Training on censored data with varying sequence length I have a sequence of daily events, along with times between events. Each such sequence is labeled by $i$ and consists of $n_i$ events. For every $i$, each event $j$ has a list of features $x_{ij}$, and I would like to predict the time $y_{ij}$ till the next event, given the entire past history. This is a pretty classic problem in censored survival time. While not explicitly relevant to what's below, the model is an LSTM. My question is more geared towards how to train this model correctly.
Here's what worries me:
In general, the last event in each such sequence is censored. When training this model, my loss function is the average loss over each consecutive pair in the sequence. So for a sequence $i$, the loss function looks something like:
$L_i=\frac{1}{n_i}\sum_{k=1}^{n_i}f(\hat{y}_{ij},y_{ij}|x_{i1},x_{i2},\cdots,x_{in_i},c_{ij}),$
where $c_{ij}$ is 1 if event $ij$ is censored and 0 otherwise and $\hat{y}_{ij}$ is the predicted time. To be clear, here's a time diagram of a given sequence $i$, with the dashes representing how many days pass between events, and the | representing the end of the sequence (so $x_{i4}$ is censored):
$x_{i1}$--------$x_{i2}$---$x_{i3}$-------------------$x_{i4}$----|
So then I feed these sequences into my model.
However, I could also generate more sequences from my model. For example the above sequence can make:
$x_{i1}$--------$x_{i2}$-|--$x_{i3}$-------------------$x_{i4}$----
where now $x_{i2}$ is censored and we throw out $x_{i3},x_{i4}$. 
I would think that this procedure would make my predictive model more robust to different sequence lengths, especially if there are imbalances of sequence length in my training set. But I'm finding it hard to convince myself that it would improve my model, as opposed to just training on existing sequences. What exactly does one do in classical survival time modeling?
 A: In classical discrete-time survival analysis, the data are often broken down even farther, to individual data points for each time point within each sub-sequence (as you propose; indexed if you think that sub-sequences might need different models) of each overall sequence (which you label here as i). The approach is explained for example in this paper by Willett and Singer. This turns the problem into modeling a binomial response as a function of the (potentially time-dependent) features and time, and allows different models for the multiple sub-sequences within a sequence. The cited paper uses logistic regression (logit link) to model the binomial response; a complementary log-log link instead might be more closely related to the proportional-hazards assumption of continuous-time Cox models.
So your breaking down long sequences into sub-sequences has ample precedent. I have two comments on your particular application that you might want to consider. First, your analysis would be most useful if you have feature values for all time points, not just for the times of events. That's not clear from your description. Second, you might want to consider whether a good model of the event probabilities at each time point could be more generally useful than your loss function based simply on observed event times. If you model the individual probabilities you can reconstruct the loss function at the event times if you wish, and perhaps more easily examine different loss functions. 
