Connecting Survival Analysis to Overall Predicted Survival Probabilities I've got a question on combining survival analysis with predicted survival probabilities that I'm sure someone must have thought about, but I just can't find anything out there.
Imagine I have a good model for predicting how well certain patients will survive a disease and a pretty good model for how likely any given patient is to die $t$ days after infection. I want to calculate how likely a certain patient is to survive given that they've survived $t$ days so far.
Abusing the standard survival notation I can write a "survival" function (that looks sort of like Kaplan-Meier function) for a patient $i$ as
$S_i(t)=lim_{h \rightarrow \infty} \prod_{j=t}^h s_{i,j}$
where $s_{i,j}$ is the probability that the patient doesn't die on day $j$. The overall probability the person survives is then $S_i(0)=p_i$. In a perfect world I would know $s_{i,j}$ for $j=1,...,h$ where $h$ was sufficiently large.
Instead, I have a pretty good model for how likely a patient $i$ is to survive, i.e., $p_i$ and a more global model for the overall survival function $S(t)$ that isn't as accurate at an individual patient level. Is there a way to connect these so I can give a probability that patient $i$ will survive given that he or she has survived $t$ days so far?
I've tried a few things I can post in the comments, but can't seem to guarantee that the probabilities $s_{i,j}$ are always in $[0,1]$ or that the function $S_i(t)$ is monotonically increasing with $t$. Any ideas, places to look, or papers to read?

It sounds like you have two models, one survival curve for the entire patient population and a second survival model for a subgroup which is considered "subject-specific."  You can simply further subgroup each of these models by conditioning on surviving at least as long as $t$.
What this shows is that probability is a proportion of many samples.  Your understanding of the sampling behavior over many samples gives you confidence in what to expect from the result of a single sample.
It also shows that the sampling frame from which you obtained a given subject is crucial for referencing the correct model.  How was a given subject selected?  If a subject was selected at random from the entire patient population without regard for a minimum survival time, then referencing a "subject-specific" model will yield the wrong inference.  Likewise, if a given subject was selected according to a specific set of covariates including a minimum survival time, then referencing a model for the entire patient population will yield the wrong inference.
