Definition
Let be a probability space, and let be a random variable, defined as a Borel-measurable function from to its state space Then a regular conditional probability is defined as a function called a "transition probability", where is a valid probability measure (in its second argument) on for all and a measurable function in E (in its first argument) for all such that for all and all
To express this in our more familiar notation:
where i.e. the topological support of the pushforward measure As can be seen from the integral above, the value of for points x outside the support of the random variable is meaningless; its significance as a conditional probability is strictly limited to the support of T.
The measurable space is said to have the regular conditional probability property if for all probability measures on all random variables on admit a regular conditional probability. A Radon space, in particular, has this property.
Read more about this topic: Regular Conditional Probability
Famous quotes containing the word definition:
“The very definition of the real becomes: that of which it is possible to give an equivalent reproduction.... The real is not only what can be reproduced, but that which is always already reproduced. The hyperreal.”
—Jean Baudrillard (b. 1929)
“The physicians say, they are not materialists; but they are:MSpirit is matter reduced to an extreme thinness: O so thin!—But the definition of spiritual should be, that which is its own evidence. What notions do they attach to love! what to religion! One would not willingly pronounce these words in their hearing, and give them the occasion to profane them.”
—Ralph Waldo Emerson (1803–1882)
“Scientific method is the way to truth, but it affords, even in
principle, no unique definition of truth. Any so-called pragmatic
definition of truth is doomed to failure equally.”
—Willard Van Orman Quine (b. 1908)