Seeing Theory: A Visual Introduction to Probability and Statistics

https://seeing-theory.brown.edu/index.html

by max10541 • 6 years ago

519 89 6 years ago

seeing-theory.brown.edu

xabush• 6 years ago

Very interesting and helpful !

chews• 6 years ago

What a beautiful study resource.

pwaivers• 6 years ago

This is a really cool website!

suyash• 6 years ago

nice use of interactive data visualization to explain concepts, would love to see more teachers use this approach.

blululu• 6 years ago

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.

qsort2• 6 years ago

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 terms of realizations (i.e. much more similarly to what this tutorial does). Intuition without theory is shallow, but theory without intuition just leads to you eventually forgetting the theory.

Eugeleo4• 6 years ago

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”.

plants• 6 years ago

I am so glad I am not the only one that feels this way. In high school, I didn't have to take a single probability/stats class. In college, as a CS major (!!), I had to take a single intro stats class that was completely insufficient. And when a stats education is insufficient, god damn is it insufficient. No motivating examples whatsoever (what distribution would I use to measure ${real world process}? why would I need to calculate ${X} about the distribution?), just formulas that you're expected to memorize and vomit onto an exam with no understanding of why you're doing what you're doing at all.

What is the deal with this? Why isn't stats commonly taught in school when it is by far one of the most prevalent disciplines? And why, on the rare occasion when it is taught, is it so abysmal? Statistics forms the basis for all of science, for god's sake. I've since developed a patchwork understanding of statistics on my own from various resources I've found the time to consume. For the record, I grew up in the US.

shekharshan• 6 years ago

A couple of years ago I was just learning Python and was playing around with matplotlib. Running simulation of a dice roll 100, 1000, 10,000, 100,000, and 1,000,000 times started to show how the distribution starts to catch up with the expected 1/6th probability of each face. I was thinking how good it would be to teach young students this way.

gbrown• 6 years ago

Definitely! Also, not just young students. If you can get over code-phobia, doing random experiments in a class can be really illustrative. When I teach hypothesis testing, I always teach it both from a simulation perspective and from a traditional perspective.

For one, by doing the simulation part directly it's easier to see the "under repeated sampling..." logic inherent in frequentist procedures. Additionally, it's possible to do simulation-based procedures where traditional methods break down (think: permutation tests).

nextos• 6 years ago

You can get quite deep this way.

Generative models map well to programming concepts. Mixtures are quite similar to composition, and hierarchical models can be understood as inheritance. Lots of classical models like HMM, LDA, etc are quite similar to those presented in the GoF book in the sense they combine composition and inheritance in some particularly interesting manner.

ArtWomb• 6 years ago

In an effort to reduce screen time, I recently tried to instigate a game of classic table-top Dungeons & Dragons. And I swear, kids were even more interested in the BigInt N-sided die function I cribbed in a python shell than any demons or demigods ;)

Seeing Theory interactivity is very interesting. I think if there is one canonical example to tie it all together it would be something akin to "estimate the likelihood of an extremely rare event". Say, you're a top astrophysicist at NASA and you have to give the President a briefing on the improbability not impossibility of an extinction level asteroid event. And you must justify how those beliefs are informed by and change with data. It ties everything together: physically based world models, event spaces, conditional probabilities, monte carlo sampling and entropy estimation. And would be really fun to boot!

harshitaneja• 6 years ago

I had a similar idea to teach physics, well mostly scientific thinking and process by using a physics game engine for our world or some esoteric one and then asking them to perform some actions by creating theories of how the system behaves. Moving from a qualitative analysis of the system to building concrete quantitative theories and then in later stages having their simple theories fail as we deal with more complex interactions with the system and having them adapt their models of this world or rethink another.

Imagine how much their thought process would change if they intimately understood how scientific modelling works.

cjauvin• 6 years ago

A couple of years ago I was also a great fan of this paradigm where you try to convince yourself that you understand a math concept by coding/simulating it (a procedural, rather than conceptual understanding, if you will). Here for instance I studied the so-called "Secretary Problem", using the tools you mention:

http://cjauvin.blogspot.com/2012/12/find-true-love-on-dating...

laddng• 6 years ago

"If you roll 2 six-sided dice, what are the chances you roll at least one dice above 5 (5 or 6)?"

