The following was written by me during the “Finding the Right Abstractions for healthy systems” research workshop, hosted by Topos Institute in January 2023. However, I invented the idea before.
Here’s an elegant diagrammatic notation for constructing new infrakernels out of given infrakernels. There is probably some natural category-theoretic way to think about it, but at present I don’t know what it is.
By “infrakernel” we will mean a continuous mapping of the form X→□Y, where X and Y are compact Polish spaces and □Y is the space of credal sets (i.e. closed convex sets of probability distributions) over Y.
Syntax
The diagram consists of child vertices, parent vertices, squiggly lines, arrows, dashed arrows and slashes.
There can be solid arrows incoming into the diagram. Each such arrow a is labeled by a compact Polish space D(a) and ends on a parent vertex t(a). And, s(a)=⊥ (i.e. the arrow has no source vertex).
There can be dashed and solid arrows between vertices. Each such arrow a starts from a child vertex s(a) and ends on a parent vertex t(a). We require that P(s(a))≠t(a) (i.e. they should not be also connected by a squiggly line).
There are two types of vertices: parent vertices (denoted by a letter) and child vertices (denoted by a letter or number in a circle).
Each child vertex v is labeled by a compact Polish space D(v) and connected (by a squiggly line) to a unique parent vertex P(v). It may or may not be crossed-out by a slash.
Each parent vertex p is labeled by an infrakernel Kp with source S1×…×Sk and target T1×…×Tl where each Si is corresponds to a solid arrow a with t(a)=p and each Tj is D(v) for some child vertex v with P(v)=p. We can also add squares with numbers where solid arrows end to keep track of the correspondence between the arguments of Kp and the arrows.
If s(a)=⊥ then the corresponding Si is D(a).
If s(a)=v≠⊥ then the corresponding Si is D(v).
Semantics
Every diagram D represents an infrakernel KD.
The source space of KD is a product X1×…×Xn, where each Xi is D(a) for some solid arrow a with s(a)=⊥.
The target space of KD is a product Y1×…×Ym, where each Yj is D(v) for some non-crossed-out child vertex.
The value of the KD at a given point x is defined as follows. Let ~Y:=∏vD(v) (a product that includes the cross-out vertices). Then, KD(x) is the set of all the marginal distributions of distributions μ∈Δ~Y satisfying the following condition. Consider any parent vertex p. Let a1,a2…ak be the (dashed or solid) arrows s.t.s(ai)≠⊥ and t(ai)=p. For each i s.t., choose any yi∈D(s(ai)). We require that Kp(x,y) contains the marginal distribution of μ∣y. Here, the notation Kp(x,y) means we are using the components of x and y corresponding to solid arrows a with t(a)=p.
The following was written by me during the “Finding the Right Abstractions for healthy systems” research workshop, hosted by Topos Institute in January 2023. However, I invented the idea before.
Here’s an elegant diagrammatic notation for constructing new infrakernels out of given infrakernels. There is probably some natural category-theoretic way to think about it, but at present I don’t know what it is.
By “infrakernel” we will mean a continuous mapping of the form X→□Y, where X and Y are compact Polish spaces and □Y is the space of credal sets (i.e. closed convex sets of probability distributions) over Y.
Syntax
The diagram consists of child vertices, parent vertices, squiggly lines, arrows, dashed arrows and slashes.
There can be solid arrows incoming into the diagram. Each such arrow a is labeled by a compact Polish space D(a) and ends on a parent vertex t(a). And, s(a)=⊥ (i.e. the arrow has no source vertex).
There can be dashed and solid arrows between vertices. Each such arrow a starts from a child vertex s(a) and ends on a parent vertex t(a). We require that P(s(a))≠t(a) (i.e. they should not be also connected by a squiggly line).
There are two types of vertices: parent vertices (denoted by a letter) and child vertices (denoted by a letter or number in a circle).
Each child vertex v is labeled by a compact Polish space D(v) and connected (by a squiggly line) to a unique parent vertex P(v). It may or may not be crossed-out by a slash.
Each parent vertex p is labeled by an infrakernel Kp with source S1×…×Sk and target T1×…×Tl where each Si is corresponds to a solid arrow a with t(a)=p and each Tj is D(v) for some child vertex v with P(v)=p. We can also add squares with numbers where solid arrows end to keep track of the correspondence between the arguments of Kp and the arrows.
If s(a)=⊥ then the corresponding Si is D(a).
If s(a)=v≠⊥ then the corresponding Si is D(v).
Semantics
Every diagram D represents an infrakernel KD.
The source space of KD is a product X1×…×Xn, where each Xi is D(a) for some solid arrow a with s(a)=⊥.
The target space of KD is a product Y1×…×Ym, where each Yj is D(v) for some non-crossed-out child vertex.
The value of the KD at a given point x is defined as follows. Let ~Y:=∏vD(v) (a product that includes the cross-out vertices). Then, KD(x) is the set of all the marginal distributions of distributions μ∈Δ~Y satisfying the following condition. Consider any parent vertex p. Let a1,a2…ak be the (dashed or solid) arrows s.t.s(ai)≠⊥ and t(ai)=p. For each i s.t., choose any yi∈D(s(ai)). We require that Kp(x,y) contains the marginal distribution of μ∣y. Here, the notation Kp(x,y) means we are using the components of x and y corresponding to solid arrows a with t(a)=p.