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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

#31

Post by Eugeleo »

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 for the probability stuff, but I’m unsure about statistics. I don’t even know what the relation between probability and statistics is — most similar questions I found online (i.e. “How to learn stats?”) are answered with a “Read this probability book and you’re good”.

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

#32

Post by mtzet »

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…

The mathematics behind probability and statistics is about as ripe for intuition as calculus and linear algebra. A lot of it really comes down to counting in probability (calculus/measure theory for the continuous case) and quantifying properties about probability distributions for statistics.

The really hard part is the modelling part, where you transform the problem to a mathematical statement and vice versa. It's very easy to misinterpret both the problem in terms of mathematics and the mathematical result in terms of the problem. All the wrong answers to brain teasers like the monty hall problem, the tuesday boy problem etc., are right answers to the wrong question.

Unfortunately, in education we do not seem to want to discuss the modelling part on equal terms with the theory. We seem to be okay with solving the entire problem, or solving just the theoretical part with no regards to the application, but expressing just the mathematical problem to be solved is never appreciated. In a calculus setting, this could be deriving the answer to some physical problem depends on the solution of some partial differential equation -- even if you do not have the tools to solve it outright.

My guess is that it's just easier to teach theory with clear cut answers. Modelling the real world is ambiguous and hard.

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

#33

Post by Eugeleo »

qsort wrote: post #9
blululu wrote: post #7

On a different thread this morning someone bemoaned the lack of statistical education - a sentiment that is widespread among people who have studied and worked with statistics and probability. It is really exciting to see pedagogical tools that help explain basic but important concepts like distributions and sampling. Great work.

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 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 for the probability stuff, but I’m unsure about statistics. I don’t even know what the relation between probability and statistics is — most similar questions I found online (i.e. “How to learn stats?”) are answered with a “Read this probability book and you’re good”.

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

#34

Post by jtsuken »

mturmon wrote: post #29
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…

Wow, strong disagree. Once you develop intuition, probability is really quite intuitive. This kind of course should be working to develop this intuition — like the conditional probability examples and the CLT examples. The computational examples inline really help here. The Monte Hall problem is more of a curiosity than a fundamental principle! (Was a TA in undergrad engineering probability for 2 years, saw my share…

I can't argue with "once you develop intuition, probability is intuitive". I was arguing that lessons starting with E(X)=... basically stop the majority of people from getting to the point, where they see how their "initial intuition" is wrong.

Convincing as many people as possible that statistical intuition is not something we are born with should be the key priority of any probability and statistics class.

Monte Hall was one example. The birthday problem and the base rate fallacy are two more [1][2]. The result seems obvious but most people get these wrong.

With a couple of papers or books by Kahneman and Tversky in hand we can generate an almost infinite list of simple statistics/probability questions, which most people get wrong. Let people make some mistakes, before dumping the theory on them.

[1]https://en.wikipedia.org/wiki/Base_rate_fallacy [2]https://en.wikipedia.org/wiki/Birthday_problem

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

#35

Post by mtzet »

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…

I kind of like M.G. Bulmer's "Principles of Statistics". It's short and to the point so there's a chance of getting through it all. I really like the discussion of distributions in terms of raw data, it makes thinking about mean, variance, higher moments etc., much easier. It also doesn't skimp on the mathematical theory, but it doesn't allow itself to get bogged down by it.

That said, there's a chance I just read it late enough in my career to be more ready for its content.

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

#36

Post by jtsuken »

wodenokoto wrote: post #26
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…

Why are you calling the Monty Hall problem for the goat problem? It is known in academic circles as Monty hall and when it pops up in popular media, it is also referred to as Monty hall.

Thanks for pointing this out. While Google and Wikipedia confidently redirect me to the Monty Hall article, which does mention a goat at some point, the common name for it in English is "Monty Hall Problem". In other languages it's a three-door-problem or the goat problem, but given that it's such a good example for so many things in psychology and maths, I should be using the most common name in each language.

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

#37

Post by will_pseudonym »

WildParser wrote: post #30
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…

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."

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

#38

Post by jtsuken »

whatever1 wrote: post #21
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…

We are very good at finding correlations. It is still very hard to prove causality in natural phenomena from experiments, specially when we cannot control them. This became blatantly obvious in the covid outbreak where nobody had a clue for months about whether masks would help or not. Edit to clarify: It is very hard to prove to causality and be sure that you did not mess up.

You are right about people confusing causality and correlation. Otherwise this site wouldn't be so funny: https://www.tylervigen.com/spurious-correlations

You are wrong about people being good at finding correlations. I rarely met people who can process a sufficiently large sample size in their memory to calculate any significant correlation results. Whereas guessing correlations from charts exposes you to a number of optical illusions, which will fool the brain into seeing things that don't exist.

There may be a propensity to make more type 2 errors and see correlations between any random things such as 5G and COVID, but I haven't seen any research on that.

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

#39

Post by wodenokoto »

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…

> I don’t even know what the relation between probability and statistics is

That's a great question, and I think the lines are more than a little blurry.

My attempt at an answer would be:

Probability: Given a set of dice and coins and an order for rolling and throwing them, what is the chance of a specific outcome?

Statistics: Given a set of outcomes, what dice where rolled?

So if you want to know if smoking kills, you tally up medical history, and use statistics to see if there is a relationship between smoking and dying.

If you want to know the probability of smoking killing you, you look at the risc each cigarette brings to the table and tally it up using probability theory.

More elegantly phrased examples can be found on Stack Overflow: https://stats.stackexchange.com/questions/665/whats-the-diff...

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

#40

Post by marksbrown »

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…

Probability & statistics by de Groot is the standard text I believe. Full of examples and questions.
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