Log-Odds Distillation into Probabilistic-Bit Classifiers with Fewer Inputs: Sharp Firing-Probability Bounds, Decision Reversal and Read Budget

Probabilistic-bit (p-bit) classifiers are trained by distilling a full-precision teacher into the stochastic binary network, usually by matching log-odds. A p-bit's firing probability is a sigmoid of its bias, so when the deployed network sees fewer sensors than its teacher, the output firing probability is whatever log-odds matching produces, which is not in general the Bayes posterior given the student's inputs. We characterise the gap for a frozen, Bayes-correct binary teacher. Given a teacher-confidence bound and the coarse posterior, a tangent construction gives the sharp attainable interval of the output firing probability; a classical third-convex-order result sharpens it given the conditional variance; a monotone optimisation map turns both into sharp log-loss and Brier envelopes, an exact reversal threshold with a safe window, and a read-budget envelope for a majority vote. A p-bit multilayer perceptron on a 16-cell exact-teacher panel, read through a simulated stochastic magnetic tunnel junction model, tracks the predicted firing probabilities up to sampling error and a small systematic shortfall at the two most extreme targets, reverses where predicted (more reads then converge to the wrong class), and shows the predicted read inflation. Four pre-specified tabular comparisons are null. The device is simulated; no energy claim is made.