Great explanation! A lot of professors just go though the formulas without providing an intuition to what's going on. I love that in the field of AI and Data Science there are so many great lectures and tutorials online. Makes you wonder how useful university really is. Keep up the good work!
I am so glad I found this channel right as I started my PhD program in biostatistics. You straddle the line between proof/mechanics and intuition perfectly. So many videos on these topics are either way too surface level that I can't immediately apply it, or way too technical that I don't develop the intuition for what's going on. This is not the first video of yours that I have felt this way about. These videos are so good that sometimes after you reach a milestone in an explanation (like explaining how the acceptance probability drives the mechanics for this method), I just sit in awe at how good of an explanation it was that I lose track of the 15 seconds that has passed and I have to rewind the video. You are doing an incredible job; please keep it up.
i'm doing my internship in a pretty heavy statistics role where i have to sample from very weird custom distributions, thanks for saving my life with these sampling vids.
I can't thank you enough. Although I understood the concept in my class, still I wasn't able to visualize how this is working until I saw your video. You are helping thousands of us students. Thank you sooo much.
Couple of tricks for sampling: If you need to sample from a normal distribution, then you can take N uniformly distributed random numbers and add them up (rand + rand + rand ...), then you can scale this result horizontally and vertically if you need it, the result will be normally distributed - also many times I need sampling from exponential distributions to have an extreme behave for the random variable, for this I take 1/rand or ln(rand)^2, these methods are pretty fast and robust.
It's really nice for tutors to promise to come back later each time they want to introduce state-of-the-art solutions. It keeps students on track and motivates thinking.
Very nice explanation! We use a similar technique in physics for molecular monte carlo simulations where we don't know the value of the partition function (i.e. normalization constant) but we do know a state's Boltzmann factor (i.e. the numerator value). So when a new molecular state is proposed during the MC sim, you take a ratio of the two states' Boltzmann factors and that gives you the accept/reject probability.
Hey, that's super cool! I'm clearly more of a math person so I always love hearing when people have an actual application of some of these topics. Thanks!
0:00 - 1:10 Why we need Accept-Reject Sampling? 1:10 - 4:00 The case problem 4:00 - 6:25 How to sample from p(s) from g(s) 6:25 - 8:20 Criteria for accept/reject the sample from g(s) 8:20 - 12:20 Mathematical explanation of why the acceptance criteria works for the samples from g(s) using Bayes Theorem 12:20 - 15:08 Intuitive explanation of why the acceptance criteria works by really understanding what f(s)/g(s) means 15:08 - 17:20 Limitations of the Accept-Reject Sampling and importance of choosing the right g(s) 17:20 - end of video Conclusions and ending Thanks for the video! I love your explanations of this concept especially the intuitive understanding part.
You are very muchhhh underrateddd man!!You deserve similar appreciation as any other highly rated channel like 3B1B or Veritasium.May be people now a days go for animated videos only but your words are very valuable I could see that. You are trying to teach us exact same way you have learned it from scartch and that helps a lot.
many articles go through the algorithm, but it never really made sense to me as to why this works. This is a crazy good explanation of why it works(especially the bayes theorem part for accepting a sample).
Thank you! After looking at six other sources that all explained it the same way and coming up short, I really appreciated the effort you took to explain it differently and intuitively. And great pajamas!
Thank you for the amazing video. It's very useful when someone gives some intuition. Just one observation. By looking at wikipedia I can tell your proof is a bit misleading. You forgot to mention that the probability of acceptance is defined using a uniform distribution, instead of just getting there using the fact that P(A) = int(g(s)*p(A|s)ds). With this you get to the same expresion you use for P(A), but also you let your audience know that you need to use a uniform distribution to decide whether you reject or accept a sample.
Magnifique ! Do another AR video with some ( one or two ) examples!. ....Maybe you did and Ijust haven't seen it....I jumped on this one since it was very good, easy to follow, and as you stress, intuitive ! "Right On" as we used to say back in the 60's out there in LA.
Great explanation! 1 question though, if we have a f(s) that is easy to sample from, why can't we just directly sample from it and be done with, rather than going through the sample from g(s) steps?
if you need a good g in ARS, why not using g for p? The aim is to find a good unknown density for your samples right? So by finding a good enough g for ARS you find your good density approx. You dont need ARS at all in my intuition. What is the advantage of ARS? Maybe make an approx even more better?
You bring up a very interesting question. Usually, the distributions, p, that we want to sample from are not very nice looking (can have many peaks, noisy, etc.), so finding a distribution g that is "similar" to p can be challenging or impossible. So, instead we use a g that is "close enough" to the target and use ARS to actually sample from the target.
@@ritvikmath thx for answering my question. So ARS can smooth the potentially noisy density or find an easy alternative for p, if you have an approx/good candidate g for p, right? But what is the advantage against kernel density estimation KDE with special kernel k?
I've been hooked and watching through your videos in the past week. Do you have any favorite books or resources that you use for reference on the mathematical side of data science?
