Consider right-censored observations, with events at times $t_1, t_2, \dots$. The number of susceptible individuals at time $i$ is $n_i$, and the number of events at time $i$ is $d_i$.
The Kaplan-Meier or product estimator arises naturally as a MLE when the survival function is a step function $S(t) = \prod_{i : t_i < t} \alpha_i$. The likelihood is then $$ L(\alpha) = \prod_i (1-\alpha_i)^{d_i} \alpha_i^{n_i-d_i} $$ and the MLE is $\widehat\alpha_i = 1 - {d_i\over n_i}$.
OK, now assume that I want to go Bayesian. I need some kind of ``natural'' prior with which I will multiply $L(\alpha)$, right?
Googling the obvious keywords I found that the Dirichlet process is a good prior. But as far as I understand, it is also a prior on the discontinuity points $t_i$?
This is surely very interesting and I am eager to learn about it, however I would settle for something simpler. I begin to suspect that’s not so easy as I first thought, and it’s time to ask for your advice...
Many thanks in advance!
PS: A few precision on what I am hoping I am interested in (as simple as possible) explanations about the way to handle the Dirichlet process prior, however I think it should be possible to use simply a prior on the $\alpha_i$ — that is a prior on the step functions with discontinuities in $t_i$.
I think that the "global shape" of the step functions sampled in the prior should not depends on the $t_i$'s -- there should be an underlying family of continuous functions which are approximated by these step functions.
I don't know if the $\alpha_i$ should be independent (I doubt it). If they are, I think this implies that the prior $\alpha_i$ depends on$\Delta t_i = t_i - t_{i-1}$, and if we denote its distribution by $A(\Delta t)$ then the product of a $A(\Delta_1)$ variable by an independent $A(\Delta_2)$ variable is a $A(\Delta_1+\Delta_2)$ variable. It seems here that log-$\Gamma$ variables can be useful.
But here basically I am stuck. I did not type this at first because I did not want to direct all answers in this direction. I would particularly appreciate answers with bibliographic references to help me justifying my final choice.