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Stream: learning: questions

Topic: dependence, posets, and ambiguity

David Egolf (Dec 05 2022 at 23:43):

I was looking a little at "Duoidal Structures for Compositional Dependence" by Shapiro and Spivak. In that paper, they talk about the idea that a poset can be used to model dependence. Assume we have a poset category $P$. We think of the objects as "events" of some kind, and we put a morphism $f: a \to b$ if $a$ must happen before $b$ happens.

I was trying to use this formalism to describe a situation that arises in imaging. The idea would be to put a morphism $f: a \to b$ to indicate that making some observations $b$ requires that the thing we are looking at satisfy some property as described by $a$. The problem is that most observations are ambiguous - there could be multiple past events describing the nature of the target being imaged that lead to the event of making a particular observation with particular data values.

In this ambiguous situation, I was thinking there would actually be no morphism with target of $b$ if $b$ was an ambiguous observation. Now that I type this all out, I realize that if we define a "union" of events - defined by $a_1 \cup a_2$ being said to happen exactly when at least one of $a_1$ or $a_2$ happens - then there would actually be morphisms with target of $b$. For example, there would need to be a morphism from the union event $\cup_i a_i$ where each $a_i$ is the presence of a target configuration that could have lead to the observation made, and the union is taken over all such configurations. This is because there was some target present which generated our observations.

I was going to ask how ambiguity could be modelled in this category, but I think the union events can be used to model this. I'll create this post anyways though, in case anyone has something they would like to add on this topic.