[MUSIC] Okay, so another thing we can look at now that we understand this notion of bias are measures of error as a function of the number of data points in our training data set. Okay, so let's start with looking at our true error or generalization error. But first, I want to make sure its clear that we are looking at these errors for a fixed model complexity. So previously we were looking at errors versus model complexity, but now we are fixing the model complexity. Looking as a function of the number of data points. Fixed model complexity. And let's look at our true error. And our true error starts somewhere. And it's somewhere high, because when we have very few data points, our fitted function is a pretty poor estimate of the true relationship between x and y. So our true error's gonna be pretty high, so let's say that w hat is not approximated well from few points. But as we get more and more data, we get a better and better approximation of our model and our true error decreases. But it decreases to some limit. And what is that limit? So this is true error, just to be clear. Well that limit is the bias plus the noise inherent in the data. Because as we get tons and tons of observations, well, we're taking our model and fitting it as well as we could ever hope to fit it, because we have every observation out there in the world. But the model might just not be flexible enough to capture the true relationship between x and y, and that is our notion of bias. Plus, of course, there's the error just from the noise in observations that other contribution. Okay, so this difference here Is the bias of the model and noise of the data. Okay so that might make sense as a plot of true error versus number of data points, but now let's look at training error. And this might look a little more surprising. So let's say our training error starts somewhere. But what ends up happening is training error goes up as you get more and more data points. Cuz remember, we're keeping the model complexity fixed. So when we have few data points, so with few data points, a fixed complexity model can fit them, I guess, reasonably well, whatever it is. Fit these points reasonably well, where reasonably of course depends on what the complexity of the model is. But as I get more and more and more data points, that same complexity of the model can't hope to fit all these points perfectly well. And what is the limit? What is the limit of training error? Let me just annotate this as training error. Well that limit is exactly the same as the limit of our true error. And why is that? Well, I have tons and tons of points there. That's all points that there could ever be possibly in the world, and I fit my model to it. And if I measure training error, I'm running it to all the possible points there are out there in the world. And that's exactly what our definition of true error is. So they converge to exactly the same point in the limit. Where that difference again, is the bias inherent from the lack of flexibility of the model, plus the noise inherent in the data. Okay, so just to write this down in the limit, I'm getting lots and lots of data points, this curve is gonna flatten out. To how well model can fit true relationship f sub true. Okay, so I feel like I should annotate here also, saying in the limit our true error equals our training error. Okay so what we've seen so far in this module are three different measures of error. Our training, our true generalization error as well as our test error approximation of generalization error. And we've seen three different contributions to our errors. Thinking about that inherent noise in the data and then thinking about this notion of bias in variance. And we finally concluded with this discussion on the tradeoff between bias in variance and how bias appears no matter how much data we have. We can't escape the bias from having a specified model of a given complexity. Okay, and in the subsequent few videos, we 're gonna look at these notions with formalism. [MUSIC]