Dual Boundary Condition Inference: A Two-Boundary Product Rule and Its Implications

Abstract

Many inference problems are constrained from two sides: forward information and a second constraint on outcomes. Dual Boundary Condition Inference (DBCI) multiplies them and renormalises. The rule is not new: it is the product-of-experts form and the unit-exponent member of the logarithmic-pooling family. The paper contributes a classification and reducibility characterisation. The algebra does not fix the exponent at which the inputs combine: given the statistic log p_fwd + log p_bwd, which already builds in the product form, maximum entropy supplies the family (p_fwd p_bwd)^lambda and selects no member. If the inputs are read as two equally weighted opinions, three requirements select lambda = 1/2: unanimity preservation, consistency under a common Bayesian update, and minimal symmetric Kullback-Leibler divergence. If they are read as separately applied factors, two requirements select lambda = 1: a neutral second input must leave the first unchanged, and an update to one input must pass through unchanged. For a rank-one intermediate projective measurement, the Aharonov-Bergmann-Lebowitz (ABL) rule realises this form with no extra parameter. Every positive exponent orders outcomes identically, so the choice is invisible to picking the likeliest outcome, but not to scoring a class by total mass. Under the relevant identifications, the same product form gives Bayes' rule, while a final effect proportional to the identity returns the ordinary forward Born probabilities. The second result is a Two-Boundary Reducibility Criterion: DBCI factors through the forward boundary exactly when the effective backward boundary, up to positive rescaling, does. So a fixed prior or model-fixed structural constraint adds no distinctions between cases sharing the same forward boundary, though it may encode substantial information and still materially affect the result.