ora-0108

8.5 Coverage is part of the categorical prior

An arbitrary distribution can make a structurally defective model appear accurate simply by assigning zero mass to every revealing probe. A PACC declaration must therefore say which categorical strata require coverage. A simple finite mixture is

\begin{equation} \begin{aligned} P={}& \lambda _0P_{\mathrm{obj}}+\lambda _1P_{\mathrm{arr}} +\lambda _2P_{\mathrm{comp}}+\lambda _{\mathrm{th}}P_{\mathrm{th}}\\ & +\lambda _3P_{\mathrm{horn}}+\lambda _UP_{\mathrm{univ}}, \qquad \lambda _i{\gt}0,\quad \sum _i\lambda _i=1. \end{aligned} \end{equation}

8.5

Only the strata required by the doctrine need appear, but none of those strata may be assigned zero weight.

Probe stratum	Question sampled	Defect made statistically visible
Objects	typing, identity, attribute, or label	missing or conflated entities
Arrows	source, target, and observed transition	mistyped mechanisms or dynamics
Composition	two-step path versus declared composite	failure of compositional prediction
Theory	generator factorization or declared equation	inadequate or overidentified presentation
Horns	existence and compatibility of fillers	unresolved or incoherent higher behavior
Universal properties	cones, mediators, and uniqueness up to equivalence	incorrect limits, colimits, or Kan extensions

Table 8.1 Structural coverage turns a numerical risk bound into a statement about a declared portion of categorical structure.

Equation 8.5 does not assert that one canonical distribution exists. It makes the choice auditable. In an automaton or PSR, for example, \(P\) may weight future action–observation tests. In a finite category, it may sample typing and partial composition-table entries. In a quasi-categorical model, it may sample boundary and inner-horn diagrams. Different mixtures certify different notions of approximation.