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Sets and Probability

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Re: Sets and Probability

#21

Post by benrbray »

Sharlin wrote: post #20

Probability theory with sets actually generalizes well to infinite and even uncountable cardinalities—rather than counting events you just have to switch to the more general notion of the measure of a (sub)set [1]. [1] https://en.wikipedia.org/wiki/Measure_(mathematics)

For anyone looking properly (i.e. with proofs) learn measure theory, I highly recommend "Introduction to Measure Theory" [1] by Terrence Tao (has an excellent collection of exercises -- they're crucial to the reading!)

I think it's OK to skim the first few chapters about constructing Lebesgue measure from Jordan measure, as it makes the whole topic appear more difficult than it needs to be. From Chapter 4 (abstract measure spaces) onwards, the book is full of great insight.

A good one on probability & stochastic processes is "Measure Theory and Probability Theory" by Athreya & Lahiri.

[1] https://terrytao.files.wordpress.com/2011/01/measure-book1.p... [2] https://www.springer.com/gp/book/9780387329031

Re: Sets and Probability

#23

Post by stopachka »

sAbakumoff wrote: post #14

References to textbooks that OP read would be a nice addition to the essay

My favorite was this textbook:

https://www.amazon.com/gp/product/1292025042/ref=ox_sc_act_t...

This one was more dense, but very good in the beginning if you take the time: https://www.amazon.com/gp/product/0070484686/ref=ppx_od_dt_b...

---

Will update post later, but wanted to share with you now : } -- thanks for taking the time to read!

Re: Sets and Probability

#24

Post by atdixon »

petters wrote: post #11

Ironically, this is the sort of probability theory Nassim Taleb hates. :-)

Exactly. This is the brand of neutered, ludic probability on which The Black Swan is a holistic dump. Very ironic indeed that it leads with Taleb as the inspiration.

Re: Sets and Probability

#25

Post by BeetleB »

I don't recall how I learned it in high school, but for my undergrad statistics & probability course, sets were very much the basis for everything. I'm surprised that this isn't the norm ...?

Re: Sets and Probability

#26

Post by zokier »

I'm bit confused, isn't counting desirable outcomes vs possible outcomes standard way of teaching probability? How is nCr/nPr stuff explained if not in the context of counting outcomes?

I'm also confused where does set theory come in here?

Re: Sets and Probability

#27

Post by t_serpico »

benrbray wrote: post #21
Sharlin wrote: post #20

Probability theory with sets actually generalizes well to infinite and even uncountable cardinalities—rather than counting events you just have to switch to the more general notion of the measure of a (sub)set [1]. [1] https://en.wikipedia.org/wiki/Measure_(mathematics)

For anyone looking properly (i.e. with proofs) learn measure theory, I highly recommend "Introduction to Measure Theory" [1] by Terrence Tao (has an excellent collection of exercises -- they're crucial to the reading!) I think it's OK to skim the first few chapters about constructing Lebesgue measure from Jordan measure, as it makes the whole topic appear more difficult than it needs to be. From Chapter 4 (abstract m…

How useful do you think measure theory is from a practical perspective? I've seemed to get by without more or less during my career (which is pretty stats heavy).

Re: Sets and Probability

#28

Post by BeetleB »

canjobear wrote: post #4

The set-based intuitions are a nice guide, but knowing how to turn the crank and churn out the calculations is also crucial for the cases where intuitive thinking doesn't scale.

It's really not just a guide. It is practically the basis. Yes, knowing how to do the calculations will make life much simpler, but I often find the usual calculation rules are easy to misuse. I often derive things from scratch using the set based axioms.

Re: Sets and Probability

#29

Post by benrbray »

t_serpico wrote: post #27
benrbray wrote: post #21

Earlier quoted context omitted.

For anyone looking properly (i.e. with proofs) learn measure theory, I highly recommend "Introduction to Measure Theory" [1] by Terrence Tao (has an excellent collection of exercises -- they're crucial to the reading!) I think it's OK to skim the first few chapters about constructing Lebesgue measure from Jordan measure, as it makes the whole topic appear more difficult than it needs to be. From Chapter 4 (abstract m…

How useful do you think measure theory is from a practical perspective? I've seemed to get by without more or less during my career (which is pretty stats heavy).

Learning measure theory has eliminated a lot of blind spots for me when it comes to stats / probability / ML. I find that I get "stuck" much less frequently when reading papers, and I'm able to better in the gaps when authors are too sparse with the mathematical details. Before, I would always wonder, "how on earth did they come up with this idea?!", but the more math I learn, the more I'm able to recognize that certain ideas are inspired by XYZ branch of math.

It's also been useful for understanding the "novelty" papers like "Neural Differential Equations" and so on.

I was lucky enough to take a good measure theory class in college, where it was fine to spend an entire day working through a proof from a textbook. Now that I'm working, though, it would be harder to justify that time commitment. So it really depends on your priorities.

Re: Sets and Probability

#30

Post by cleansingfire »

jhncls wrote: post #9
canjobear wrote: post #4

The set-based intuitions are a nice guide, but knowing how to turn the crank and churn out the calculations is also crucial for the cases where intuitive thinking doesn't scale.

It's indeed a nice guide. One place where the intuition stops unexpectedly, is calculating the probability of getting 2 reds balls out (without replacement). In the situation with all balls together, the chance would be 1/4. But with two separate boxes, the probability would change to 4999/20000 (1/2 * (1/100 * (0 + 1/2 * 99/100) + 99/100 * (1/2 * 1/100 + 1/2 * 98/99))).

There's no 1/4 here. (1/2) (99/199) = 99/398 First ball removed changes the odds for second draw.
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