Thank you for this! Ive been waiting for 3blue1brown's 3rd part of the probabilities of probabilities series. There are a lot of very comprehensive videos on beta distribution out there but this has been the most intuitive so far.
To Luis Serrano- The most humble and articulated teacher I know so far. I am a data scientist who is self taught through videos/blogs and books and do not have any academic training in this field. I wish I could roll back time and start learning these watching your videos. You are an institution yourself. I request you if you could have a series of videos in the field of optimization in general.
For years I've been trying to understand the beta distribution. I'm out of college and studying for a probability exam, and it finally makes sense thanks to your video.
i stuck in beta baz they do not explain gamma to me now i knew which step they sliped you are the grokking machine learning‘s writer! i just be recommend this book.
For those who don't know why we use the Beta function as the normalizing factor: We need to sum up all the probabilities of all the coins. Since we have infinite numbers of coins, that summation means the area under the distribution curve. In calculus, the area under the curve is defined by the integral of the function of that curve. In this case, the function is x^(q-1)*(1-x)^(r-1) (q=8, r=4 for the example in the video; substituting them in the equation, you will have x^7*(1-x)^3). The integral of x^(q-1)*(1-x)^(r-1) from 0 to 1 (x is a probability, it can only take on a value from 0 to 1) is exactly the Beta function.
Dr Serrano thank you for your well explained video. Watching your videos understood in 1 hour more than I have been studying for months. that's a talent and thank you so much for your analysis. EVERYTHING IS MUCH APPRECIATED
Thank you Luis, that was a very good video. Clearly presented, easy to follow, it has helped me enormously in understanding this distribution. I'm studying Bayesian Statistics so it has helped me with that as well. I've subscribed, so I'll look at other videos in your collection. Much appreciated.👍
Just have to say, I've watched the binomial distribution vids of 3Bl1Br and your explanation made more sense!👍🏼 Granted we're still waiting for his Part 3; pls help prod him for it!😊
Very good video. Did a much better explanation than my statistics professor. One tiny suggestion though, at 11:16, I think it would be more intuitive to the viewers if you didn't vertically scale each probability graph but instead maintained the same scale. That way it is easier to see how a more spread-out distribution results in less probability density around the mode of 0.7.
Luis, I have a request. It may be something too difficult (I have been trying to find a good, understandable answer for many years now), but on the other hand, it is absolutely *fundamental* to the topics you cover on your channel. The question is: How would one go about calculating (or approximating numerically) values for the Gamma Function without the aid of a black-box tool like a spreadsheet gamma(x) or gammaln(x), or Wolfram Alpha, or anything like that? And not just for special values like for integers n or (n + 1/2), but for a general rational/real number r. Everywhere I look there are either a) appeals to an existing black-box function that will return a value for you, without understanding where it came from, b) resorting just to the general properties of the Gamma function (such as Gamma(x + 1) = x * Gamma(x)) and special pre-known values such as Gamma(1/2) being some value involving Pi, I believe, or c) virtually impenetrable mathematical formulas that assume the reader knows way more advanced calculus and advanced theorems that themselves require other advanced theorems to even *understand* what they're *for!* It's very frustrating, because the Gamma function is used everywhere when you start digging down into the fundamentals of probability theory, Bayesian statistics (and classical stats, too), information theory, machine learning, and a whole host of other areas. But *nowhere* have I found an explanation of how to calculate it that goes from 1) here's the definition and some properties of it, to 2) here's how you can actually calculate it for yourself if you really want to. Is it really that opaque??? Compare the situation with say the exponential or natural logarithm. If you have the patience, you can use the Taylor series to calculate them for whatever input you like. Likewise for most other common functions you can think of, trig functions, even things like the Normal distribution can be approximated in various fairly straightforward ways. Surely someone can open up the black boxes and explain how they work to us non-super-advanced-math folks?!?! Especially for such a fundamentally important function! Anyway, if you have any ideas you could mention or perhaps if you get inspired to even make a video about it, I personally would very much appreciate it, and I suspect I'm not the only person in the world who would! 😅
Gamma is defined as an integral, gamma(x) = integral from 0 to infinity: t^(x - 1) * e^(-t) dt You can use a substitution to make these bounds finite, e.g. let t = -ln u: gamma(x) = integral from 0 to 1: (-ln u)^(x - 1) du Now you can use numerical integration. You have to be a bit careful though since the lower bound is improper. A good strategy might be to start at u = 1 and integrate backwards, taking smaller and smaller steps as you get to u --> 0. Writing this as a summation is possible but not very useful as you'd want to program it, and at that point why not just use the black-box function anyway?
3 года назад
A very neat way to introduce this topic. Thanks sir
Excellent explanation, one minor comment though. At the end you present 3 graphs with the beta distribution at different values, and it's clear the one with a=701, b=301 is the more likely one to land close to 0.7, but you didn't show the *height* which I believe is the maximum likelihood estimator and it's much higher in the 3rd one.
I thought data was incorporated into the likelihood. But in your last example, with larger sample, the beta prior reflects the data increase. I’m not clear on that part.
I don't understand something. Why do we multiply the probability of heads to the probability of tails? Eg: coin 3 of the second example has P(H)=0.3, why do we multiply it with P(T)=0.7? Edit: I also didn't understand why is it (1-p) still
Luis, really great video as always. I would like to ask to make sure I understood, If I have a problem where I want to get a likelihood score for 3/5 and 60/100, and I want the second one to yield a higher score then I use the Beta Distribution right? The second question will the probability which is are under curve will differ a lot from f(x) from a=7, b=3 and a=700, b=301
Best video I have found on this topic - I have been waiting for 3b1b to finish the binomial topic by talking about using the beta. One question - let's say I have a coin and have done 10 throws and so I use beta distribution where a=8 and b=4. Can I then say something concrete like p(h) is between 0.6 and 0.8 with confidence of 54% (54% is based on cumulative dist at 0.8 minus cumulative dist at 0.6)? And with a=71 b=31 confidence would be more like 77% for the same range?
Excellent video. Thanks so much. One picky note: sometimes you pronounce it "beta" and sometimes "bee". The Roman letter B (bee) and the Greek letter B (beta) look alike, but aren't pronounced the same.
What is the intuitive explanation of the asymmetry of the curve? And why is the curve getting more symmetric as the number of coin flips rises? I would be nice to see these discussed/explained.
that's a great question, I hadn't thought about the symmetry... I don't know if the one with more flips looks more symmetric simply because it's skinnier, or because it's actually more symmetric for a combinatorial reason. I'll give it more thought to see if it means something, keep me posted if anything comes to mind!
Thanks! THere are some eigenvectors in this video: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-g-Hb26agBFg.html But soon there'll be a video on generalized eigenspaces, stay tuned!