# Conditional probability

Discussion in 'Probability' started by Vedran, Jun 15, 2010.

1. ### VedranGuest

How can p(A|B|C) be expanded?

Are p(A|(B|C)), p((A|B)|C) and p(A|B|C) all different or are they the
same thing.

Vedran, Jun 15, 2010

2. ### Bastian ErdnuessGuest

Do you know of any of them what it means?

Cheers,
Bastian

Bastian Erdnuess, Jun 15, 2010

3. ### Frederick WilliamsGuest

I know what p(X|Y) means but what does X|Y mean?

Frederick Williams, Jun 15, 2010
4. ### VedranGuest

Here is the explanation of what x,y and z mean.

****

Mutual information I(x;y) in information theory is defined as:

I(x;y) = p(x|y)/p(x) (1)

Conditional mutual information is written as: I(x;y|z)

By that rationale I would guess that conditional mutual information would
be:
I(x;y|z) = p(x|(y|z))/p(x) (2)

Actual definition is:
I(x;y|z) = p(x|y,z)/p(x|z) (3)

And I really do not have I clue how (3) is obtained. I(x;y|z) is defined
as "the information provided about the event x by the occurrence of the
event y given z".

Vedran, Jun 16, 2010
5. ### HenryGuest

If p(X|Y) is short for p(X|Y=y), e.g. the conditional density
of the random variable X given that another random variable Y
takes the value y, e.g. the derivative of the conditional cdf
Prob(X<=x|Y=y), then I would assume X|Y is short for X|Y=y
and so is a (conditional/constrained?) random variable which
has the conditional density.

Going back to p(A|B|C), I would take that to be short for
p(A|B=b|C=c) which looks as if it should be p(A|B=b,C=c).
Isn't this the same as p(A|(B=b|C=c)), p((A|B=b)|C=c)?

The only thing which makes me hesitate is the

Henry, Jun 16, 2010
6. ### Bastian ErdnuessGuest

= p(x,y)/[p(x)*p(y)]
Read this as I((x;y)|z) not as I(x;(y|z)).
That doesn't really make sense to me.
= p(x,y|z)/[p(x|z)*p(y|z)]

That makes sense to me.
Read this as "the information provided about the event x by the
occourrence of the event y, assuming that all the time z was already
given".

Cheers,
Bastian

Bastian Erdnuess, Jun 16, 2010