Thank you, I tried to understand GP via papers, but only you could help me to build up understanding the idea. That is great that you took time to explain gaussian distribution and the important operations! You're the best!
Learning a new subject via papers isn’t very helpful indeed :) They expect you to understand basic principles of GP. However lectures like these or books start with the basic principles💪🏻
This lecture is so amazing! The hand drawing part is really helpful to build up intuition reagarding GP. This is a life-saving video to my finals. Many thanks!
Wow, you saved my life with this genius lecture ! I think it's a pretty abstract idea with GP and it's nice that you can walk one through from scratch !
Thank you, thank you thank you!! I was stuck on a homework problem and still figuring out what it means to be a testing vs. training data set and how the play a role in the Gaussian Kernel function. I was stuck for the last 3 days, and your video from about 45min - 1 hour mark made the lightbulb go off!
I started learning about multivariate Gaussian processes in 2011, but it's terrible that I just got to this video when 2021 is ending. He explained things in a way that even a layperson could grasp. He first explains the meaning of the concepts, followed by an example/data, and last, theoretical representation. Typically, mathematic's presenters/writers avoid using data to provide examples. I'm always on the lookout for lectures like these, where the theoretical understanding is demonstrated through examples or data. Unless the concepts are not difficult to grasp, but the presenter/writer has made us go deep in order to open up complex notations without providing any examples.
Thanks a lot Prof. Just a minor correction for the people following the lectures. You made a mistake while writing out the formulae at 22:10 You were writing out mean and variance of P(X1|X2) whereas the diagram was to find P(X2|X1). Since this is symmetric, you can just get them by appropriate replacements, but just letting slightly confused people know
+akshayc113 I think all that should change is the formula for the given graphs. It should read: mu_21 = mu_2 + sigma_21 sigma_11*-1 (x_1 - mu_1). Everything else can stay the same.
Very clearly explained. The dependencies for learning the framework is concisely and incrementally given while details that make the framework harder to understand is elaborately evaded (You will understand what I mean if you try to dig through Rasmussen's book on GP).
1:04:08 - Would be good to emphasize that the test set is actually used for generating prior ... I had a hard time making sense out of it because the test set is usually provided separately (but in this case we are generating it !!)
Hello Nando, thank you for your excellent course. Following the bell example, the muy12 and sigma12 you wrote should be for the case that we are giving X2=x2 and try to find the distribution of X1 given X2=x2. Am I correct? Other understanding is welcomed. Thanks a lot!
Am I correct to think that the "f" notation in 30':30" is not the same "f" in 1:01':30"? In the latter case, each f consists of all the 50 f distributions that are exemplified in the former case? If that understanding is correct, then in sampling from the GP, each sample is a 50by1 vector from the 50D multivariate Gaussian distribution. This 50by1 vector is what Dr. Nando refers to as "distribution over functions". In other words, given the definition of a stochastic process as "indexed random variables", each random variable of GP is drawn from a multivariate Gaussian distribution. In that viewpoint, each "indexed" random variable is a function in 1:01':30". This lecture from 2013 is truly an amazing resource.
Thanks for the lecture, I have a problem with the discussion around 11 - from my understanding, a spherical case does represent some correlation between X and Y, as X is a sub-component of the max radius calculation, meaning larger x leads to smaller possible values of y (or at least lower probability for higher values). In other words, the covariance can be approximated to something like E[x*sqrt(r^2-x^2)]. Are we saying that ends up being zero, i.e. correlation is unable to express such a dependency? My intuition currently understands a square to express 0 correlation
Basic summary of lecture video: 1) Recap on multivariate Normal/Gaussian distribution (MVN). - some info on conditional probability 2) Some information on how sampling can be done from Univariate/Multivariate Gaussian distribution. 3) 39:00 - Introduction to Gaussian Process (GP) It is important to note that GP is considered as a Bayesian non-parametric approach/model
Thanks for the helful lecture! The only thing I want to point out is that if you put labels on the axises on your plots, it would be more helful for the listener to understand from the begging what you describe
1:05:58 Analog computers existed way before the first digital circuits. A WWII vintage electrical analog computer, for example, consisted of banks of op amps, configured as integrators and differentiators.
For the noisy GP case, we assume the noise is sigma^2 * the identity matrix, which assumes iid. What if the noise is correlated, can we incorporate the true covariance matrix?
