"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.
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?
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.
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.)
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.
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.
> 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.
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.
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.
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).
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!
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:
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.
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 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.
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.
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”.
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]
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.
> 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.
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.
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!
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
"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.
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.
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.
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?
jtsuken|5 years ago
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...
will_pseudonym|5 years ago
mtzet|5 years ago
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.
mturmon|5 years ago
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.)
mnky9800n|5 years ago
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.
whatever1|5 years ago
1e-9|5 years ago
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.
WildParser|5 years ago
wodenokoto|5 years ago
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.
unwaryquerier|5 years ago
shekharshan|5 years ago
gbrown|5 years ago
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).
ArtWomb|5 years ago
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!
cjauvin|5 years ago
http://cjauvin.blogspot.com/2012/12/find-true-love-on-dating...
nextos|5 years ago
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.
blululu|5 years ago
qsort|5 years ago
plants|5 years ago
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.
Eugeleo|5 years ago
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”.
roenxi|5 years ago
> 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
giarc|5 years ago
Here is one on the Odds Ratio for example https://www.bmj.com/content/bmj/320/7247/1468.1.full.pdf
wodenokoto|5 years ago
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...
mtzet|5 years ago
That said, there's a chance I just read it late enough in my career to be more ready for its content.
emmanueloga_|5 years ago
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/
blablablerg|5 years ago
Maybe it is too basic for you, but It is focused on the intuition part and I can recommend it!
marksbrown|5 years ago
Layvier|5 years ago
ivansavz|5 years ago
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
laddng|5 years ago
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|5 years ago
And here is the board: https://www.edcollins.com/backgammon/diceprob.htm
will_pseudonym|5 years ago
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.
jp0d|5 years ago
unknown|5 years ago
[deleted]
unknown|5 years ago
[deleted]
mettamage|5 years ago
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?
suyash|5 years ago
eithed|5 years ago
johndoe42377|5 years ago
Numbers do not exist outside of human cultures.
Aeolun|5 years ago
chews|5 years ago
pwaivers|5 years ago
xabush|5 years ago
kyrers|5 years ago
nicetryguy|5 years ago
non-entity|5 years ago
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