nLab observational monad

Idea

An observational monad is a monad where, intuitively, one can distinguish two monadic (for example probabilistic) values by means of taking repeated independent observations.

It is of particular importance in categorical probability, where these observations correspond to iid random variables.

Definition

Let $C$ be a category with products, and let $P$ be a symmetric monoidal monad (or commutative monad) on $C$.

Denote also by $C_T$ its Kleisli category, which is then a copy-discard category. As usual, we have an adjunction $L:C\to C_T, R:C_T\to C$ such that $P=R\circ L$.

Denote the comonad $L\circ R$ on $C_T$ again by $P$. (Note that this equips $C_T$ with a thunk-force structure.) Finally, denote $\varepsilon_X:P X\to X$ the components of the counit of the adjunction. (In categorical probability, this is known as the sampling map.)

The monad $P$ is called observational if and only if for every object $X$, the family of morphisms of $C_T$

$$\Big\{ P X \xrightarrow{\quad\mathrm{copy}\quad} (P X)^{\otimes n} \xrightarrow{\quad\varepsilon^{\otimes n}\quad} X^{\otimes n}\;,\; n \in \mathbb{N} \Big\}$$

is jointly monic.

The intuition is that one can distinguish two morphisms (for example, states) into $P X$ by sampling independently as many times as needed.

A representable Markov category whose probability monad is observational is called an observationally representable Markov category.

Properties

• The Giry monad is observational thanks to the pi-lambda theorem? or, equivalently, the monotone class theorem?. (Moss-Perrone’22)

• The Kleisli category of an observational monad, with its canonical copy-discard and thunk-force structures, is such that every deterministic morphism is thunkable (the converse is always true). (Moss-Perrone’22)

• Every observationally representable Markov category is necessarily almost-surely-compatibly representable. That is, in the bijection

  $$f:X\xrightarrow{\qquad} Y \qquad\leftrightarrow\qquad f^\sharp:X \xrightarrow{\;(det)\;} P Y$$

  we have that for every state (probability measure) $p$ on $X$, two morphisms $f,g:X\to Y$ are $p$-almost surely equal? if and only if $f^\sharp$ and $g^\sharp$ are. (FGLPS)

• Observationality is related to de Finetti's theorem: stating the theorem as a limit, observationality is closely related to the uniqueness part in the universal property. It coincides exactly with this uniqueness whenever the necessary Kolmogorov products exist. (Moss-Perrone’22)

See also

• probability theory, categorical probability
• Giry monad, probability monad, Markov category
• strong monad, monoidal monad, commutative monad
• de Finetti's theorem, iid random variables
• pi-lambda theorem?, monotone class theorem?
• sober measurable space, zero-one measure
• infinitary tensor product, Kolmogorov extension theorem
• thunk-force category, monad in computer science

References:

The concept, in the form presented here, was first introduced in:

• Sean Moss, Paolo Perrone, Probability monads with submonads of deterministic states, LICS 2022. (arXiv:2204.07003)

It also appears in:

• Tobias Fritz, Tomáš Gonda, Antonio Lorenzin, Paolo Perrone and Dario Stein, Absolute continuity, supports and idempotent splitting in categorical probability, 2023. (arXiv)

• Noé Ensarguet, Paolo Perrone, Categorical probability spaces, ergodic decompositions, and transitions to equilibrium, arXiv:2310.04267

• Tobias Fritz and Antonio Lorenzin, Involutive Markov categories and the quantum de Finetti theorem, 2023. (arXiv)

Related papers:

• Tobias Fritz, Tomáš Gonda, Paolo Perrone, Eigil Fjeldgren Rischel, Representable Markov categories and comparison of statistical experiments in categorical probability, Theoretical Computer Science 961, 2023. (arXiv:2010.07416)

• Tobias Fritz, Tomáš Gonda, Paolo Perrone, De Finetti’s theorem in categorical probability. Journal of Stochastic Analysis, 2021. (arXiv:2105.02639)

category: probability