Thank you for the lecture, and I appreciate the way you presented, spending a reasonable amount of time explaining Multivariate Gaussian distribution and building up from basics. My question to you is the following: If I happen to anticipate that the underlying distribution is Poisson (say), instead of Gaussian, WHAT will be the appropriate changes (I have an understanding its the likelihood which is modified, but not sure!). Will it still be called a Gaussian Process (or Poisson Process)?
Correct me if I am wrong, but isn't the whole cluster of examples starting at 36:35 flawed? Nando shows three points in a single dimension: x1, x2, x3 and their corresponding f-values: f1, f2, f3. It seems these points are three samples from a univariate normal distribution with a scalar variance, rather than what he shows, i.e. a vector from R^3 with a 3x3 covariance matrix.
On further reflection, perhaps you're doing a non-parametric approach in which you assign a Gaussian per point... ...since the distribution you're forming is empirical, it seems it would be more precise to to say the mean vector of the f-distribution is [f1, f2, f3], yes?
you want to describe the similarity between the f's via the given x's. The multivariate gaussian summerizes the connection/correlation between these (three) points
I have a question about the regression part which I spend a lot of time thinking. In the beginning, we are assume f_i ~ N(0,K). I think this is because for the prior purpose. At the Noiseless GP regression, we are using f as the mu. My understanding is if we had a measurement, we consider that as mu for that specific x. Is that correct? What if there are multiple measurements for same x? Thank you.
Sum of two Gaussian r.v with means mu1 and mu2 is gaussian with mean mu1+mu2. Isn't it? Multiple measurements are multiple draws from the Gaussian process, so the means must be added.
As far as I understood, Gaussian process (regression) serves two purposes: refining the prior (and posterior) and predicting the response for new points. If you collect new observations for the same points you are refining the posterior and if you extend your new point to a new dimension, you're predicting. In the former case, the confidence interval between two points remains relatively fat. Querying for points in new dimensions (given that practically you can do that) squeeze the confidence interval. Theoretically, it doesn't matter I guess. Think of an experiment in which you keep the x the same in every iteration but you read different y's. Think of another experiment in which your x values are changing from one iteration to another and you receives y's. From GP point of view, both are the same.
It looks like that the notation of the axis in the graph on the right side of the presentation, @ around 20:39, is not correct. It could probably be the x1 on x-axis. I.e: it would make sense if μ12 refered to the mean of variable x1, rather than x2, judging from the equation shown on the next slide.
5 minutes in and its already better than all 3 hours at class earlier today!
This is absolutely the best explanation of the Gaussian!
I feel so fortunate to find this video. It's like walking in a fog and finally be able to see things clearly.
The best tutorial for GP among all the materials I've checked.
Thank you, I tried to understand GP via papers, but only you could help me to build up understanding the idea. That is great that you took time to explain gaussian distribution and the important operations! You're the best!
Learning a new subject via papers isn’t very helpful indeed :) They expect you to understand basic principles of GP. However lectures like these or books start with the basic principles💪🏻
Thanks, this helped me a lot. By the time you got to the hour mark, you had covered sufficient ground for me to finally understand gaussian processes!
This lecture is so amazing! The hand drawing part is really helpful to build up intuition reagarding GP. This is a life-saving video to my finals. Many thanks!
The way you started from basics and built up on it to explain the Gaussian Processes is very easy to understand. Thank you :)
Finally! This is gold for beginners like me! Thank you Nando!! Saw you o the committee at the MIT defense, great questions!
This is indeed an Awesome lecture! I liked the way the complexity is slowly built over the lecture. Thank you very much!
Thank you for sharing this wonderful lecture! Gaussian process was so confusing when it was taught in my university. Now it is crystal clear!
I've found so many lectures for understanding gaussian process. Until now you are the only one I think can make me understand it.. Thanks a lot man
I tried to understand GP via blog article, paper and a lot of videos. Best video ever on GP! Thank you !
Wow, you saved my life with this genius lecture ! I think it's a pretty abstract idea with GP and it's nice that you can walk one through from scratch !
Very clear lectures ! Thanks for make them publicly available !
Thank you, thank you thank you!! I was stuck on a homework problem and still figuring out what it means to be a testing vs. training data set and how the play a role in the Gaussian Kernel function. I was stuck for the last 3 days, and your video from about 45min - 1 hour mark made the lightbulb go off!
Thank you ! Your videos are so much awesome than any ML lecture series I have seen so far ! -- Grad Student from CMU
What an amazing lecture. It is much clearer than lectures taught in my university.
This becomes very easy to understand with your thorough explanation. Thank you very much!
This lecture is amazing Professor. From the bottom of my heart, I say thank you.
