Welcome to The Nonlinear Library, where we use Text-to-Speech software to convert the best writing from the Rationalist and EA communities into audio. This is: Finite Factored Sets in Pictures, published by Magdalena Wache on December 11, 2022 on LessWrong.
Finite factored sets are a new paradigm for talking about causality. You can use them to do some cool things you can’t do with Pearl’s causal graphs, for example inferring a causal arrow between two binary variables.
Also, finite factored sets are a really neat mathematical structure: they are a way of taking a set and expressing it as a product of some factors. Set factorizations are analogous to integer factorizations, in the same way that set partitions are analogous to integer partitions.
So, here is my current understanding of finite factored sets, in pictures.
1. What are Set Factorizations?
What do these “factored sets” look like? Let’s start with a set S and factor it.
The first concept we need is a partition of a set S. A partition is a way of chopping up S into subsets (called parts). Here are a few examples of partitions:
We usually call the partitions X,Y,Z,U,V, or W, and their parts xi,yi,. like this:
U is called the trivial partition. It only has one part.
We can think of partitions as properties, or variables over our set. For example, consider a set like this:
and compare it to the partitions X,Y and Z from above:
Then
The partition X is the property “color”, with x1 = blue and x2 = orange.
The partition Y is the property “form” with y1 = square, and y2 = circle.
The partition Z is the property “filled” with z1 = yes, and z2 = no.
Exercise
Consider these two partitions X and Y on the set S. What would it look like to represent them as properties (e.g. X = shape, Y = color) instead?
Spoiler space
It could look something like this:
Here X = shape = {x1,x2,x3} = {star, circle, square}, and Y = color ={y1,y2} = {green, orange}.
I hope you can see how partitions and properties are basically the same thing. In the rest of this post, I will use “partitions” and “properties” interchangeably. Sometimes I will use the ring-style visualization of partitions, and sometimes the property style, depending on what I find more intuitive in any given example.
Now we can define set factorizations:
A factorization B of our set S looks like this:
A factorization is a set B={b1,b2,.,bn} of partitions (called factors). In this case B={b1,b2}={X,Y}={{x1,x2},{y1,y2}}.
But it can’t just be any set of partitions. In the following sections, I will explain the two conditions that B needs to fulfill in order to count as a factorization:
There is a unique element for all combinations of properties
No factor is trivial
1. There is a unique element for all combinations of properties
Let’s look at our partitions in terms of properties again:
What we need in order for B to be a factorization, is that for all combinations(xi,yj) of properties (for example (x1,y2), which is (blue, circle)), there is a unique element with these properties.
We can see that this is the case here: We have exactly one blue square, exactly one orange square, exactly one blue circle, and exactly one orange circle.
To express it more mathematically: For B={X1,X2,.,Xn} to be a factorization, we need that for all x1∈X1,x2∈X2,.xn∈Xn, it holds that the intersection of {x1,x2,.,xn} contains exactly one element. This means the cartesian product of our factors is bijective to the set S, which justifies that we say we can “express S as the product of our factors”.
2. No factor is trivial
Here is an example of a non-factorization:
B = {U, V} is not a factorization here, because U is trivial and factors aren’t allowed to be trivial.
This is analogous to integer factorization, where we don’t count 1 as a factor. For example, for the integer 6 we say the factorizations are {6} and {2,3}, and don’t mention {6,1} and {6,1,1} and {6,1,1,1} and so on.
Exercise
What about this? Is B = {X, Y} a factorization here? (take a moment to t...
