Sampling as a Counit

If the Dirac unit builds a point mass, sampling is its formal dual, a counit that collapses a distribution to a single draw.

For a distribution monad , the Kleisli category has a canonical functor embedding the deterministic base morphisms. It is faithful but not full, since every function is a channel whereas not every channel is a function. The monad unit is the Dirac family . Dually, sampling is a counit, with components in and the triangle law

so that forming a point mass and then sampling is the identity.

Definition of via the Kleisli hom-isomorphism

The Kleisli adjunction supplies a natural bijection . Setting and transporting the identity gives sampling. Thus is the image of under this iso, the generic sample of , a Kleisli map that reads a distribution and returns a draw.

Remark on the counit being only up to the surviving triangle

Calling a counit is by analogy with the Dirac unit , not a claim that is a comonad. Only one triangle holds on the nose, (build a point mass, sample it back). The reverse composite samples and then treats the draw as a point mass. It forgets the distribution’s spread and is not the identity in general. That asymmetry is where the stochasticity lives.