Where Does Probability Live?

Abstract

The frequentist-Bayesian divide is usually introduced as a dispute about the meaning of probability, the status of parameters, priors, repeated sampling, or the interpretation of intervals. Those disagreements are real, but they begin after an earlier statistical problem has already been solved or silently assumed: what empirical evidence exists, what formal target it bears on, how the evidence could arise, what connects it to the target, and what world-facing claim the result is meant to support. This paper uses the Statistical Bridge to relocate that shared work. It distinguishes four analytical obligations: bridge constitution, probability source, inference certificate, and claim license. The obligations are dependencies rather than a mandatory chronology. Probability sources may be evidence-side, such as randomized designs or observation models, or target-side, such as Bayesian priors. Frequentist and Bayesian inference are then treated as two historically dominant certificate bases rather than an exhaustive binary, alongside likelihoodist, confidence-distribution, inferential-model, fiducial, betting/e-value, and multi-certificate constructions. The paper stress-tests the architecture against de Finetti, Birnbaum, calibrated Bayes, fiducial inference, and empirical Bayes; explains why large-sample or computational agreement does not imply bridge identity; and argues that Bayesian computational fertility is largely an advantage in realizing a standardized conditional inference certificate rather than an automatic advantage in bridge constitution or claim licensing. The central conclusion is simple: probability has an address. It enters at a particular relation in the bridge, under particular assumptions or instituted mechanisms; its inferential authority is carried by a particular certificate; and that authority extends only as far as the source, certificate, and claim license support.