{R}_+^2$ (modeling edge sets of graphs), any jointly exchangeable, symmetric simple point process can be constructed from a triple $(I, S, W)$ (a "graphex"), where
  • $I$ is the intensity of isolated edges,
  • $S$ is an integrable "star" function,
  • $W: \mathbb{R}_+^2 \to [0,1]$ is a symmetric kernel (generalizing the notion of a graphon),
  • subject to integrability and local-finiteness constraints (Veitch et al., 2015, Caron et al., 2014, Borgs et al., 2019).

3. Measure-Valued Pólya Sequences and Dirichlet Process Mixtures

Measure-valued Pólya urn sequences (MVPS) extend the classical finite-color Pólya urn scheme to scenarios with infinitely many colors, with urn compositions represented as finite measures on a space $\mathbb{X}$ (Sariev et al., 2023, Chorbadzhiyska et al., 2 May 2025). For an exchangeable MVPS with parameters $(\theta, \nu, R)$ (mass, base measure, reinforcement kernel), the evolution is: $$\mathbb{P}(X_{n+1}\in A | X_1,\dots,X_n) = \frac{\theta \nu(A) + \sum_{i=1}^n R_{X_i}(A)}{\theta + \sum_{i=1}^n R_{X_i}(\mathbb{X})}.$$ Exchangeable MVPS admit a stick-breaking Dirichlet mixture representation for their directing random measures. Specifically, under mild regularity, the directing measure $\widetilde{P}$ has the form

$\widetilde{P} = \sum_{j=1}^\infty V_j R_{U_j},$

with $V_j = W_j \prod_{i<j}(1-W_i)$ for i.i.d. $W_i \sim \mathrm{Beta}(1,\theta)$ and $U_j \sim \nu$. If $R_x$ is dominated by $\nu$ or $\mathbb{X}$ is countable, $\widetilde{P}$ is a Dirichlet process mixture over a family of distributions with disjoint supports (Chorbadzhiyska et al., 2 May 2025, Sariev et al., 2023).

Classification results show that non-balanced MVPS (where $R_x(\mathbb{X})$ is not constant) are forced to be i.i.d. sequences. For all balanced and exchangeable MVPS, the predictive distributions converge in total variation to $\widetilde{P}$ almost surely.

4. Exchangeable Random Measures in Random Graph Modeling

Random exchangeable measures underlie modern generative models for random graphs, both dense and sparse (Caron et al., 2014, Veitch et al., 2015, Todeschini et al., 2016). For example, the Caron–Fox model represents an undirected graph as a symmetric point process on $\mathbb{R}_+^2$, driven by completely random measures (CRMs). The associated random measure is jointly exchangeable, and graphs are obtained as adjacency measures via random marking of edges, with the link function $f(w_i,w_j) = 1 - \exp(-2 w_i w_j)$.

Kallenberg’s theorem provides that any jointly exchangeable, symmetric, simple random measure on $\mathbb{R}_+^2$ is determined by a graphex $(I, S, W)$, which unifies classical dense exchangeable graph models (graphons; when $I=S=0$ and $W$ is supported on a bounded domain) and sparse models (unbounded support, infinite-activity CRMs). This framework enables the derivation of explicit formulas for edge/vertex counts, degree distributions, and thresholds for the emergence of a giant component, including the presence of power-law degree phenomena and sparsity when the CRM’s Lévy measure is infinite-activity and regularly varying (Veitch et al., 2015, Caron et al., 2014, Todeschini et al., 2016).

5. Applications to Model Theory and Invariant Measures

The interaction between probabilistic symmetries and model theory is exemplified in applications to the study of invariant Keisler measures (Braunfeld et al., 2024). If $M$ is a countable homogeneous structure, automorphism-invariant probability measures on expansions of $M$ may, under combinatorial and growth-rate criteria (e.g. $k$–overlap closure), be forced to be fully exchangeable. In such contexts, every invariant Keisler measure is exchangeable and thus admits the Aldous–Hoover–Kallenberg representation. These results generalize classical theorems (de Finetti, Aldous–Hoover) and yield the first higher-arity classification of invariant measures in certain homogeneous structures since Albert–Ensley. They reveal deep connections between model-theoretic amalgamation properties, probabilistic symmetry, and probabilistic classification problems.

6. Generalizations and Related Symmetry Notions

Several notions generalize or weaken exchangeability:

• Swap-invariance: Zonoid-equivalent invariance under permutations (in moments, not full distribution), yields ergodic theorems and mixture representations with a random scaling factor and exchangeable component, but diffuse swap-invariant measures are trivial (proportional to Lebesgue/Borel measure) (Nagel, 2016).
• Relative Exchangeability: Invariance under only partial group actions, important in the representation of structures with partial symmetries or "partial exchangeability" (Braunfeld et al., 2024, Austin, 2013).
• Finitely Exchangeable Measures: Concerns when finitely exchangeable (but not infinitely) distributions admit extensions to longer exchangeable sequences, with a full functional-analytic criterion based on symmetrization operators (Konstantopoulos et al., 2015).

7. Further Directions and Open Problems

Open questions remain regarding the scope and limits of exchangeable random measure theory:

• The exact characterization of local-finiteness in the Kallenberg representation requires subtle integrability conditions that have only recently been fully clarified (Borgs et al., 2019).
• Extensions to higher-order arrays (hypergraphs), more general Polish product spaces, signed measures, and functional representations for conditional or partial invariance are all rich current research areas.
• The interplay of combinatorial model-theoretic properties with exchangeability poses deep structural questions, especially regarding when partial automorphism-invariance forces full exchangeability (Braunfeld et al., 2024).

Random exchangeable measures thus form the core language connecting probabilistic symmetries, combinatorics, nonparametric Bayesian statistics, stochastic processes, and model theory. They provide both practical inference tools (as in random network models and Bayesian nonparametrics) and broad structural insight into the effects of invariance under infinite or finite symmetries.