Discrete Bayesian Inference as a Structure of Paths

Bayesian inference is predominantly formulated in a continuous framework, in which posterior beliefs are represented by smooth probability densities. However, an alternative discrete representation—already implicit in Bayes’s original construction—remains conceptually distinct and structurally informative. This paper develops a representation-level analysis of Bayesian updating in the binomial setting and shows that discrete and continuous posteriors may exhibit qualitatively distinct behavior under finite parameter resolution. In particular, coarse discretization can induce regime-dependent divergence from the continuous posterior, even when the algebraic form of the likelihood is identical. The analysis further demonstrates that divergence is not determined solely by grid resolution but also by the balance between prior strength and sample size. By introducing a scale-dependent perspective in which representational resolution and prior magnitude jointly define distinct regimes of inference, the paper clarifies how structural and analytic descriptions interact under finite conditions.