Estimates

Why

We want to guess uncertain outcomes. We model these uncertain outcomes as random variables.

Definition

Suppose $(\Omega , \mathcal{A} , \mathbfsf{P} )$ is a probability space and $x: \Omega \to V$ is a random variable. An estimate (or prediction, guess) for $x$ is a value $v \in V$. Some authors call the selection of an estimate estimation or an estimation problem.

Cost functions

A cost function for an estimate in $V$ is a function $C: V \times V \to \R $. A cost function is also known as a risk function. The cost of an estimate is itself a random variable $c: \Omega \to \R $ which is $c(\omega ) = C(x(\omega ), \xi )$.