# Binary logistic regression with only positive training examples - does that even make sense?

(I have learned about polynomial linear regression, logistic regression, and neural networks.)

I have a binary logistic regression problem. I need to classify things to be true or false. What makes this hard for me to understand is that I only have y=1 training examples. Is it possible to get a decision boundary for this data set?

I simplified the problem by only thinking about 2 of my features.

I experimented a little bit by placing fake y=0 training examples everywhere.

Gradient descent with the real and fake training sets seems to fit the data ok. It is a bit underfit, taking into account that this is a degree 6 curve and regularization is turned off.

There has to be a better way to approach this problem. Heck, I could even calculate the convex hull! (Not sure how well that will work in 15+ dimensions.) Any advice?

• What you're doing is called density estimation, and there are purpose-designed methods for the task. Clustering is related, and both fall under the purview of unsupervised learning. – Hong Ooi May 2 '15 at 1:58

Do you have a test set with $y = 0$ examples? If not, it will be pretty hard to evaluate your classifier!