A reformulation of Finite Factored Sets

post by Matthias G. Mayer (matthias-georg-mayer) · 2023-07-24T13:02:25.382Z · LW · GW · 1 comments

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I present my bachelor's thesis that reformulates Finite Factored Sets:

Causality with Deterministic Relationships

Abstract:
We present a reformulation of Finite Factored Sets that uses functions and cartesian products instead of partitions and factorizations. In the new setting, we motivate assuming conditional orthogonality from observed conditional independence and show that inference is decidable. We show that Pearlian causal structures have analogs in the developed framework and propose an extension to the infinite unconditional case using measurable spaces.

I will quote Section 1.3 to give an overview of the additions to the original paper.

Most of the concepts in this work were first introduced in [Finite Factored Sets (FFS)], where the author starts with a set $S$ and factorizes it using a set of partitions $\{F_i:i\in I\}$ of $S$. Then partitions on $S$ are analyzed using this factorization. This is called a (finite) factored set.

In this work, we instead start from the factors $F_i$ as sets and construct $F={\times }_{i\in I}{F}_i$ as a cartesian product. Here we call $F$ a (finite) factor space. Instead of a partition on $S$, we analyze features $X:F\to {\Omega }_X$, where a feature would correspond to the partition $\left\{X^{-1}\right(\omega ):\omega \in {\Omega }_X\}$ of $F$.

In the same manner, we can translate all concepts using functions. In the authors' opinion, this is a more natural way to frame things. We hope that this reframing is helpful to understand the framework and for doing further work.

To compare this work to the original paper [FFS], we give a detailed list of the main additions, apart from the reformulation.

• In Section 2.4, we provide an alternative way to define conditional histories using a factor space induced on a subset $A\subseteq F$.
  This is helpful to understand the concepts more deeply and leads, in some cases,
  to shorter and more intuitive proofs.
• In Section 3.3, we generalize the examples provided in [FFS] (Example 1 and 2)
  to groups.
• In Section 3.2 and the appendix, we spell out the implications carried by
  the fundamental theorem of finite factored sets [FFS] (Theorem 3)
  to justify assuming conditional orthogonality from observed conditional independence.
• In Section 3.4, we prove that inference is decidable, thus solving
  this open problem raised in [FFS] (Section 7.1).
• In Section 4, we convert Pearlian causal structures and models to factor spaces.
  We prove that d-separation in a causal structure is equivalent to
  conditional orthogonality in the corresponding factor space.
  We further give a definition of interventions in such a factor space and prove that they lead to the same probabilities a causal structure provides.
• In Section 5, we provide a generalization of the framework to arbitrary
  measure spaces in the unconditional case.

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