model with only positive responses
This sounds like a one-class or unary classification problem where you can build a model of normality using the class that you have. 

Take a look at novelty and outlier detection documentation for scikit learn. They discuss the one-class SVM there, which attempts to model the decision boundary given the observations that you have. The example that is implemented is pretty straight forward. Sadly, this implementation does not provide you with a probabilistic measure (although, there are papers on how to add this functionality).

Alternatively, you can consider an auto-associative neural network (replicator network). 

There is a nice discussion on time-series anomaly detection here on DS.