Hough transform

In probability theory, a conditional expectation (also known as conditional expected value or conditional mean) is the expected value of a real random variable with respect to a conditional probability distribution.

The concept of conditional expectation is extremely important in Kolmogorov's measure-theoretic definition of probability theory. In fact, the concept of conditional probability itself is actually defined in terms of conditional expectation.

Introduction

Let X and Y be discrete random variables, then the conditional expectation of X given the event Y=y is a function of y over the range of Y

$\mathrm{E}(X|Y=y)=\sum _{x\in \mathcal{𝒳}}x\mathrm{P}(X=x|Y=y)=\sum _{x\in \mathcal{𝒳}}x\frac{\mathrm{P}(X=x,Y=y)}{\mathrm{P}(Y=y)},$

where $\mathcal{𝒳}$ is the range of X.

If now X is a continuous random variable, while Y remains a discrete variable, the conditional expectation is:

$\mathrm{E}(X|Y=y)=\underset{\mathcal{𝒳}}{\int }x{f}_X(x|Y=y)dx$

where $f_X(\cdot |Y=y)$ is the conditional density of $X$ given $Y=y$.

A problem arises when Y is continuous. In this case, the probability P(Y=y) = 0, and the Borel–Kolmogorov paradox demonstrates the ambiguity of attempting to define conditional probability along these lines.

However the above expression may be rearranged:

$\mathrm{E}(X|Y=y)\mathrm{P}(Y=y)=\sum _{x\in \mathcal{𝒳}}x\mathrm{P}(X=x,Y=y),$

and although this is trivial for individual values of y (since both sides are zero), it should hold for any measurable subset B of the domain of Y that:

$\underset{B}{\int }\mathrm{E}(X|Y=y)\mathrm{P}(Y=y)\mathrm{d}y=\underset{B}{\int }\sum _{x\in \mathcal{𝒳}}x\mathrm{P}(X=x,Y=y)\mathrm{d}y.$

In fact, this is a sufficient condition to define both conditional expectation and conditional probability.

Formal definition

Let $(\mathrm{\Omega },\mathcal{ℱ},\mathrm{P})$ be a probability space, with a random variable $X:\mathrm{\Omega }\to {\mathbb{ℝ}}^n$ and a sub-σ-algebra $\mathcal{\mathscr{H}}\subseteq \mathcal{ℱ}$.

Then a conditional expectation of X given $\mathcal{\mathscr{H}}$ (denoted as $\mathrm{E}\left[X\right|\mathcal{\mathscr{H}}]$) is any $\mathcal{\mathscr{H}}$-measurable function ($\mathrm{\Omega }\to {\mathbb{ℝ}}^n$) which satisfies:

$\underset{H}{\int }\mathrm{E}\left[X\right|\mathcal{\mathscr{H}}](\omega )\mathrm{d}\mathrm{P}(\omega )=\underset{H}{\int }X(\omega )\mathrm{d}\mathrm{P}(\omega )\text{for each}H\in \mathcal{\mathscr{H}}$.

Note that $\mathrm{E}\left[X\right|\mathcal{\mathscr{H}}]$ is simply the name of the conditional expectation function.

Discussion

A couple of points worth noting about the definition:

• This is not a constructive definition; we are merely given the required property that a conditional expectation must satisfy.
  • The required property has the same form as the last expression in the Introduction section.
  • Existence of a conditional expectation function is determined by the Radon–Nikodym theorem, a sufficient condition is that the (unconditional) expected value for X exist.
  • Uniqueness can be shown to be almost sure: that is, versions of the same conditional expectation will only differ on a set of probability zero.
• The σ-algebra $\mathcal{\mathscr{H}}$ controls the "granularity" of the conditioning. A conditional expectation $E\left[X\right|\mathcal{\mathscr{H}}]$ over a finer-grained σ-algebra $\mathcal{\mathscr{H}}$ will allow us to condition on a wider variety of events.
  • To condition freely on values of a random variable Y with state space $(\mathcal{𝒴},\mathrm{\Sigma })$, it suffices to define the conditional expectation using the pre-image of Σ with respect to Y, so that $\mathrm{E}\left[X\right|Y]$ is defined to be $\mathrm{E}\left[X\right|\mathcal{\mathscr{H}}]$, where

$\mathcal{\mathscr{H}}=\sigma (Y):=Y^{-1}\left(\mathrm{\Sigma }\right):=\{Y^{-1}(S):S\in \mathrm{\Sigma }\}$

This suffices to ensure that the conditional expectation is σ(Y)-measurable. Although conditional expectation is defined to condition on events in the underlying probability space Ω, the requirement that it be σ(Y)-measurable allows us to condition on Y as in the introduction.

