math.PR — Probability

Causal prediction of rare recovery responses and the optimal scale of observation time

Contributed by Man Huang

We determine the information and physical waiting needed to predict a rare intervention response with error smaller than its occurrence probability in a one-dimensional growth process with renewed Poisson interval catastrophes. Deletions of initial sources in macroscopic intervals define a mixed response, with all intervention arms sharing the same stationary past and future forcing. A fixed number of growing submacroscopic gaps determines its rare-event probability scale. We identify the response probability and amplitude, characterize when terminal extinction bits lose predictive information, and construct a gate that recovers this information from the recovery trajectory. For absolute and squared loss, a deterministic-age Bayes-risk expansion holds uniformly with additive error smaller than the rare-event probability. Our main waiting theorem covers every honest causal stopping rule, allowing the complete realized catastrophe history. Its average-time benchmark is the sum of each ordered gap length weighted by the extinction probabilities of preceding gaps. Prediction error smaller than the rare-event probability requires mean waiting divided by this benchmark to diverge; every prescribed diverging multiplier is achievable using actual recovery records. With gap scales $n^{2/5}$ and $n^{4/5}$ and integrated catastrophe tail $A_x\sim C/x$, deterministic observation requires age larger than the $n^{4/5}$ scale, whereas a stopping rule achieves mean waiting $O(n^{2/5}\log n)$. The proofs preserve the rare-event scale through weighted response approximation, risk transfer over arbitrary information fields, and posterior lower bounds on rare history branches.

Author comments: Complete English manuscript with Technical Supplement Appendices A–E. Full English–Chinese edition, sources and original MIT-licensed diagnostic code: https://doi.org/10.5281/zenodo.24138927. Mathematical content version 2026-11-11.2. Companion preprints: https://doi.org/10.5281/zenodo.23911961 and https://doi.org/10.5281/zenodo.23912425.