A nice trick to visually solve this in your head I heard once is:

If you think of rolling two dice as a square. X and Y are each dice. You get a 36 square board. Getting 1 six is just the upper boarder. 6 on the top, 6 on the right (6 and 6 overlap). So 11 out of the 36 squares.

borishn• 6 years ago

Chances of rolling below 5 are 4x4/36 = 16/36, so above are 20/36

And here is the board: https://www.edcollins.com/backgammon/diceprob.htm

astatine• 6 years ago

There is the small matter of "equal to 5"

will_pseudonym7• 6 years ago

Another way to think about it using the square board concept would be to figure out how many ways you can get not the result you're looking for, and take 36 minus that number for the number of possible squares out of 36 squares. So getting "no 6's" on either die would be the square on the board of 1 through 5, by 1 through 5, or 25 squares. So inverting that we'd arrive at the 11 squares.

In studying probability, I found that accounting for the "overlap" as you described it was more tedious in more complicated problems than just always calculating the joint probabilities and inverting them.

jtsuken5• 6 years ago

"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 trajectories, acceleration and distances (derivatives and integrals), areas, volumes, amounts, etc. But on probability your gut-feel will always fool you.

In the end, you see a handful of math bloggers bemoaning the lack of education in probability and the nonsense being discussed by journalists and politicians. And it hardly matters whether it's an election or a pandemic. The lack of understanding of uncertainty and the false belief that one can reason about these without looking at the numbers too closely is dangerous.

Sorry about the rant. But...

Dear creator of seeing-theory.brown.edu, if there is one thing you could change about the project to make it different and infinitely more useful: Please start the first chapter with the goat problem[1], then go through a couple of examples from chapter 10 in Thinking Fast and Slow[2], the discuss information (maybe with a simplified version of Mendel's pea experiment[3]), discuss distributions and leave expectations and variances for much-much later.

[1] https://en.wikipedia.org/wiki/Monty_Hall_problem [2]: https://en.wikipedia.org/wiki/Thinking,_Fast_and_Slow [3]: https://www.sciencelearn.org.nz/resources/1999-mendel-s-expe...

whatever1• 6 years ago

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.

jtsuken5• 6 years ago

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.

wodenokoto2• 6 years ago

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.

will_pseudonym7• 6 years ago

From Wikipedia[0]

... It became famous as a question from a reader's letter quoted in Marilyn vos Savant's "Ask Marilyn" column in Parade magazine in 1990 (vos Savant 1990a):

Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?

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

jtsuken5• 6 years ago

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.

will_pseudonym7• 6 years ago

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?

em5002• 6 years ago

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.

jibal5• 6 years ago

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.

st1x7• 6 years ago

I see this argument a lot and for some reason it doesn't help me with the intuition at all. If you (wrongly) get caught up on the fact that the remaining door and your pick have the same initial probabilities of being a car, then you'll still think that switching doesn't make a difference even in the million-door case.

Here's what works for me:

- the switching strategy always gives you the opposite of your initial choice

- the initial probabilities are 2/3 goat and 1/3 car so by switching you get 2/3 car and 1/3 goat

jhncls• 6 years ago

What's unintuitive about the Monty Hall problem is the difference between mathematical Monty and a psychological Monty. It is easy to imagine Monty almost only opening doors in case the guest choose the car, and not opening anything in case they choose a goat. So, when presented with the choice, a cautious guest will hesitate to change. If, however, the guest knows from previous shows that Monty will ALWAYS open a goat door, it is still mentally hard to change the cautious strategy.

cecilpl2• 6 years ago

What gave me the intuition for it was to realise that Monty is giving you the choice to either stick with your original door, or take the sum of the prizes between the other two doors.

(He also opens one of those two doors to reveal a goat, but you already knew that one of them had a goat so that doesn't give you any additional information.)

TurtleDove03• 6 years ago

Rather than going through this convoluted kinda similar problem, I find it easier to stick to the original one.

Get a piece of paper. Draw all possible outcomes, 9 total. ( Car is behind door 1 you pick 1, Car is behind door 1 you pick 2...). 3 of the 9 result in success.

Now draw the outcomes again but switch every time. 6 out of 9 outcomes are a success.

mturmon3• 6 years ago

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 of learners.)

jtsuken5• 6 years ago

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

jibal5• 6 years ago

"Once you develop intuition, probability is really quite intuitive"

That's a tautology.

Plenty of studies, such as the work by Kahneman and Tversky, show that humans by default have incorrect statistical intuitions. These faulty intuitions are hard to overcome, even by a considerable amount of training.