Definition of conditional probability

For any event $A\in \mathcal{𝒜}\supseteq \mathcal{ℬ}$, define the indicator function:

${\mathbf{1}}_A(\omega )=\{\begin{array}{ll}1& \text{if }\omega \in A,\\ 0& \text{if }\omega \notin A,\end{array}$

which is a random variable with respect to the Borel σ-algebra on (0,1). Note that the expectation of this random variable is equal to the probability of A itself:

$\mathrm{E}({\mathbf{1}}_A)=\mathrm{P}(A).$

Then the conditional probability given $\mathcal{ℬ}$ is a function $\mathrm{P}(\cdot |\mathcal{ℬ}):\mathcal{𝒜}\times \mathrm{\Omega }\to (0,1)$ such that $\mathrm{P}(A|\mathcal{ℬ})$ is the conditional expectation of the indicator function for A:

$\mathrm{P}(A|\mathcal{ℬ})=\mathrm{E}({\mathbf{1}}_A|\mathcal{ℬ})$

In other words, $\mathrm{P}(A|\mathcal{ℬ})$ is a $\mathcal{ℬ}$-measurable function satisfying

$\underset{B}{\int }\mathrm{P}(A|\mathcal{ℬ})(\omega )\mathrm{d}\mathrm{P}(\omega )=\mathrm{P}(A\cap B)\text{for all}A\in \mathcal{𝒜},B\in \mathcal{ℬ}.$

A conditional probability is regular if $\mathrm{P}(\cdot |\mathcal{ℬ})(\omega )$ is also a probability measure for all ω ∈ Ω. An expectation of a random variable with respect to a regular conditional probability is equal to its conditional expectation.

• For the trivial sigma algebra $\mathcal{ℬ}=\{\mathrm{\varnothing },\mathrm{\Omega }\}$ the conditional probability is a constant function, $\mathrm{P}\left(A\right|\{\mathrm{\varnothing },\mathrm{\Omega }\})\equiv \mathrm{P}(A).$

• For $A\in \mathcal{ℬ}$, as outlined above, $\mathrm{P}(A|\mathcal{ℬ})=1_A.$.

See also conditional probability distribution.

Conditioning as factorization

In the definition of conditional expectation that we provided above, the fact that Y is a real random variable is irrelevant: Let U be a measurable space, that is, a set equipped with a σ-algebra $\mathrm{\Sigma }$ of subsets. A U-valued random variable is a function $Y:(\mathrm{\Omega },\mathcal{𝒜})\mapsto (U,\mathrm{\Sigma })$ such that $Y^{-1}(B)\in \mathcal{𝒜}$ for any measurable subset $B\in \mathrm{\Sigma }$ of U.

We consider the measure Q on U given as above: Q(B) = P(Y⁻¹(B)) for every measurable subset B of U. Then Q is a probability measure on the measurable space U defined on its σ-algebra of measurable sets.

Theorem. If X is an integrable random variable on Ω then there is one and, up to equivalence a.e. relative to Q, only one integrable function g on U (which is written $g=\mathrm{E}(X\mid Y)$) such that for any measurable subset B of U:

$\underset{Y^{-1}(B)}{\int }X(\omega )d\mathrm{P}(\omega )=\underset{B}{\int }g(u)d\mathrm{Q}(u).$

There are a number of ways of proving this; one as suggested above, is to note that the expression on the left hand side defines, as a function of the set B, a countably additive signed measure μ on the measurable subsets of U. Moreover, this measure μ is absolutely continuous relative to Q. Indeed Q(B) = 0 means exactly that Y⁻¹(B) has probability 0. The integral of an integrable function on a set of probability 0 is itself 0. This proves absolute continuity. Then the Radon–Nikodym theorem provides the function g, equal to the density of μ with respect to Q.

