Disintegration theorem
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,
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
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 . For example, for the motivating example above, one can define , which gives that , a slice we want to capture.
- Let ∈ P(X) be the pushforward measure ν = π∗(μ) = μ ∘ π−1. This measure provides the distribution of x (which corresponds to the events ).
The conclusion of the theorem: There exists a -almost everywhere uniquely determined family of probability measures {μx}x∈X ⊆ P(Y), which provides a "disintegration" of into , such that:
- the function is Borel measurable, in the sense that is a Borel-measurable function for each Borel-measurable set B ⊆ Y;
- μx "lives on" the fiber π−1(x): for -almost all x ∈ X, and so μx(E) = μx(E ∩ π−1(x));
- for every Borel-measurable function f : Y → [0, ∞], In particular, for any event E ⊆ Y, taking f to be the indicator function of E,[1]
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 in P(X2) (which is (π1)∗(μ)-almost everywhere uniquely determined) such that
which is in particular
and
The relation to conditional expectation is given by the identities
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.CS1 maint: multiple names: authors list (link)
- ^ 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.
- Theorems in measure theory
- Probability theorems