Structural Bayesian Inference: Theory-Space Learning in the Structural Descent Framework

Bayesian inference provides one of the most widely used mathematical frameworks for reasoning under uncertainty. In its classical formulation, inference proceeds by updating probability distributions over a fixed hypothesis space using observed data. Although this approach has proven highly successful in statistics and machine learning, it presupposes that the representational structure of the hypothesis space remains fixed throughout the inferential process. In many scientific and computational settings, however, progress requires modification of the underlying representational structure itself. This paper develops a mathematical extension of Bayesian inference within the theory-space formulation of the Structural Descent Framework. In this formulation, a theory is represented as an ordered pair $T=(S,\theta)$, where $S$ denotes structural specification and $\theta$ denotes parameters internal to that structure. Classical Bayesian inference is shown to correspond to descent along fixed structural fibers of theory-space. The analysis demonstrates that such fixed-structure inference can converge to structural plateaus in which parametric optimization cannot reduce the incompleteness functional. To overcome this limitation, a form of Structural Bayesian inference is introduced in which posterior distributions are defined over the full theory-space. Several theoretical results are established. First, the Bayesian Embedding Theorem shows that classical Bayesian inference is a special case of theory-space descent. Second, the Structural Plateau Theorem demonstrates that fixed-structure inference can terminate at suboptimal points relative to the full theory-space. Third, a Structural No-Free-Lunch theorem shows that inference confined to a single structural fiber cannot achieve globally optimal adequacy when superior structures exist elsewhere in theory-space. Fourth, a Structural PAC-Bayesian bound is derived, separating parametric and structural sources of model complexity. Finally, a Structural Convergence Theorem establishes that admissible structural descent sequences produce an asymptotic reduction of the incompleteness functional, yielding a formal notion of dynamic reducibility. Taken together, these results provide a unified mathematical framework connecting Bayesian inference, structural learning, and theory-space dynamics within the Structural Descent Framework. The theory suggests that inferential systems capable of modifying their own representational structures may play an important role in scientific discovery and advanced machine learning.