Expectation Algebra

An Eilenberg–Moore algebra of the Giry monad computes an expectation, a coherent rule that collapses a distribution over back to a single point of .

Fix the Giry monad on measurable spaces, or a distribution monad on , with unit the Dirac embedding and multiplication averaging a distribution of distributions. An expectation algebra is an Eilenberg–Moore algebra for . Its structure map is the barycenter or expectation , and it satisfies the two coherence laws

The first says the expectation of a point mass is ; the second says averaging a distribution of distributions in one step equals averaging its inner and outer levels separately.

Example. Barycenter on a simplex

When carries an affine convex structure, sends a distribution to its probabilistic barycenter . On a simplex this is the centre of mass of the mass assignment. Affineness of is what lets that integral name a point of , so without it there is no coherent barycenter and no algebra.

Definition. -magma

Dropping the two coherence laws leaves a -magma , an operation whose arity functor is itself, so its inputs are whole distributions rather than finite tuples. An expectation algebra is a -magma that also respects the monad laws, one categorical level up from an $F$-algebra that happens to be a monad algebra.

Example. The algebras depend on the arity functor

Over the Eilenberg–Moore category of the finite-support distribution monad is , since closing convex combinations under the laws forces full linear structure, and more generally the algebras are modules. Changing the arity functor changes the algebras. For the non-monadic the initial -algebra is , the natural numbers, so “algebra of a functor” runs from arithmetic to expectation.

Noise on an expectation algebra

The unbiased Kleisli endomorphisms of form a monoid under composition. Work in , where an endomorphism is a channel that spreads a point into a distribution. Such an is a noise when it is unbiased against the expectation,

that is , so averaging the perturbed point returns the original and the noise has mean zero without writing a subtraction. The Dirac unit is unbiased by the first coherence law , and unbiasedness is closed under Kleisli composition since collapses through the second coherence law . Therefore is a submonoid of the endomorphism monoid . It is a monoid rather than a group, since a channel only spreads mass further and unbiasedness is preserved but not reversed, so is the unique deterministic element.

References

• Michèle Giry, “A categorical approach to probability theory”, in Categorical Aspects of Topology and Analysis, Lecture Notes in Mathematics 915 (1982), pp. 68-85