What's the purpose of randomness in causal discovery techniques?

Roulette
Wheel (Wikipedia)

Preamble 

In this short exposition, we inquire about the purpose of randomness and how this related to discovering or testing causal inferential problem solving using data and causal models. In his seminal work by Holland (1986) point out something striking that was not put in such form earlier works. He stated the "obvious" that almost all data sets addressing interventional nature, such as treatment vs. non-treatment, that a person or unit we study, can not be treated and not-treated at the same time. We delve question of randomness from this perspective, i.e., so called fundamental problem of causal inference.

Group assignment for causal inference   

Group assignment probably one of the most fundamental approach in statistical research, such as in the famous Lady tea tasting problem.  The idea of assignment in causal inference, we need to find a matching person or unit that is not-treated if we have a treated sample or the other-way around, so called a matching or balancing.  

Randomness in causality: Removal of  pseudo-confounders

Randomness doesn't only allow fair representation of control and treatment group assignments, reducing bias essentially. The primary effect of randomness is removal of  pseudo-confounders, this is not well studied in the literature. What it means, if we don't randomise there would be other causal connections that would really shouldn't be there. 

Conclusion

Here, we hint about something called  pseudo-confounders.  Randomisation in both matching and other causal techniques primarily removes bias but  removal of pseudo-confounders is not commonly mentioned and an open research.

Further reading

• looper : Causality Resource 

Please cite this article as: 

 @misc{suezen23ran, 

     title = {What's the purpose of randomness in causal discovery techniques?}, 

     howpublished = {\url{https://memosisland.blogspot.com/2023/11/causal-inference-randomisation.html}, 

     author = {Mehmet Süzen},

     year = {2023}

}

  

  

Why should there be no simultaneity rule for causal models?

Dominos in motion
(Wikipedia)

Preamble 

The definition of weighted directed graphs (wDAGs) provides a great opportunity to express causal relationships among given variates. Usually this is expressed as SCMs, Structural Causal Model or in more generally causal model. A given causal model can be expressed as set of simultaneous equations, given a direction for the equality, right to left , meaning $A=B$ implies B causes A to happen  $B \to A$ . Then what happens if A is a function of B and C, $A=f(B,C)$, then we say $ B \to A$ and $C \to A$ occurs simultaneously.  In this post we discuss this situation that there should be no simultaneity rule in causal models, regardless of if they are not time-series models.

Understanding  causal models

Basic definition of a causal model follows a functional form with set of equations, realistically with added noise. The models forms a weighted Directed Acyclic Graphs (wDAGs) visually. Here is the mathematical definition due to Pearl (Causality 2009), we made it a bit more coarser in this definition: 

Definition (Causal Model) : Given set of $n$-variables $X \in \mathbb{R}^{n}$, two subsets of $X= x_{1} \cup x_{2}$, they can form set of equations $x_{2}=f(x_{1}; \alpha; \epsilon)$,  $\alpha$ being the causal effect sizes on $x_{1}$ as causes of $x_{2}$ with some noise $\epsilon$. This corresponds to a $wDAG$ formed among $X$ with weights $\alpha$. So that there is a graph $\mathscr{G}(X, \alpha)$ representing this set of equations, where by equality put direction from right to the left side of the equation. 

However this definition does not set any constraints on the values of $\alpha$. Any two or more values of $\alpha$-s can be equivalent on the same path within $X$. This implies an interestingly that there would be set of variates simultaneously causes the same thing. It sounds plausible and physically possible to a degree within Planck-time. However, this brings an ambiguity of breaking ties in ordering events.

Perfect Causal Ordering

Given wDAG as a causal model induces causal ordering among all members of $X$. As we defined how this can be achieved in a recent post: Practical causal ordering. In this context, perfect causal ordering implies  $\alpha$ values within the first order paths to a given end variable are different. Mathematically speaking a definition follows. 

Definition (No simultaneity rule) Given all $k$ triplets ($x_{i}, y, \alpha_{i}$), that $x_{i}$ is one of the causes of $y$, and $\alpha_{i}$ causal effect sizes, all $\alpha_{i}$ are different numbers, inducing a perfect causal ordering.

This rule ensures we don't need to break ties randomly as causal ordering is established uniformly.

Conclusion: Importance of no simultaneity 

By this definition we ruled out any simultaneous causes. This may sound too restrictive for modelling but this impacts decision making significantly; ranking causes of an outcome will impact how to prioritise the policy in addressing the outcome, i.e., such as medical intervention to prevent first cause. Also, it may not be feasible to intervene simultaneous causes. Hence, establishing primary causes in order is paramount in decision making and execution of any reliable policy.

Further reading

• looper : Causality Resource 
• Looper Nuggets 1 (LN1) : Definition of wDAGs with no simultaneity rule.

Please cite this article as: 

 @misc{suezen23nos, 

     title = {Why should there be no simultaneity rule for causal models?}, 

     howpublished = {\url{https://memosisland.blogspot.com/2023/11/causal-model-simultaneous.html}, 

     author = {Mehmet Süzen},

     year = {2023}

}