Uniform mixtures of jointly dissociated array laws #
Every exchangeable probability law on arrays over a standard Borel space is a measurable mixture of jointly dissociated ones, with the same array symmetry as the original law and with one uniform variable on the unit interval as the mixing variable. This isolates the global noise in the Aldous--Hoover representation: it remains to represent the resulting measurable family by vertex (row and column) and cell noise.
Both array symmetries are covered. JointlyExchangeable.exists_dissociated_kernel and
SeparatelyExchangeable.exists_jointlyDissociated_kernel give the mixture as a Markov kernel from
the unit interval, with its composition against volume equal to the original law;
JointlyExchangeable.exists_dissociated_coding and
SeparatelyExchangeable.exists_jointlyDissociated_coding realize that kernel using a second
independent uniform variable. Their sections retain both the symmetry and joint dissociation, and
their joint pushforward is the original array law. The second variable samples a whole array; it is
not yet resolved into vertex (row and column) and cell variables.
The components of a separately exchangeable law are asserted to be jointly dissociated, which is
what dropping the global variable of a separate coding asks of them
(AldousHoover.exists_map_separateArray_snd_eq_of_jointlyDissociated).
The construction samples an array from the original law using uniform noise, then takes its
conditional law given the corner tail. Thus it only randomizes a standard Borel array space;
no standard Borel instance for the Giry space of probability measures is needed. Nothing in it
mentions a symmetry, so it is carried out once for an arbitrary property of the conditional laws
(exists_kernel_of_ae_condExpKernel_arrayTail, exists_coding_of_ae_condExpKernel_arrayTail) and
specialized afterwards.
References #
- D. Aldous, "Representations for partially exchangeable arrays of random variables", Journal of Multivariate Analysis 11 (1981), 581--598.
- O. Kallenberg, Probabilistic Symmetries and Invariance Principles, Springer, 2005, Chapter 7.
- O. Kallenberg, Foundations of Modern Probability, 3rd ed., Lemma 4.22 (randomization).
The corner-tail conditional laws of an array law, sampled by a uniform variable. Drawing
the conditioning variable through the canonical uniform coding presents those conditional laws as a
Markov kernel from the unit interval whose composition with the uniform law is the original law.
The supplied property P, when it holds for almost every conditional law, holds for almost every
component of the kernel.
An array law is sampled by two independent uniform variables, the first selecting a
corner-tail conditional law and the second sampling from it. The coding is jointly measurable, and
one almost-sure set of first variables works for the given property P of the conditional laws.
Taking P to be a conjunction gives one set on which all conjuncts hold.
A jointly exchangeable array law is a uniform mixture of jointly exchangeable, dissociated probability laws. The component laws form a measurable Markov kernel, so the same global parameter can be retained in subsequent conditional representations.
A jointly exchangeable array can be sampled by two independent uniform variables: the first selects an exchangeable, dissociated component law, and the second samples from that law. The coding is jointly measurable, and one almost-sure set of first variables works for both properties of the component law.
A separately exchangeable array law is a uniform mixture of separately exchangeable,
jointly dissociated probability laws. The components keep the two-axis symmetry of the original
law, so this is the reduction of the separate Aldous--Hoover representation to its ergodic form;
joint dissociation is the hypothesis that the global-variable-free separate coding
(AldousHoover.exists_map_separateArray_snd_eq_of_jointlyDissociated) asks of a component.
A separately exchangeable array can be sampled by two independent uniform variables: the
first selects a separately exchangeable, jointly dissociated component law, and the second samples
from that law. This is the separate-symmetry counterpart of
JointlyExchangeable.exists_dissociated_coding; the second variable samples a whole array and is
not yet resolved into row, column and cell variables.