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Seeing Theory: A Visual Introduction to Probability and Statistics

seeing-theory.brown.edu

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Re: Seeing Theory: A Visual Introduction to Probability and Statistics

#41

Post by qsort »

Eugeleo wrote: post #33
qsort wrote: post #9

Earlier quoted context omitted.

Agreed, this is extremely well-done. Even worse than the general lack of statistical education, I feel the teaching of statistics and probability suffers of the same problems as calculus/real analysis. Introductory statistics classes ramble at length about how random variables are functions from a probability space to a measurable space, but everyone who actually 'gets' the concept behind it eventually thinks in term…

I’ll copy my comment from other place in this thread, because I think it might be relevant here. I feel that most math subjects are treated either as full-on “fluff” (e.g. calculus, all computing, no theory building) or full-on theory (real analysis). A combination of intuition AND rigor is hard to come by. With that said... What textbook(s?) would you recommend for a thorough self-learning of statistics? I’m looking…

> I feel that most math subjects are treated either as full-on “fluff” (e.g. calculus, all computing, no theory building) or full-on theory (real analysis)

My background is computer science and I had a similar experience. Just a caveat: I'm not arguing that we should stop teaching theory, quite the contrary: most of the times we err on the side of the fluff. In particular, the fact that many reputable institutions are cutting formal logic, computability theory, etc. from their CS curriculums is an absolute disgrace. Intuition is hard to teach (easy to fall into the 'monads are burritos' trap) and it's something you have to work for yourself if you want to develop. My point is just that lack of intuition/operative knowledge will lead to your theoretical knowledge of the field being less in-depth and generally less helpful to you.

I honestly don't think it really matters what book you are studying as an introduction to a subject, as usually introductory courses are teaching well-established theory that everyone knows/agrees on. If you have no prior knowledge, a decent starting point is this: https://www.amazon.com/gp/product/1981369198/ the author's website has similar content: https://www.statlect.com

Re: Seeing Theory: A Visual Introduction to Probability and Statistics

#42

Post by roenxi »

Eugeleo wrote: post #31

What textbook(s?) would you recommend for a thorough self-learning of statistics? I’m looking for both intuition _and_ mathematical rigor — not all proofs, but not all fluff either. I’m a bioinformatics student and I will have a semester of combined probability/stats some time this year, but I think that won’t be enough to support me given my preference for DS-based bioinformatics jobs. I’m reading Feller right now f…

Rather than a textbook, I've had success getting a copy of the course notes directly from the stats department. The best textbooks I've read where history of statistics and philosophy of statistics.

> I’m reading Feller right now for the probability stuff, but I’m unsure about statistics.

Probability is the study of mathematical objects, and nobody is totally sure if any of them exist even in the approximate. Is anything in the universe random? The question is open, and likely to eternally remain so. Lots of things look similar to a random variable if viewed from the right perspective, but most of them aren't actually random. Not really a problem for the mathematicians, they feel no special need to study things that exist.

Statistics is roughly the study of how to deal with actual results. If you do a census, those results exist. Statisticians then need to make decisions about how to think about their results, and usually fall back on models rooted in probability. Technically speaking, "a statistic" is "any quantity computed from values in a sample". [0]

Basically, statistics is probability + data.

[0] https://en.wikipedia.org/wiki/Statistic

Re: Seeing Theory: A Visual Introduction to Probability and Statistics

#46

Post by WildParser »

will_pseudonym wrote: post #37
WildParser wrote: post #30

Earlier quoted context omitted.

My professor for statistics (he was quite famous in the field) talked about Monty Hall, but made clear that he will not give a solution because of science-political reasons.

I am incredibly curious what he meant by "science-political reasons."

Me too, but he refused to explain. I think there must have been a time where choosing a side was able to end friendships.