The defining condition of conditional expectation then is the equation

$\underset{Y^{-1}(B)}{\int }X(\omega )d\mathrm{P}(\omega )=\underset{B}{\int }\mathrm{E}(X\mid Y)(u)d\mathrm{Q}(u),$

and it holds that

$\mathrm{E}(X\mid Y)\circ Y=\mathrm{E}(X\mid Y^{-1}\left(\mathrm{\Sigma }\right)).$

We can further interpret this equality by considering the abstract change of variables formula to transport the integral on the right hand side to an integral over Ω:

$\underset{Y^{-1}(B)}{\int }X(\omega )d\mathrm{P}(\omega )=\underset{Y^{-1}(B)}{\int }(\mathrm{E}(X\mid Y)\circ Y)(\omega )d\mathrm{P}(\omega ).$

This equation can be interpreted to say that the following diagram is commutative in the average.

                  E(X|Y)= goY
Ω  ───────────────────────────> R
          Y                        g=E(X|Y= ·)
Ω  ──────────>   R    ───────────> R
  
ω  ──────────> Y(ω)  ───────────> g(Y(ω)) = E(X|Y=Y(ω))
  
                        y    ───────────> g(  y ) = E(X|Y=  y )

The equation means that the integrals of X and the composition $\mathrm{E}(X\mid Y=\cdot )\circ Y$ over sets of the form Y⁻¹(B), for B a measurable subset of U, are identical.

Conditioning relative to a subalgebra

There is another viewpoint for conditioning involving σ-subalgebras N of the σ-algebra M. This version is a trivial specialization of the preceding: we simply take U to be the space Ω with the σ-algebra N and Y the identity map. We state the result:

Theorem. If X is an integrable real random variable on Ω then there is one and, up to equivalence a.e. relative to P, only one integrable function g such that for any set B belonging to the subalgebra N

$\underset{B}{\int }X(\omega )d\mathrm{P}(\omega )=\underset{B}{\int }g(\omega )d\mathrm{P}(\omega )$

where g is measurable with respect to N (a stricter condition than the measurability with respect to M required of X). This form of conditional expectation is usually written: E(X | N). This version is preferred by probabilists. One reason is that on the Hilbert space of square-integrable real random variables (in other words, real random variables with finite second moment) the mapping X → E(X | N) is self-adjoint

$\mathrm{E}(X\cdot \mathrm{E}(Y\mid N))=\mathrm{E}(\mathrm{E}(X\mid N)\cdot \mathrm{E}(Y\mid N))=\mathrm{E}(\mathrm{E}(X\mid N)\cdot Y)$

and a projection (i.e. idempotent)

$L_{\mathrm{P}}^2(\mathrm{\Omega };M)\to L_{\mathrm{P}}^2(\mathrm{\Omega };N).$

Basic properties

Let (Ω, M, P) be a probability space, and let N be a σ-subalgebra of M.

• Conditioning with respect to N  is linear on the space of integrable real random variables.

• $\mathrm{E}(1\mid N)=1.$ More generally, $\mathrm{E}(Y\mid N)=Y$ for every integrable N–measurable random variable Y on Ω.

• $\mathrm{E}(1_B\mathrm{E}(X\mid N))=\mathrm{E}(1_{B}X)$   for all B ∈ N and every integrable random variable X on Ω.

• Jensen's inequality holds: If ƒ is a convex function, then

$f(\mathrm{E}(X\mid N))\le \mathrm{E}(f\circ X\mid N).$

• Conditioning is a contractive projection

$L_P^s(\mathrm{\Omega };M)\to L_P^s(\mathrm{\Omega };N),\text{ i.e. }\mathrm{E}|\mathrm{E}(X\mid N){|}^s\le \mathrm{E}|X{|}^s$

for any s ≥ 1.

See also

• Law of total probability
• Law of total expectation
• Law of total variance
• Law of total cumulance (generalizes the other three)
• Conditioning (probability)
• Joint probability distribution
• Disintegration theorem

Notes

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References

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534Template:Page needed
  • Translation: 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
    
    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534Template:Page needed
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534Template:Page needed
• William Feller, An Introduction to Probability Theory and its Applications, vol 1, 1950 Template:Page needed
• Paul A. Meyer, Probability and Potentials, Blaisdell Publishing Co., 1966 Template:Page needed
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534, pages 67-69

External links

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