From Risk Sets to Martingales: A Counting-Process Framework for Event-History Learning

Counting-process notation separates predictable risk-set information from observed event jumps through decompositions of the form dN(t)=Y(t)alpha(t)dt+dM(t). This article develops a unified event-history learning framework for censored, truncated, recurrent, multistate, and covariate-dependent data. Rather than cataloguing survival methods, the treatment translates each partially observed learning target into five recurring objects: risk process, jump process, compensator, estimating equation, and limiting argument. The framework connects right-censored survival curves, product-integral estimators, bivariate and interval-censored survival estimators, log-rank tests, Cox-Andersen-Gill regression, additive hazards, accelerated failure-time models, panel-count data, landmark prediction, semi-Markov models, Bayesian nonparametric transition models, and instrumental-variable methods. The original contribution is threefold. Computationally, the article turns risk-set sweeps, product-integral updates, interval-likelihood calculations, semi-Markov elapsed-time bookkeeping, Bayesian transition-hazard updating, and cross-fitted validation into reusable algorithms and simulation diagnostics. Theoretically, it gives proof templates for the recurring martingale, likelihood, product-integral, and empirical-process arguments, and proves a new out-of-fold compensator validation identity for cross-fitted censored learners. For applications, it maps biomedical, reliability, operational, economic, financial, literary, historical, and causal survival examples onto the same risk-set and compensator language. The resulting account provides a common mathematical language for deriving, checking, and comparing classical and machine-learning methods for censored, recurrent, and multistate event-history data.