I once found a good, i.e. intuitive, explanation for the law of iterated expectations. I since lost/ forgot it but would need it for a presentation for non quants I am working on. I am only prepared to use this as part of my presentation if I can remember/find the easy explanation; otherwise, I will lose the audience. It is just to make a point, rather than a particular application of it. Any help appreciated!thanks

When you take an expectation you are throwing away information. Whether you throw it all away at once or in two pieces doesn't make any difference.

When your random variables are square-integrable, you can think of conditioning as an orthogonal projection onto a smaller subspace, and so you can project all at once or do it in a couple steps. For just L1 random variables you need a couple more lines of math, but the intuition is still the same.

RDK, thanks for that. That, despite my limited maths knowledge, makes sense. It also sheds light on mj's remark.I think I am learning more on this forum than from my books at the moment.thanks guysPS: RDK, what do you mean by L1 random variables; haven't come across this term before.

refers to the space of random variables such that is finite. For finite measure spaces, when . In particular . To define the notion of an orthogonal projection, your space needs to have some geometry, which means it needs to have an inner product, ie it needs to be a Hilbert space. has such an inner product (the inner product of two square-integrable random variables is just the integral of their product), so it makes sense to think of conditional expectation as such a projection. isn't a Hilbert space, so a priori you can't think of conditioning this way. You can get around this without too much trouble. It is easy to show that for some sigma algebra , for X square-integrable, which means that conditioning is continuous with respect to the notion of distance. You use this fact as well as the fact that is dense in to define conditioning of strictly random variables as an appropriate limit of orthogonal projections of random variables. I hope this helps.

Why not just think of it as taking an average of averages as opposed to averaging it all at once?

Yeah i do usually think about conditioning as partitioning the sample space and then averaging over those partitions, as you mention. The only drawback is that this way typically only makes sense when the sample space is finite.

There is no such thing as an intuitive argument when it comes to infinite sample spaces that don't rely on some finite to infinite jump through something like... the axiom of choice.

Jeez, some of you guys have a weird idea of what constitutes "intuitive"! The original question was for an "intuitive, explanation for the law of iterated expectations...would need it for a presentation for non quants". I don't know which non-quants y'all deal with but I'm willing to bet a decent sum that they won't like having to deal with the axiom of choice, orthogonal projection or sigma algebras. If they really found those concepts intuitive, they would not need to employ you at all!I thought mj did rather a good job.

crmorcom,RDK asked about intuition for infinite sample spaces... Someone non-technical wouldn't care about infinte sample spaces... and I was merely explaining why you can use the same intuition. And mj's explanation tacitly assumes the axiom of choice. if you are talking about getting technical.

I suspect that someone non-technical would not care how many dimensions the space has. The axiom of choice (or equivalents) tends to be precisely that which you need to make sure your intuition carries over from finite to infinite: the point is that the intuition is the same. I completely agree with you, technically. My only issue is that what you, as an axiom-of-choice-aware person, find helpful for your intuition is definitely not going to be the same as what would be helpful to an infinity-virgin.Try, as an experiment, explaining the axiom of choice or Zorn's lemma to a non-technical friend. I have tried this a couple of times, and you are likely to find, as I did, that they can't understand what the fuss is: they tend to think it really is kind-of a no-brainer. It's only after you have spent a while twisting your brain with how fucked-up infinity can really be that you can understand why it's an issue at all. But, by that point, you are definitely not in Kansas anymore, Dorothy.

Let me be perfectly clear, I have and never will use the axiom of choice to explain things to a nontech... I was merely saying that: do the "simple example" using numbers, talk about averaging... then if one of the bright sparks complains that it is a finite sample space (which if they really aren't quanty ppl they won't) I would say the intuition is the same - it is a bit tricky to prove rigourously, but I can give you a reference if you like.