I am trying to solve for what seems to be a multivariate survival model, but am getting stuck as there are both recurrent events and also jumps back to 100% alive.
Rephrasing the larger project in everyday terminology: Assume the survival probability is not the probability of actual physical death, but the probability of actively participating in some activity (measure of engagement).
Assume three possible events which an occur
- Joins an activity
- Changes activities
- Leaves all activities
Thinking of each event as a 'risk' and engagement as being 'alive', the following describes the problem:
- You can not be “at risk” for (1) without having experienced event (3) first
- Likewise you can not be “at risk” for (2) without experiencing (1)
- You can not be "at risk" for (3) without having experienced (1)
- Not everyone may do event (2), though most do.
- At events (1) or (2) you have 100% chance of being “alive” at that particular moment. (possible discontinuity from any previous value)
- At events (3) you have 0% chance of being “alive” – until event (1) happens again.
- Event (2) can happen multiple times between (1) and (3)
- There exist a survival curve of probability of being alive ("engagement") inbetween events of (1 or 2) and (3), representing the two extremes of 100% and 0% engagement respectively. In other words, you may have changed activities five hours ago, but still 'registered' that you join - however, your engagement could be near 0%.
The timestamps for each event for each person is known.
The reason for the Bayesian tag was that realized your probability of being engaged in the activity also depends on what everyone may be doing. If everyone is consistently changing activities (therefore engagement being 100% spikes), and you haven't done any event in past T time that should lower your probability of being engaged as conditioned on all current behavior most people are.
Not sure if phrased it the right way. It seems a mix of survival analysis, competiting risks, and bayesian analysis; only that your probability can spike back up to 100% again and again.
EDIT: Tom Minka point out good point; I haven't actually asked a question!
Question: Given the constraints above, and the history of activity of one person; what is the probability they are engaged in their current activity? Example: If they last chose an activity five hours ago, and never left nor changed; good probability they lost interest and are inactive, but never officially changed. (e.g. they could have left the building if a physical activity). Likewise the probability of engagement might be different if taking everyone elses current activity into account (hence the Bayesian tag).
