33.Decision boundary

In the last video, you learned about the logistic regression model. Now, let's take a look at the decision boundary to get a better sense of how logistic regression is computing these predictions. To recap, here's how the logistic regression models outputs are computed in two steps. In the first step, you compute z as w.x plus b. Then you apply the Sigmoid function g to this value z. Here again, is the formula for the Sigmoid function. Another way to write this is we can say f of x is equal to g, the Sigmoid function, also called the logistic function, applied to w.x plus b, where this is of course, the value of z. If you take the definition of the Sigmoid function and plug in the definition of z, then you find that f of x is equal to this formula over here, 1 over 1 plus e to the negative z, where z is wx plus b. You may remember we said in the previous video that we interpret this as the probability that y is equal to 1 given x and with parameters w and b. This is going to be a number like maybe a 0.7 or 0.3. Now, what if you want to learn the algorithm to predict. Is the value of y going to be zero or one? Well, one thing you might do is set a threshold above which you predict y is one, or you set y hat to prediction to be equal to one and below which you might say y hat, my prediction is going to be equal to zero. A common choice would be to pick a threshold of 0.5 so that if f of x is greater than or equal to 0.5, then predict y is one. We write that prediction as y hat equals 1, or if f of x is less than 0.5, then predict y is 0, or in other words, the prediction y hat is equal to 0. Now, let's dive deeper into when the model would predict one. In other words, when is f of x greater than or equal to 0.5....