Re: Seeing Theory: A Visual Introduction to Probability and Statistics

#47

Post by em500 »

will_pseudonym wrote: post #28
jtsuken wrote: post #14

"Don't trust your intuition". This should be the basis for all teaching in statistics and probability. If all goes wrong, it should be the one thing everyone remembers from their statistics education. And yet year after year, everyone is starting with E(X)=sum(x*P(x)) and has no idea what it was about afterwards. With calculus and linear algebra your gut feel is about right no average. You can quickly get a feel for…

On the topic of the Monty Hall problem, what helped me "believe" it more was if you change it to 1,000,000 doors, still with only 1 car, and the rest goats. You choose 1 door. The host then opens up 999,998 other doors, which all contain goats. So there are 2 doors left. Your door, and the only other door the host didn't open. Do you feel at a gut level that you should switch?

Many people have suggested this "intuitive" explanation. But it's not at all clear or intuitive that jumping from 3 to 1,000,000 doors should lead the host to open 999,998 other doors rather than 1 other door.

Re: Seeing Theory: A Visual Introduction to Probability and Statistics

#48

Post by jibal »

will_pseudonym wrote: post #28
jtsuken wrote: post #14

"Don't trust your intuition". This should be the basis for all teaching in statistics and probability. If all goes wrong, it should be the one thing everyone remembers from their statistics education. And yet year after year, everyone is starting with E(X)=sum(x*P(x)) and has no idea what it was about afterwards. With calculus and linear algebra your gut feel is about right no average. You can quickly get a feel for…

On the topic of the Monty Hall problem, what helped me "believe" it more was if you change it to 1,000,000 doors, still with only 1 car, and the rest goats. You choose 1 door. The host then opens up 999,998 other doors, which all contain goats. So there are 2 doors left. Your door, and the only other door the host didn't open. Do you feel at a gut level that you should switch?

But this raises a different problem with intuition:

If Monty doesn't know where the car is, then if 999,998 doors were opened showing goats, leaving two doors, the odds that the car is behind your door or behind the remaining door is 1:1 ... this defies many people's intuition.

The difference between the two cases is that, if Monty knows where the car is, then his opening 999,998 doors with goats behind them is exactly what we expect, whereas if he doesn't know where the car is, then his opening 999,998 doors with goats behind them is an extraordinarily unlikely event. But if that does happen despite being extraordinarily unlikely, then there's still a 50% chance that the car is behind your door.

Re: Seeing Theory: A Visual Introduction to Probability and Statistics

#49

Post by mnky9800n »

jtsuken wrote: post #14

"Don't trust your intuition". This should be the basis for all teaching in statistics and probability. If all goes wrong, it should be the one thing everyone remembers from their statistics education. And yet year after year, everyone is starting with E(X)=sum(x*P(x)) and has no idea what it was about afterwards. With calculus and linear algebra your gut feel is about right no average. You can quickly get a feel for…

I'm always wondering the answer to this question:

Why does multiplication coupled with some sort of integral calculus work the way it does? We multiply to get moments of a distribution, we multiply to convolve, we multiply to get the work done on an object. I suppose the answer is multiplication allows us to scale some function f(x) with some function g(x). But I guess I want something deeper and I feel like I'm missing it.

Re: Seeing Theory: A Visual Introduction to Probability and Statistics

#50

Post by jibal »

em500 wrote: post #47
will_pseudonym wrote: post #28

Earlier quoted context omitted.

On the topic of the Monty Hall problem, what helped me "believe" it more was if you change it to 1,000,000 doors, still with only 1 car, and the rest goats. You choose 1 door. The host then opens up 999,998 other doors, which all contain goats. So there are 2 doors left. Your door, and the only other door the host didn't open. Do you feel at a gut level that you should switch?

Many people have suggested this "intuitive" explanation. But it's not at all clear or intuitive that jumping from 3 to 1,000,000 doors should lead the host to open 999,998 other doors rather than 1 other door.

"But it's not at all clear or intuitive that jumping from 3 to 1,000,000 doors should lead the host to open 999,998 other doors rather than 1 other door."

It SHOULD be clear, because you have two givens: 1) Monty never reveals the car. 2) He opens all the doors except 1.

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