Hey, thanks for the kind words. I get this question often and the admittedly unsatisfying answer is no. I've found that different resources out there do a really good job at different things or at least offer different ways of viewing the same problem. When I try and learn a new topic, or review an old topic when making a video, I'll look around at lots of different resources to see which path I want to take in explaining it. That said, I think the most important part for learning (in my opinion) is to write some basic code, doesn't have to be pretty, which implements the method. That way, you can do sanity checks to see if your understanding matches to how real data would behave. Plus, you get some coding experience out of it. Sorry to not have an answer to your initial question but I hope this helps regardless!
I think this looks kinda similar (-ish) to MCMC algorithm? Maybe it's a good idea to cover MCMC in one of the next videos, since they are related. Anyway, that was a great video, I really enjoyed it. Keep up the good work!
You're reading my mind. I put this video out first so that in the MCMC videos (releasing next week onward), we can compare it against this. Stay tuned :)
The ratio you get from f(X)/Mg(X) gives you the probability of acceptance. Then you use a uniform distribution from 0 to 1 and if the value is less than the ratio, you accept. If it's bigger you reject.
Love the video. This may sound like a silly question but do you use some sort of threshold to decide whether you accept or reject something ? I get the intuition behind the ratio but whats the process of actually accepting or rejecting ?
Hi, the threshold is "f(s)/(Mg(s))", which is in (0,1). Since the larger result, say, f(s) greater, g(s) smaller, indicates g(s) can represent f(s) better, this sample should be accepted with greater probability. So now we can just generate u~U(0,1), and s~g(s), if u < the threshold, we accept. The process means, the more f(s)/(Mg(s)) close to 1, the higher probability u
Great video! Wouldn’t it make sense to always choose uniform distribution as g(s). Ofcourse it could be uniform within a large interval for practical purposes. What would be the reasons to choose any other g(s)?
This is a great explanation! I do have a specific practical question though. In the student score example, how do we practically get f(s)? Since the issue is we know f(s) yet we don't know but want to know p(s), it seems very curious to me how we could get f(s) in such an abstract math form or any math form?
This is a valid question and indeed something that I also had confusion over for a long time. This is a common case in Bayesian stats. For example, P(A|B) is proportional to P(B|A)P(A) / P(B) by Bayes theorem. We might care about sampling from P(A|B) which is the posterior but don't know its full form since the denominator P(B) might be difficult to compute. So, we can use Accept-Reject sampling to still sample from the posterior given only the numerator in Bayes theorem.
I am a little bit confused about inverse sampling. If I already had a pdf, why does I still need to bother inverse transformation to simulate a random number of the pdf I already obtained?
Maybe we don't need f(s) to always be lower than Mg(s), if we allow outputting the sample multiple times. E.g. if f(s)/Mg(s) = 2, then we'd output s twice.
So, [f(x)/Mg(x)] will be some number, say its 0.1. Then, we accept that sample with probability 0.1. That means, we generate some random number "u" from the Uniform distribution and if it is
So just to be sure, M*g(s) isn't technically a probability density because integrating M*g(s) over s would give a value greater than 1 right? ie: M scales probabilities of s, not the observable values of s? [edit] Actually the scaling makes sense I think, I was confusing your f(s) which isn't a pdf, for p(s)
Great video but I have a follow-up question: we were told to assume that our equation for P(A|s) can be interpreted as a probability. Why? Can you point to a proof for this?
Thank you for the great explanation. I am studying this concept for an actuarial exam, and my textbook says the probability of accepting a sample is 1/M "on average." Is this just because they are assuming f(x) is a pdf already? The book doesn't mention normalizing constants at all.
13:10 , dude, if the ratio is high doesnt mean that f(s) is high, its a ratio. and u started with we know f, which i think would be the problem, that we dont. and g and f modes and the whole curvature of f and g are kind of parallel, in the case of multimodal g, and unimodal f, i think this is not a good way to calculate p(s), it would damp the data beneath one of the modes of g. still good explanation
Why do we know the pdf? Can you provide a real example of how we know the pdf, even though it may be hard to sample from it? Like the example you provided above, with exponentials. How did we even know that the pdf had that analytical form?
You say we need g(s) to be close to p(s), but you also say we don't know p(s) and in the illustration you show g(s) close to f(s), not p(s). Also in most of the other materials I see covering this method they seem to say that f(s) is in fact the known target distribution you want to sample from, just that it might be difficult to sample from directly.
With this method, we're trying to find p(s) right? and p(s) is f(s)/constant. To use this method we need to know what f(s) is. Then don't we already know what p(s) is? Can someone explain?
I dont get where the numerator assumption comes from. In real life I would just have the scores of the students like 75, 63, 91 etc... Yes I can create an histogram from them but what is the numerator here?
Hi, for that last case(where M would be large if we choose Normal), how would you suggest solving that? Do we consider another distribution or can we consider multiple distributions, like piece-wise for sample creation?
Both. There's something called adaptive rejection sampling which creates some "proposal" distribution on the fly (See bishop book for more details). In general, rejection sampling has issues in high dimensions, Bishop's book explains it well (explained at the end of the section on adaptive rejection sampling)
