What techniques should I use to predict binary events with repeated observations per person? My problem is structured as follows: for each user, I have some number of binary observations, whether or not someone repays a loan. The reason this number is arbitrary is because people can choose to not take future loans, or can default and not be able to take more. Currently, my model is only trained on the first such observation for each user, because then it can be structured as a "vanilla", well-defined classification problem. 
The basic intuition I'm trying to represent is: if someone repays their first loan, but then defaults on their second, I don't want to penalize the model as strongly for assessing that person to be high risk, relative to someone who ends up repaying, say, 10 loans. Notionally, if we think of everyone as having a hidden true-repayment-probability that governs their actions through some kind of internal bernoulli draw, then having many successful positive draws should make me more confident that that true probability is high. 
Does anyone know of a principled way to incorporate that intuition into a modeling technique? I've considered predicting "number of successful loans", but what I ultimately want out of my model is a scaled probability, which I'd have to translate back out of the regression framework. 
Is observation weighting a good idea here? (i.e. weighting positives with many observed subsequent positives more strongly) And, if so, what kind of numeric weighting framework do you think would work? 
 A: Binary effects suggest a logistic regression. Repeated measures on each subject suggest a repeated measures or a mixed effects model. (Essentially, a repeated measures model is a special case of a mixed effects model.)
Put the two together and look for a mixed effects logistic regression, modeling the propensity to repay using a logistic model, with a random effect per subject, which would model exactly the kind of latent trait (in a non-technical sense) you describe.
Plus, depending on your software, you might be able to model covariances between successive loan repayments using an appropriate covariance structure. In R, for instance, you could look at a corCAR() covariance, which is a Continuous AutoRegressive covariance. The idea is that successive loan repayments correlate (that's the autocorrelation), but that this autocorrelation depends on how much time has elapsed between them (that's the continuous part).
A: This sounds rather like discrete survival analysis.. Where logistic regression is used to predict survival, given no default in previous time period, for each loan. So you can model each loan independently or add eg number of previous loans as an input (instead of time)... 
