Scale-Free Priors and Survival Dynamics: A Bayesian Framework for Conflict Duration

Abstract

We have developed a fully Bayesian survival-analysis framework that reformulates inference about system lifetimes in terms of hazard and survival functions, and extends this representation to interacting actors. Starting from J.~Richard Gott's Copernican principle, we express the scale-free prior as a baseline hazard $\lambda(t)=1/t$, thereby linking a static prior over lifetimes to the dynamic language of survival analysis. In this formulation, Bayesian updating corresponds to conditioning on survival, while the resulting posterior distribution admits a natural representation in terms of hazard and survival functions. The approach is intended for settings where data are sparse or unreliable, and where a scale-free, assumption-light baseline is preferable to heavily parameterized models. Building on this foundation, we derive general expressions for two-actor systems that characterize joint survival, conditional lifetimes, and comparative outcomes without requiring a specific parametric form of interaction. This yields a flexible and modular framework in which baseline dynamics are separated from interaction effects, allowing different mechanisms to be incorporated transparently. Thus, the primary contribution is a general hazard-based formulation of Bayesian updating and its extension to interacting systems To illustrate the framework, we consider a multiplicative resource-depletion specification in which interaction modifies the baseline hazard through cumulative engagement intensity. This example demonstrates how interaction terms can be embedded while preserving analytical tractability, including closed-form expressions under simplifying assumptions. We further provide a stylized application to an asymmetric two-actor conflict, the 2026 US/Israel--Iran hostilities, to highlight the qualitative implications of the approach.