Bayesian classification with informative survival

When the distribution of an observation horizon depends on an unknown state, survival to the present time is itself informative. We study Bayesian classification with a finite latent state, a continuously observed diffusion, and a termination intensity of the form $\lambda(t,x)+\varepsilon\theta_k$. We derive global $L^1$ and relative-entropy bounds for the posterior error caused by ignoring survival. Under a linear posterior-margin condition, the expected excess Bayes cost is $O((\varepsilon t)^2)$, and a local positive-density condition yields a matching lower bound. We verify these conditions and calculate the leading constant in a Gaussian model. When $\varepsilon$ is unknown, we propose a recursive estimator and show that under stability and moment conditions, its mean-square error is $O(n^{-1})$, yielding $O(\log N)$ cumulative classification regret. Numerical experiments support the theoretical rates and constants.