Probabilistic Observation and Relative Entropy of Retrieval Hypotheses: A FinStat Formulation of Loopback Observation Theory

This preprint develops a deliberately minimal finite probabilistic model for Loopback Observation Theory (LOT), as a sequel to the deterministic finite-set model of observation-map adjunction and retrieval ambiguity. A finite latent state space, a finite record space, a prior distribution, and an observation channel are used to define classical instruments, record probabilities, Bayes re-expansion, and retrieval hypotheses. The paper does not claim to prove a probabilistic adjunction. The deterministic adjunction between direct image and inverse image in the first paper is used only as a reference point. In the probabilistic setting, Bayes re-expansion depends on both the observation channel and the prior distribution, and is defined only for positive-probability records. Using a FinStat-style representation, the observation channel is lifted to a joint distribution on the state-record space, and stochastic right inverses are read as retrieval hypotheses. The main loss is defined as a relative-entropy mismatch between the true joint distribution and the distribution reconstructed from records by a retrieval hypothesis. The zero condition is characterized as agreement between the retrieval hypothesis and the Bayes posterior for every positive-probability record. The mathematical components used here are standard: finite probability theory, Bayesian updating, classical instruments, FinStat, stochastic right inverses, and relative entropy. The contribution of this paper is to organize these structures as a finite probabilistic model of record formation, Bayes re-expansion, retrieval hypotheses, and information loss within LOT.