I am curious to know if there are methods that exists for sequential modeling of binary outputs? Let me give an example to help further clarify the question:
I have a problem where I have binary outcomes and some covariate information that I want to model using some sort of statistical model for binary outputs (think logistic regression type models). My setup is the following, I start with some initial set of data, say of size $n_0$, at time $t=0$ and for each subsequent time point $t=1,...,k$ I run an experiment and then receive an additional output which is binary. Now, in my situation, I get binary outcomes, however, in the beginning I am almost surely guaranteed to have all of my $n_0$ outcomes equal to 0 (in my case say the binary outcome is either 0 or 1). Furthermore as I collect more data, I may not see an outcome of a 1 for some time. So in this situation logistic regression becomes infeasible because I can't fit the model if I have only observed one type outcome.
So now that the problem setup has some context, is there a sequential method that is appropriate for modeling this type of data scheme? Or is there a suggestion of how this kind of data should be handled? Ultimately at each iteration of the algorithm I want to be able to fit a model, then use that model to predict something, and based on that prediction I then get a new data outcome so being able to build a model that predicts well is also tantamount.