Exact Conditioning

Dario Stein’s account of Bayesian inference as a categorical operation. Conditioning is not an external procedure but a morphism inside a symmetric monoidal category of open inference problems.

Exact conditioning treats an observation as an effect, a cap into the monoidal unit, rather than as a normalisation applied from outside the diagram. Composing a state with this cap imposes the constraint, and the surrounding structure supplies the renormalised posterior. The category of open inference problems is symmetric monoidal, so states, channels, and observations all live as morphisms in one calculus and compose as string diagrams. Conditioning is then the Bayesian face of the state–predicate duality, since the effect pairs a predicate against a state to reshape it.

The cap induces a conditioning product together with an uninformative unit , the improper everywhere-flat prior. The pair satisfies the special commutative Frobenius laws. A category in which every object carries such a Frobenius structure is a hypergraph category, so conditioning has exactly the algebra of a corelation, and constraints can be freely merged and copied.

Property. Frobenius conditioning

The cap , product , and unit make each a special commutative Frobenius object. The product is associative and commutative with unit , and the copy/cap pieces satisfy the Frobenius equation . Conditioning is therefore the hypergraph, not merely Markov, fragment of probabilistic reasoning.

Remark. Zero-probability and improper priors

Conditioning handles probability-zero observations, such as conditioning a continuous variable on an exact value, without an ill-defined , because it is an effect rather than a division. A refinement to Gaussian and linear-relations models further admits improper priors. The uniform distribution on appears as , a legitimate object of the diagram rather than a limit taken by hand.