The Monte Hall problem is more of a curiosity than a fundamental principle!

It's quite straightforward conditional probability. That so many people, including trained mathematicians, get it wrong is quite illustrative. And it's not unique ... the coins and drawers problem is similar, and one can craft many others. MH is not a mere curiosity, it's simply well known.

WildParser2• 6 years ago

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.

will_pseudonym7• 6 years ago

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

mtzet2• 6 years ago

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.

swiley• 6 years ago

It's because math courses are crazy expensive and teaching both modelling and theory together would mean taking way longer (read costing way more) or making the failure rate (cost) way higher.

mnky9800n• 6 years ago

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.

1e-9• 6 years ago

But on probability your gut-feel will always fool you.

Yes, this is especially true when first learning; however, one can still develop intuition so that it serves as an invaluable motivator and guide through difficult problems.

unwaryquerier• 6 years ago

"The law of Small Numbers" chapter 10?

nicetryguy• 6 years ago

Gorgeous web design

Layvier2• 6 years ago

So so cool ! And it goes to show how poorly probabilities and statistics are usually taught, it's such a waste. I'm working on a non profit project aiming in parts to aggregate this kind of pedagogical tools into a collaborative learning map and serving it in a personalised way: https://sci-map.org. Early phases still, but if people are interested to contribute please hit me up!

ivan_ah• 6 years ago

sci-map sounds very interesting. Have you looked at metacademy.org before? They did a lot of good work on the data model (concepts, resources, learning pathways, etc), and also collected a lot of content, mostly on computer topics. https://metacademy.org/graphs/concepts/bayesian_logistic_reg...

Sadly the project is no longer actively developed but if you haven't seen it yet, you should definitely check out for inspiration: https://github.com/metacademy

Layvier2• 6 years ago

I didn't know about, seems very similar indeed! I will check it out in details, thanks a lot for sharing !

Eugeleo4• 6 years ago

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.

mtzet2• 6 years ago

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.

wodenokoto2• 6 years ago

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...

marksbrown• 6 years ago

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

roenxi• 6 years ago

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

tauist• 6 years ago

Probability Theory is a branch of mathematics. Statistics is the art of processing data to extract information suitable for the human cognitive system or a computer algorithm. Statistics use mathematical tools like physics or chemistry do.

ayewo• 6 years ago

Can you explain this sentence a bit more: "The best textbooks I've read where history of statistics and philosophy of statistics." ?

Are these names of actual books (Google doesn't help) or merely the themes of the stats textbooks you benefited from the most?

Thank you.

blablablerg• 6 years ago

I am doing harvard 110 now (free lectures on youtube) and I am working through his book Introduction in probability.

Maybe it is too basic for you, but It is focused on the intuition part and I can recommend it!

emmanueloga_• 6 years ago

I cannot recommended them because I have them on my back burner, but I would like to start working through the "Think" Stats series soon [1].

I love the premise: "if you know how to program, you can use that skill to learn other topics."

Perhaps someone here can speak to their experience with some of these books?

1: http://www.allendowney.com/wp/books/

giarc• 6 years ago

Not a text, but I highly recommend the Bland and Altman Statistics Notes in the BMJ. They are usually 1 page, easy to read explanation on a single statistic topic.

Here is one on the Odds Ratio for example https://www.bmj.com/content/bmj/320/7247/1468.1.full.pdf

johndoe42377• 6 years ago

Brain does not count. It updates/alters its structures - the process which humans model with numbers and back propagation.

Numbers do not exist outside of human cultures.

kyrers• 6 years ago

Amazing. Thanks for sharing!

Aeolun• 6 years ago

Got stuck at chapter 2 since there was zero indication what a valid syntax for a set was.

eithed• 6 years ago

I must say that this website/way of reading a book is simply gorgeous

mettamage• 6 years ago

How come that an undergraduate person (at the time) makes one of the most compelling statistic textbooks?

Is it because there are many more amateur statistic textbooks in existence, or published attempts at one (so more chance for a runaway success to be picked up)?

Or is it because people in the statistic textbook industry don't feel this frustration and/or don't dare to take any risk?

jp0d• 6 years ago

This looks brilliant. This can be very useful to grasp the concepts of probability and statistics in a visual way. I've been struggling to understand some of the concepts and I hope to use this as a supplement. Although, I don't believe it can replace a university course or a proper text book.

Seeing Theory: A Visual Introduction to Probability and Statistics

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