2.3 Conditional probabilities 27
is also adapted. Afiltered probability spaceis the tuple (Ω,F,F,P), where
(Ω,F,P) is a probability space andF= (Ft)tis filtration ofF.
2.3 Conditional probabilities
Conditional probabilities are introduced so that we can talk about how
probabilities should be updated when one gains some partial knowledge about a
random outcome. Let (Ω,F,P) be a probability space and letA,B∈Fbe such
thatP(B)>0. Theconditional probabilityP(A|B)ofAgivenBis defined
as
P(A|B) =
P(A∩B)
P(B)
.
We can think about the outcomeω∈Ω as the result of throwing a many-sided
dice. The question asked is the probability that the dice landed so thatω∈A
given that it landed withω∈B. The meaning of the conditionω∈Bis that we
focus on dice rolls whenω∈Bis true. All dice rolls whenω∈Bdoes not hold
are discarded. Intuitively, what should matter in the conditional probability ofA
givenBis how large the portion ofAis that lies inBand this is indeed what
the definition means.
The importance of conditional probabilities is that they define a calculus of
how probabilities are to be updated in the presence of extra information.
The probabilityP(A|B)is also called thea posteriori(‘after the fact’)
probability ofAgivenB. Thea prioriprobability isP(A). Note thatP(A|B)is
defined for everyA∈Fas long asP(B)>0. In fact,A7→P(A|B)is a probability
measure over the measure space (Ω,F) called the a posteriori probability measure
givenB(see Exercise 2.7). In a way the temporal characteristics attached to
the words ‘a posteriori’ and ‘a priori’ can be a bit misleading. Probabilities are
concerned with predictions. They express the degrees of uncertainty one assigns
to future events. The conditional probability ofAgivenBis a prediction of
certain properties of the outcome of the random experiment that results inω
given a certain condition.Everything is related to a future hypothetical outcome.
Once the dice is rolled,ωgets fixed and eitherω∈A,Bor not. There is no
uncertainty left: predictions are trivial after an experiment is done.
Bayes rulestates that provided eventsA,B∈ Fboth occur with positive
probability,
P(A|B) =
P(B|A)P(A)
P(B)
. (2.2)
Bayes rule is useful because it allows one to obtainP(A|B)based on information
about the quantities on the right-hand side. Remarkably, this happens to be