Thank you very much Prof. de Freitas, excellent introduction
by far the best structured lecture on gaussian processes. love it :D
Thank you so much Professor De Freitas. What a clear explanation of GP
All time one of the best videos on TH-cam
That was a great lecture Mr.Freitas, thank you very very much!
I watched it to study my Computational Biology course, and it really helped.
Absolutely superb lecture! Everything is clearly explained even with source code.
Very good lecture, full of intuitive examples which deepens the understanding. Thanks a lot
The lecture is quite clear and it inspires me about the the key ideas of gaussian process.
Many thanks!
This is an amazing video! Clear and digestible.
wow, this is probably the best lecture I've ever watched. on any topic.
One of the best teachers in ML out there!
Here I am in 2021, yet your explanation is the easiest one to understand from all the sources I gathered! Thank you very much 😍
me in 2023, still the same
I started learning about multivariate Gaussian processes in 2011, but it's terrible that I just got to this video when 2021 is ending.
He explained things in a way that even a layperson could grasp.
He first explains the meaning of the concepts, followed by an example/data, and last, theoretical representation. Typically, mathematic's presenters/writers avoid using data to provide examples.
I'm always on the lookout for lectures like these, where the theoretical understanding is demonstrated through examples or data.
Unless the concepts are not difficult to grasp, but the presenter/writer has made us go deep in order to open up complex notations without providing any examples.
That is clear and flows naturally, Thank you very much.
Lucidly explained. Great video
FYI Notation @22:05 is wrong. since he selected an x1 to condition on, he should be computing mu2|1 but he is computing mu1|2
Great explanation and pace. Very legit.
after one hour of smooth explanation, he says and this brings us to Gaussian processes :)
thanks a lot prof. Very clean and easy to understand explanation.
Thanks a lot Prof. Just a minor correction for the people following the lectures. You made a mistake while writing out the formulae at 22:10
You were writing out mean and variance of P(X1|X2) whereas the diagram was to find P(X2|X1). Since this is symmetric, you can just get them by appropriate replacements, but just letting slightly confused people know
+akshayc113 I think all that should change is the formula for the given graphs. It should read:
mu_21 = mu_2 + sigma_21 sigma_11*-1 (x_1 - mu_1). Everything else can stay the same.
I think he actually meant to draw x_1 where x_2 is in the diagram. This switch would agree with the KPM formulae on the next slide.
Brilliant lecture. One could not have taught GPs better.
Very clearly explained. The dependencies for learning the framework is concisely and incrementally given while details that make the framework harder to understand is elaborately evaded (You will understand what I mean if you try to dig through Rasmussen's book on GP).
After 30mins, I am sure that he is top 10 teacher in my life
really clear and comprehensive. thanks so much.
Really great video for reading Gaussian Processes for Machine Learning!
Finally understood how this idea is explained and applied using mathematical language
Excellent introduction to the subject! Thank you :)
Great explanation. I wish that the title mentioned that it was part one of two, so that I would have known it was going to take twice as long.
amazing lecture.
An extremely good lecture! Thank you for recording this :) :)
The best lecture for gaussian processes
Awesome lecture, very well explained!
This helped me so much! Thanks!
Extremely good lecture. Well done.
a master doing his work
Gaussian processes start at 1:01:15
pin this
Great lesson! Thank you!
Amazing lecture!
well done! Very intuitive!
Great Teacher! Thanks!
Wow! Great Lecture!
BEAST MODE teaching
Great lecture!
Thank you for sharing this vdo, it was really helpful
Thank you! It is a wonderful lecture
Chapeau, good lecture!
great lecture !
Thank you so much for this amazing lecture. I wanted to applaude at the end but I realised I was in front of my computer.
amazing lecture, thanks a lot
1:04:08 - Would be good to emphasize that the test set is actually used for generating prior ... I had a hard time making sense out of it because
the test set is usually provided separately (but in this case we are generating it !!)
Thanks!!! Easy to understand👍👍👍
Great lecture. Thanks.
Very helpful. Thanks!
Hello Nando, thank you for your excellent course.
Following the bell example, the muy12 and sigma12 you wrote should be for the case that we are giving X2=x2 and try to find the distribution of X1 given X2=x2. Am I correct?
Other understanding is welcomed. Thanks a lot!
Am I correct to think that the "f" notation in 30':30" is not the same "f" in 1:01':30"? In the latter case, each f consists of all the 50 f distributions that are exemplified in the former case?
If that understanding is correct, then in sampling from the GP, each sample is a 50by1 vector from the 50D multivariate Gaussian distribution. This 50by1 vector is what Dr. Nando refers to as "distribution over functions".
