In mathematics, the disintegration theorem is a result in measure theory and probability theory. It rigorously defines the idea of a non-trivial "restriction" of a measure to a measure zero subset of the measure space in question. It is related to the existence of conditional probability measures. In a sense, "disintegration" is the opposite process to the construction of a product measure.
Motivation
Consider the unit square in the Euclidean plane R2, S = [0, 1] × [0, 1]. Consider the probability measure μ defined on S by the restriction of two-dimensional Lebesgue measure λ2 to S. That is, the probability of an event E ⊆ S is simply the area of E. We assume E is a measurable subset of S.
Consider a one-dimensional subset of S such as the line segment Lx = {x} × [0, 1]. Lx has μ-measure zero; every subset of Lx is a μ-null set; since the Lebesgue measure space is a complete measure space,
\( E\subseteq L_{{x}}\implies \mu (E)=0. \)
While true, this is somewhat unsatisfying. It would be nice to say that μ "restricted to" Lx is the one-dimensional Lebesgue measure λ1, rather than the zero measure. The probability of a "two-dimensional" event E could then be obtained as an integral of the one-dimensional probabilities of the vertical "slices" E ∩ Lx: more formally, if μx denotes one-dimensional Lebesgue measure on Lx, then
\( \mu (E)=\int _{{[0,1]}}\mu _{{x}}(E\cap L_{{x}})\,{\mathrm {d}}x \)
for any "nice" E ⊆ S. The disintegration theorem makes this argument rigorous in the context of measures on metric spaces.
Statement of the theorem
(Hereafter, P(X) will denote the collection of Borel probability measures on a metric space (X, d).) The assumptions of the theorem are as follows:
Let Y and X be two Radon spaces (i.e. a topological space such that every Borel probability measure on M is inner regular e.g. separable metric spaces on which every probability measure is a Radon measure).
Let μ ∈ P(Y).
Let π : Y → X be a Borel-measurable function. Here one should think of π as a function to "disintegrate" Y, in the sense of partitioning Y into \( {\displaystyle \{\pi ^{-1}(x)\ |\ x\in X\}} \). For example, for the motivating example above, one can define \( {\displaystyle \pi ((a,b))=a,(a,b)\in [0,1]\times [0,1]}, \) which gives that \( {\displaystyle \pi ^{-1}(a)=a\times [0,1]} \) , a slice we want to capture.
Let \( \nu \) ∈ P(X) be the pushforward measure \( \nu \) = π∗(μ) = μ ∘ π−1. This measure provides the distribution of x (which corresponds to the events π \( {\displaystyle \pi ^{-1}(x)}) \).
The conclusion of the theorem: There exists a \( \nu \) -almost everywhere uniquely determined family of probability measures {μx}x∈X ⊆ P(Y), which provides a "disintegration" of \( \mu \) into \( {\displaystyle \{\mu _{x}\}_{x\in X}}) \), such that:
the function \( x\mapsto \mu _{{x}} \) is Borel measurable, in the sense that \( x\mapsto \mu _{{x}}(B) \) is a Borel-measurable function for each Borel-measurable set B ⊆ Y;
μx "lives on" the fiber π−1(x): for \( \nu \) -almost all x ∈ X,
\( \mu _{{x}}\left(Y\setminus \pi ^{{-1}}(x)\right)=0,\)
and so μx(E) = μx(E ∩ π−1(x));
for every Borel-measurable function f : Y → [0, ∞],
\( \int _{{Y}}f(y)\,{\mathrm {d}}\mu (y)=\int _{{X}}\int _{{\pi ^{{-1}}(x)}}f(y)\,{\mathrm {d}}\mu _{{x}}(y){\mathrm {d}}\nu (x).\)
In particular, for any event E ⊆ Y, taking f to be the indicator function of E,[1]
\( \mu (E)=\int _{{X}}\mu _{{x}}\left(E\right)\,{\mathrm {d}}\nu (x). \)
Applications
Product spaces
The original example was a special case of the problem of product spaces, to which the disintegration theorem applies.
When Y is written as a Cartesian product Y = X1 × X2 and πi : Y → Xi is the natural projection, then each fibre π1−1(x1) can be canonically identified with X2 and there exists a Borel family of probability measures \) \{\mu _{{x_{{1}}}}\}_{{x_{{1}}\in X_{{1}}}} in P(X2) \) (which is (π1)∗(μ)-almost everywhere uniquely determined) such that
\( \mu =\int _{{X_{{1}}}}\mu _{{x_{{1}}}}\,\mu \left(\pi _{1}^{{-1}}({\mathrm d}x_{1})\right)=\int _{{X_{{1}}}}\mu _{{x_{{1}}}}\,{\mathrm {d}}(\pi _{{1}})_{{*}}(\mu )(x_{{1}}),\)
which is in particular
\( \int _{{X_{1}\times X_{2}}}f(x_{1},x_{2})\,\mu ({\mathrm d}x_{1},{\mathrm d}x_{2})=\int _{{X_{1}}}\left(\int _{{X_{2}}}f(x_{1},x_{2})\mu ({\mathrm d}x_{2}|x_{1})\right)\mu \left(\pi _{1}^{{-1}}({\mathrm {d}}x_{{1}})\right)\)
and
\( \mu (A\times B)=\int _{A}\mu \left(B|x_{1}\right)\,\mu \left(\pi _{1}^{{-1}}({\mathrm {d}}x_{{1}})\right).
The relation to conditional expectation is given by the identities
\( \operatorname E(f|\pi _{1})(x_{1})=\int _{{X_{2}}}f(x_{1},x_{2})\mu ({\mathrm d}x_{2}|x_{1}), \)
\( \mu (A\times B|\pi _{1})(x_{1})=1_{A}(x_{1})\cdot \mu (B|x_{1}). \)
Vector calculus
The disintegration theorem can also be seen as justifying the use of a "restricted" measure in vector calculus. For instance, in Stokes' theorem as applied to a vector field flowing through a compact surface Σ ⊂ R3, it is implicit that the "correct" measure on Σ is the disintegration of three-dimensional Lebesgue measure λ3 on Σ, and that the disintegration of this measure on ∂Σ is the same as the disintegration of λ3 on ∂Σ.[2]
Conditional distributions
The disintegration theorem can be applied to give a rigorous treatment of conditional probability distributions in statistics, while avoiding purely abstract formulations of conditional probability.[3]
See also
Joint probability distribution
Copula (statistics)
Conditional expectation
References
Dellacherie, C.; Meyer, P.-A. (1978). Probabilities and Potential. North-Holland Mathematics Studies. Amsterdam: North-Holland. ISBN 0-7204-0701-X.
Ambrosio, L., Gigli, N. & Savaré, G. (2005). Gradient Flows in Metric Spaces and in the Space of Probability Measures. ETH Zürich, Birkhäuser Verlag, Basel. ISBN 978-3-7643-2428-5.
Chang, J.T.; Pollard, D. (1997). "Conditioning as disintegration" (PDF). Statistica Neerlandica. 51 (3): 287. CiteSeerX 10.1.1.55.7544. doi:10.1111/1467-9574.00056.
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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