[Paper Review] A Monad for Probabilistic Point Processes

Swaraj Dash, Sam Staton|arXiv (Cornell University)|Jan 26, 2021

Point processes and geometric inequalities4 citations

TL;DR

This paper introduces a monad $GB = G \circ B$ for probabilistic point processes by composing the Giry monad $G$ (for probability measures) and the bag monad $B$ (for multisets), proving it forms a monad via a distributive law. The key contribution is showing that the intensity measure map from point processes to their expected counts is a monad morphism, enabling compositional computation of intensity measures and yielding Wald’s Lemma as a special case.

ABSTRACT

A point process on a space is a random bag of elements of that space. In this paper we explore programming with point processes in a monadic style. To this end we identify point processes on a space X with probability measures of bags of elements in X. We describe this view of point processes using the composition of the Giry and bag monads on the category of measurable spaces and functions and prove that this composition also forms a monad using a distributive law for monads. Finally, we define a morphism from a point process to its intensity measure, and show that this is a monad morphism. A special case of this monad morphism gives us Wald's Lemma, an identity used to calculate the expected value of the sum of a random number of random variables. Using our monad we define a range of point processes and point process operations and compositionally compute their corresponding intensity measures using the monad morphism.

Motivation & Objective

• To provide a categorical, compositional framework for modeling probabilistic point processes using monads.
• To formalize point processes as probability measures over multisets (bags) of points in a measurable space.
• To establish that the intensity measure of a point process is a monad morphism, enabling compositional reasoning.
• To demonstrate that this framework yields Wald’s Lemma as a special case via monad morphism properties.
• To bridge category theory with practical statistical modeling, particularly in probabilistic programming and databases.

Proposed method

• Formalize point processes as elements of $G(B(X))$, the Giry monad applied to the bag monad on a measurable space $X$.
• Prove $GB$ is a monad using Beck’s theory of distributive laws between monads.
• Define the intensity measure as a composite morphism $GB \hookrightarrow MM \xrightarrow{\mu} M$, where $M$ is the monad of all measures.
• Show that the intensity measure map $\mathbb{E}: GB \to M$ is a monad morphism by verifying commutativity of relevant diagrams.
• Use the monad morphism property to compute intensity measures compositionally, e.g., via do-notation in a functional programming style.
• Leverage naturality and functoriality to derive identities such as Wald’s Lemma for compound distributions.

Experimental results

Research questions

• RQ1 Can point processes be systematically modeled as a monad in category theory, enabling compositional construction?
• RQ2 Is there a natural monad morphism from point processes to their intensity measures, preserving monadic structure?
• RQ3 Can Wald’s Lemma for the expectation of a random sum be derived as a consequence of monad morphism properties?
• RQ4 How can the intensity measure of a composite point process be computed compositionally using the monad structure?
• RQ5 Can this categorical framework be applied to probabilistic logic programming and probabilistic databases for structured modeling?

Key findings

• The composition $GB = G \circ B$ forms a monad on the category of measurable spaces, providing a categorical foundation for probabilistic point processes.
• The intensity measure map $\mathbb{E}: GB \to M$ is a monad morphism, ensuring that intensity computation commutes with monadic composition.
• Wald’s Lemma is recovered as a special case of the monad morphism property, showing that $\mathbb{E}(\underline{N} \gg t. \underline{X}) = \mathbb{E}(\underline{N}) \gg t. \mathbb{E}(\underline{X})$.
• For a Poisson point process on the unit square with rate $\Lambda$, the intensity measure is $\lambda W \mapsto \Lambda \cdot |W|$, computed compositionally via the monad morphism.
• The framework supports compositional simulation and analysis of complex point processes using functional programming syntax such as do-notation.
• The result provides a principled, compositional method for computing intensity measures without manual integration, leveraging category theory.