probability - Conditional expectation as a projection
Jun 24, 2024 · I don't understand how conditional expectation is a projection. I have a probably space $(\Omega, \mathcal{F}, \mathbb{R})$, and a random variable $X: \Omega $ to $\mathbb{R}$ then conditionnal expectation is defined as: $$E[X | \sigma(K)] = p_{L^2(\Omega, \sigma(K), \mathbb{P})}(X)$$
We prove existence of conditional expectations using orthogonal projection in Hilbert spaces. The following theorem is a basic result in Hilbert space theory, and is proved in the Appendix. Theorem 11 (Existence and uniqueness of orthogonal projections).
We isolate some useful properties of conditional expectation which the reader will no doubt want to prove before believing E(jG ) is positive: Y 0 ! E(YjG) 0) E(jG ) is linear: E(aX +bYjG) = aE(XjG)+bE(YjG) E(jG ) is a projection: E(E(XjG)jG) = E(XjG) More generally, the \tower property". If H ˆ G then E(E(XjG)jH) = E(E(XjH)G) = E(XjH)
Because conditional expectation is defined by orthogonal projection, all of the elementary properties of orthogonal projection operators translate to corresponding properties of condi-tional expectation. Following is a list of some of the more important of these.
Proposition 2.4 (Jensen’s inequality for conditional expectation). If ’is convex and Ej’(X)j<1, then ’(E[XjG]) E[’(X)jG]; a.s. Theorem 2.2 (Conditional expectation as a projection). Let L2(;G;P) = fY: Y 2G;EY2 <1g. If EX 2<1, then inf Y2L2(;G;P) E(X Y) is attained by Y = E[XjG]. Theorem 2.3 (Conditional expectation as a contraction ...
Conditional expectation 9.1 Inner product spaces There are several ways to introduce the notion of conditional expectation. We begin by introducing inner-product spaces and motivate a definition of conditional expectation by using the Projection Theorem. Definition 9.1. ArealvectorspaceX is called an inner-product space if for all x,y 2 X,
As you will soon see, projections can be used to de ne a conditional expectation map K : M+(X Y;A B) !M+(X;A) with analogous properties: (i) K x0 = 0 and K x1 = 1 a.e.[P]; (ii) K x(c 1f 1 + c 2f 2) = c 1K xf 1 + c 2K xf 2 a.e.[P] for constants c i 2R+; (iii) K xf 1 K xf 2 a.e.[P] if f 1(x;y) f 2(x;y) for all (x;y); (iv)if f n(x;y) "f(x;y) then ...
CONDITIONAL EXPECTATION AND MARTINGALES 7.1 Conditional Expectation. Throughout this section we will assume that random variables X are defined on a probability space (Ω,F,P) and have finite second moments so E(X2) < ∞. This allows us to define conditional expectation through approximating one
Uniqueness: If it exists, the conditional expectation is unique. General property: For all A ∈ F, we have E[Y 1A] = E[Y 0 1A]. ≡ (Y − Y 0 > ). ⇒ P(A ) = 0. we have P(A−) = 0. Definition 2. Let μ, ν two σ-finite measures on (Ω, F). Theorem 3. μ, ν σ -finite measures on (Ω, F), such that ν μ. Then there exists f ∈ F such that for all A ∈ F we have.
22.1 Conditional Expectation As a Projection · GitBook
Conditional Expectation As a Projection¶ Suppose we are trying to predict the value of a random variable $Y$ based on a related random variable $X$. As you saw in Data 8, a natural method of prediction is to use the "center of the vertical strip" at the given value of $X$.