In this video, I'd like to talk about how to evaluate a hypothesis that has been learned by your algorithm. In later videos we'll build on this to talk about how to prevent the problems of over fitting and under fitting as well. When we fit the parameters of our learning algorithm, we think about choosing the parameters to minimize the training error. One might think that getting a really low value of training error might be a good thing, but we've already seen that just because a hypothesis has low training error that doesn't mean it's necessarily. [INAUDIBLE] hypothesis. And we've already seen the example of how hypotheses can overfit, and therefore fail to generalize to new examples, not in the training set. So, how do you tell if a hypothesis might be overfitting? In this simple example, we could plot the hypothesis, H of X, and just see what was going on. But in general, for problems with more features than just one feature. For problems with a large number of features like these, it becomes hard, or pr, you know maybe impossible to plot what the hypothesis. What does this function look like, and so we need some other way to validate our hypothesis. The standard way to evaluate a learned hypothesis is as follows. Suppose we have a data set like this. Here, I've just shown ten training examples, but of course, usually we may have dozens or hundreds or maybe thousands of training examples. In order to make sure we can evaluate our hypothesis, what we are going to do is split the data we have into two portions. The first portion is going to be our usual training set. And the second portion is going to be our test set. And a pretty typical split of this, of all the data we have into a training set and test set might be around, say, a 70%-30% split with more of the data going to the training set and relatively less to the test set. And so now. If we have some data set, we've got assigned only, say 70% of the data to the I training set. Right here m is as usual on over training examples. And the remainder of our data might then be assigned to become our test sets. And here I'm going to use the notation m subscript tests to denote the number of test examples and so in general this. Subsequent test is going to denote examples that come from my test set. So that X1 subsequent test, comma Y1 subsequent test, is my first test example, which I guess, in this example, might be this example over here. Finally, one last detail. Whereas here I've drawn this as though the first 70% goes to the training set and the last 30% to the test set, if there is any sort of ordering to the data that should be better to send a random 70% of your data to the training set and a random 30%. Be a deterrent to the test set and so if your data were already randomly sorted, you could just take the first 70% and last 30%. But if your data were not randomly ordered, it'd be better to randomly shuffle or to randomly reorder the examples in your training set before, you know, sending the first 70% to the training set and the last 30% to the test set. Here then, is a very typical procedure for how you would train and test a learning algorithm, maybe linear regression. First, you learn the parameters theta from the training sets. So you minimize the usual training error objective j of theta, where j f theta here was defined using that 70% of all the data you have. There's only the training data. And then you would compute the test error. And a way to denote the test error as j subscript test. And so what you do is you take your parameter theta that you've learned from the training set. And plug it in here and compute your test set error, which I'm going to write as follows. So this is basically the average squared error as measured on your test set. It's been what you would expect so run every test example through your hypothesis with parametered data and just measure the squared error the hypothesis on your M subscript test test examples. And of course, this is the definition of the test set error if we are using linear regression and using the squared error metric. How about if we were doing a classification problem. And say, using logistic regression instead. In that case, the procedure for training and testing, say logistic regression is pretty similar. First, we will learn the parameters from the training data. That first 70% of the data. And it will compute the test error as follows. As the same objective function as we always use for logistic Russian. Except it now is defined using our m sub script test, test examples. While this definition of the test set error J sub script test is perfectly reasonable sometimes there's an alternative, test set metric that might be easier to interpret and that's the, misclassification error, it's also called the zero one misclassification error, with zero one denoting that, you either get an example right, or you get an example wrong. Here's what I mean, let me define the error, of a prediction, that is H of X. And given the label Y as, equal to one, if my hypothesis, outputs a value greater than equal to five, and Y is equal to zero or if my hypothesis outputs a value less than O.5 and Y is equal to one. Alright. So both of these cases basic respond, to if, your hypothesis mislabeled the example, assuming your threshold it at 1.5. So either thought it was more likely to be one but it was actually zero or, your hypothesis was more likely the zero, but that label was actually one, and otherwise we define, this error function to be zero, if your hypothesis basicly classify the, example Y directly. We could then define the test error, using the misclassification error metric to be, one over M test of sum from I equals one to M, so scrute test, of the error. Of h of x I test comma y i. And so that's just, my way of writing out that this is exactly the fraction of the examples in my test set that my hypothesis has mislabeled. And so that's the definition of the test set error using the misclassification error or the zero one misclassification error metric. So that's the standard technique for evaluating how good a learned hypothesis is. In the next video we'll adapt these ideas, to helping us do things like, choose what features like the degree of polynomial to use with the learning algorithm or choose the regularization parameter for learning algorithm.
