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 Post subject: Conditional expectation with respect to a sigma algebraPosted: Fri, 2 Dec 2011 07:52:19 UTC

Joined: Tue, 3 May 2011 06:53:24 UTC
Posts: 8
I'm going to bump this topic because I am also having trouble getting an intuitive grasp, but this seems intuitive to the contributing posters.

Basically, if you have something like , where is a random variable and is an event, then you get a scalar. However, is not a scalar, but in fact, a random variable, but I don't know what this random variable represents. In the first case, the intuition is: if I know this event happens, what is the new expected value of . However, in the second case, I'm confused by the intuition. It doesn't make sense to say: what is the expected value of if happens (because what does it mean for to happen? This is a collection of events, not just one.)

Can anyone explain this?

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 Post subject: Re: Conditional expectation with respect to a sigma algebraPosted: Fri, 2 Dec 2011 16:20:04 UTC
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Joined: Mon, 29 Dec 2008 17:49:32 UTC
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Split topic from here.

sum wrote:
I'm going to bump this topic because I am also having trouble getting an intuitive grasp, but this seems intuitive to the contributing posters.

Basically, if you have something like , where is a random variable and is an event, then you get a scalar. However, is not a scalar, but in fact, a random variable, but I don't know what this random variable represents. In the first case, the intuition is: if I know this event happens, what is the new expected value of . However, in the second case, I'm confused by the intuition. It doesn't make sense to say: what is the expected value of if happens (because what does it mean for to happen? This is a collection of events, not just one.)

Can anyone explain this?

It may be useful to think of it this in the other way:

Conditional expectation is about the "best prediction" you can make with incomplete information (especially if you are thinking about stochastic processes --- we know about what happened in the past (well, sort of), but we in general don't know what will happen in the future, but to write means you know what happens at time t for all t). So represents the maximal information contained in about (again, in stochastic processes your will typically be something like and you want ). Since itself is a random variable, we should still get a random variable back, but this time it has to be -measurable since all we know is (so, for example, if doesn't distinguish between two points, then there is no reason we can make different prediction about at these two points).

Then as a scalar quantity, where A is a nonnull event, becomes a special case. It is the value of at points of (recall is the sigma algebra generated by ), since all you know is whether A happened or not (it has). And of course, the "best" prediction we got about in this case is its expected value "when A happens", which is your scalar (if is scalar-valued) .

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 Post subject: Re: Conditional expectation with respect to a sigma algebraPosted: Sat, 10 Dec 2011 08:38:35 UTC

Joined: Tue, 3 May 2011 06:53:24 UTC
Posts: 8
Thanks for feeding some intuition in this. I still can't get a good explicit picture of it, but I'm understanding little tidbits. Let me make some statements and do an example problem to see if they make sense.

Say we have , where are random variables. Then, I'm assuming the random variable represents ? If not, then disregard the rest.

Now, let with the typical Borel sets, so . Define and . Then, I want to see if I compute and correctly.

We have since the sigma algebra generated by is just all the Borel sets, so is certainly measurable.

Is it right that (since essentially the points are equivalent in the sigma algebra generated by )?

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 Post subject: Re: Conditional expectation with respect to a sigma algebraPosted: Sat, 10 Dec 2011 10:23:16 UTC
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Joined: Mon, 29 Dec 2008 17:49:32 UTC
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sum wrote:
Thanks for feeding some intuition in this. I still can't get a good explicit picture of it, but I'm understanding little tidbits. Let me make some statements and do an example problem to see if they make sense.

Say we have , where are random variables. Then, I'm assuming the random variable represents ? If not, then disregard the rest.

Now, let with the typical Borel sets, so . Define and . Then, I want to see if I compute and correctly.

We have since the sigma algebra generated by is just all the Borel sets, so is certainly measurable.

Is it right that (since essentially the points are equivalent in the sigma algebra generated by )?

No. There is a subtle but important difference between and .

means the -function evaluated at (so there is an "almost every " or "almost surely" or ... that you need to add).

On the other hand, is the conditional expectation of given occurs, i.e. where is the event . Unfortunately for you, is null, so this is not defined for any .

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 Post subject: Re: Conditional expectation with respect to a sigma algebraPosted: Sat, 10 Dec 2011 20:31:51 UTC

Joined: Tue, 3 May 2011 06:53:24 UTC
Posts: 8
Okay, I guess I shouldn't have said disregard the rest. Maybe I would understand better with example. As such, can you tell me how to compute and defined above?

On a sidenote, isn't the event rather the event (We have so doesn't make sense)? In this case, I don't see how this is null (probability zero, yes) since if we take , the event .

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 Post subject: Re: Conditional expectation with respect to a sigma algebraPosted: Sun, 11 Dec 2011 11:42:38 UTC
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Joined: Mon, 29 Dec 2008 17:49:32 UTC
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Location: 127.0.0.1, ::1 (avatar courtesy of UDN)
sum wrote:
Okay, I guess I shouldn't have said disregard the rest. Maybe I would understand better with example. As such, can you tell me how to compute and defined above?

On a sidenote, isn't the event rather the event (We have so doesn't make sense)? In this case, I don't see how this is null (probability zero, yes) since if we take , the event .

Oops, yes, I mean .

There is no problem with the (i.e. your argument is -measurable is fine), so I'll do from first principle here.

Recall the Dynkin's lemma (also known as the lemma) and Carath\'{e}odory/Hahn-Kolmogorov extension theorem, so to find for borel B, it suffices to find

since the Borel sigma algebra is generated by the -system (together with of course, but that can be done as a limit).

Obviously you only need by the range of X. Then and so

Now, what is the unique (in ) that will give ? Obviously we differentiate with respect to to find out:

(abusing notation slightly: really mean where ). Note the factor that you gained here! It is the local scaling factor induced by (or its inverse, depending on which way you think of it, c.f. coarea formula). So finally:

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