In other words, given the definition of a stochastic process as "indexed random variables", each random variable of GP is drawn from a multivariate Gaussian distribution. In that viewpoint, each "indexed" random variable is a function in 1:01':30".
This lecture from 2013 is truly an amazing resource.
I like the way you teach
Thanks for the lecture, I have a problem with the discussion around 11 - from my understanding, a spherical case does represent some correlation between X and Y, as X is a sub-component of the max radius calculation, meaning larger x leads to smaller possible values of y (or at least lower probability for higher values). In other words, the covariance can be approximated to something like E[x*sqrt(r^2-x^2)]. Are we saying that ends up being zero, i.e. correlation is unable to express such a dependency?
My intuition currently understands a square to express 0 correlation
Amazing lecturer
Awesome explanation. thanks
Basic summary of lecture video:
1) Recap on multivariate Normal/Gaussian distribution (MVN).
- some info on conditional probability
2) Some information on how sampling can be done from Univariate/Multivariate Gaussian distribution.
3) 39:00 - Introduction to Gaussian Process (GP)
It is important to note that GP is considered as a Bayesian non-parametric approach/model
Thanks for the helful lecture!
The only thing I want to point out is that if you put labels on the axises on your plots, it would be more helful for the listener to understand from the begging what you describe
nando you are wondeful...
Love the content.
Great lecture
You are the best
Thank you!
This was a good explanation.
1:05:58 Analog computers existed way before the first digital circuits. A WWII vintage electrical analog computer, for example, consisted of banks of op amps, configured as integrators and differentiators.
Thankyou sir for a clear explanation
For the noisy GP case, we assume the noise is sigma^2 * the identity matrix, which assumes iid. What if the noise is correlated, can we incorporate the true covariance matrix?
Thank you for this
Round of Applause
Thank you for the lecture, and I appreciate the way you presented, spending a reasonable amount of time explaining Multivariate Gaussian distribution and building up from basics.
My question to you is the following: If I happen to anticipate that the underlying distribution is Poisson (say), instead of Gaussian, WHAT will be the appropriate changes (I have an understanding its the likelihood which is modified, but not sure!). Will it still be called a Gaussian Process (or Poisson Process)?
Correct me if I am wrong, but isn't the whole cluster of examples starting at 36:35 flawed? Nando shows three points in a single dimension: x1, x2, x3 and their corresponding f-values: f1, f2, f3. It seems these points are three samples from a univariate normal distribution with a scalar variance, rather than what he shows, i.e. a vector from R^3 with a 3x3 covariance matrix.
On further reflection, perhaps you're doing a non-parametric approach in which you assign a Gaussian per point...
...since the distribution you're forming is empirical, it seems it would be more precise to to say the mean vector of the f-distribution is [f1, f2, f3], yes?
I agree. That took me a while as well.
Fantastic
Great lecture......A minor claification at 38:25 minute of the video, it is said that given X's you want to model f's. What do you exactly mean there?
you want to describe the similarity between the f's via the given x's. The multivariate gaussian summerizes the connection/correlation between these (three) points
The best 👍
I have a question about the regression part which I spend a lot of time thinking. In the beginning, we are assume f_i ~ N(0,K). I think this is because for the prior purpose. At the Noiseless GP regression, we are using f as the mu. My understanding is if we had a measurement, we consider that as mu for that specific x. Is that correct? What if there are multiple measurements for same x? Thank you.
Sum of two Gaussian r.v with means mu1 and mu2 is gaussian with mean mu1+mu2. Isn't it? Multiple measurements are multiple draws from the Gaussian process, so the means must be added.
when estimating 'f', why each point is treated as a separate dimension and not different points in the same dimension?
As far as I understood, Gaussian process (regression) serves two purposes: refining the prior (and posterior) and predicting the response for new points. If you collect new observations for the same points you are refining the posterior and if you extend your new point to a new dimension, you're predicting. In the former case, the confidence interval between two points remains relatively fat. Querying for points in new dimensions (given that practically you can do that) squeeze the confidence interval. Theoretically, it doesn't matter I guess. Think of an experiment in which you keep the x the same in every iteration but you read different y's. Think of another experiment in which your x values are changing from one iteration to another and you receives y's. From GP point of view, both are the same.
It looks like that the notation of the axis in the graph on the right side of the presentation, @ around 20:39, is not correct. It could probably be the x1 on x-axis. I.e: it would make sense if μ12 refered to the mean of variable x1, rather than x2, judging from the equation shown on the next slide.