Seeing as there is little secondary literature for the Finite Factored Set formalism, I thought I’d write up my experience of exploring it through some toy examples that are classic examples in the Pearlian paradigm. My goal was to see how these models that I understood very well in the Pearlian paradigm would work in Finite Factored Sets.

As a warning, this doesn’t make any use of the more exciting properties of Finite Factored Sets. It’s just an exploration of how this formalism handles the mundane stuff. This also means that I’m using the factored set directly, without the abstractions of the orthogonality database. Which I think is fine here, because these are tiny toy examples whose structure is fully known. (However, it’s possible that I’ve missed the entire point of Finite Factored Sets.)

The first example is the 3-variable collider that is very central to Pearl’s formalism. It is given by the following Causal Diagram:

$A$ , $B$ , and $C$ are all binary variables (0=false, 1=true).

The intended meaning of a Causal Diagram (or rather, function causal model) is that the value of a node $x_{i}$ is given by a deterministic function that takes as input the parents, $p a (x_{i})$ , (indicated by the arrows) and an “error”, or “noise”, variable $u_{i}$ that is governed by a probability distribution that is independent from the other error/noise variables: $x_{i} = f_{i} (p a (x_{i}), u_{i})$ . Thus, the value of $C$ is given by $c = f_{c} (a, b, u_{C})$ where $u_{C}$ is noise or uncertainty that is not explicitly modeled, which we can visualize like this:

We could also split up $A$ and $B$ into a deterministic and a random part, but as they are root nodes, there is little point. It would just be $a = f_{a} (u_{A})$ .

The Pearlian formalism runs on graphs, but Finite Factored Sets run on the set $S$ of all possible outcomes – the sample space. So, the goal is now to construct a sample space that is consistent to the above graph. After that, we’ll find a factorization of that sample space.

I think it should be clear that to cover the whole sample space $S$ , it is sufficient to consider all possible combinations of the outcomes of $A$ , $B$ , and $U_{C}$ (but not $C$ ), because if we know the value of these three, then we also know the value of $C$ , via $f_{c}$ .

So, we simply define $S$ as the Cartesian product of the sets of possible values of $A$ and $B$ : $A = {0, 1}$ , $B = {0, 1}$ , and the possible values of $U_{C}$ : $U_{C}$ , which I’ll leave und...

Posts

Wikitag Contributions

Comments