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Stream: theory: categorical probability

Topic: Is Prevalence a Weighted Colimit?

Ellis D. Cooper (Jul 13 2025 at 15:32):

$W$ denotes a non-void finite set of ``words." A sentence is an assignment of words to positions, $\tilde{w}:S\to W$. If thinking of $\tilde{w}$ as a word-valued random variable defined at positions (like a "field" of words), then $\mathbb{P}[\tilde{w}=k]$ is the probability of word $k$ occurring at some position of the sentence.

The triple $(S,\tilde{w},W)$ with dom$$\tilde{w}=S$$ and cod$$\tilde{w}=W$$ is exactly an object of the slice category $\mathcal{S}et/W$. Call it the category of sentences over $W$.

A document is a ``field" of sentences over a discrete category of locations, as in $\tilde{S}:D\to \mathcal{S}et/W$. The sentence $\tilde{S}q$ in the document is denoted by $\tilde{w}_q:\tilde{S}_q\to W$. By the universal property of coproduct in $\mathcal{S}et$, there is a random variable

$$\bigcup\limits_{q\in D}\tilde{S}_q\xrightarrow{\tilde{W}}W$$

such that if $p\in\tilde{S}q$ for $q\in D$, then $\tilde{W}p=\tilde{w}_{q}p$ is the word at position $p$ in the sentence at location $q$. And $\mathbb{P}[\tilde{W}]=k$ is the probability of word $k$ occurring at some position of some location in the document.

For natural language understanding it is of interest to determine, for a sample $A\subseteq D$ of sentences (such as the consecutive sentences in a paragraph), the ``density of highest-probability words" in $A$. The density of word $k\in W$ in sample $A$ is

$$\frac{\#\{\,p\in\tilde{S}_q\,|\,q\in A \wedge \tilde{W}p=k\,\}}{\#A}$$

In applications a document is subdivided into consecutive intervals $A$ of fixed length so the denominator $\#A$ equal to $\#\tilde{W}_A^{-1}k$.

Define the $\mathbf{prevalence}$ $\operatorname{Prev}_NA$ relative to a set $K\subset W$ of words of highest rank (according to some threshold $N\in\mathbb{N}$) by

$$\operatorname{Prev}_N A:=\sum_{k \in K} \#\tilde{W}_A^{-1}k\cdot \mathbb{P}[\tilde{W}=k]\,.$$

The question is whether prevalence is a weighted colimit. Evidence is a detailed formal correspondence:

$$\frac{\operatorname{colim}^W F}{\operatorname{Prev}_{N}{A}}\quad \frac{\int^{c \in C}}{\sum\limits_{k\in K}}\quad \frac{W( \underline{\hspace{0.5em}} )}{\#\tilde{W}_A^{-1}\underline{\hspace{0.5em}}}\quad \frac{F\underline{\hspace{0.5em}}} {\mathbb{P}[\tilde{W}=\underline{\hspace{0.5em}}]}$$

$$\frac{ \operatorname{colim}^W F \cong \int^{c \in C} W c \cdot F c} { \operatorname{Prev}_N{A}:=\sum\limits_{k\in K} \#\tilde{W}_A^{-1}k\cdot\mathbb{P}[\tilde{W}=k] }$$

Statement and proof of a theorem to substantiate this claim and evidence might involve ``enriched category theory." Of course, the best would be references to the literature.