## Is a causal interpretation of Bayes Nets fundamental?

**This is post 5 in a sequence exploring formalisations of causality entitled “Reasoning with causality”. This post continues to summarise the paper “Causal Reasoning with Causal Models”.
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**The previous post in the sequence is An introduction to Bayesian networks in causal modeling**

Is a causal interpretation of Bayes Nets fundamental in some way or is it simply accidental. To what extent can Bayes Net be considered to be suitable causal models?

**Bayes Nets as representations of probability distributions**

A common argument that the causal interpretation of Bayes Nets isn’t fundamental is that Bayes Nets are designed to represent probability distributions. And given any Bayes Net with a causal interpretation, another one can be generated that represents the same probability distribution but does not have the causal interpretation. On the surface then, the Bayes Net which is a causal model seems no more fundamental than the other.

Chickering’s Arc Reversal Rule can be used to support the above assertion. This rule basically just reverses the direction of all the arrows. A technical addition to how this works means that the rule can introduce arrows but not remove them. From this, there seems to be an obvious way that Bayes Nets do fundamentally represent causality: The causal model is the one with least arrows that still manages to capture the relevant probability distribution.

Unfortunately, this doesn’t work.

**Why the simplest Bayes Net isn’t the causal model**

There are circumstances under which the causal model is not the simplest Bayes Net that captures the probabalistic dependencies. Imagine the following situation: Sunscreen decreases instances of cancer but increases the time people spend in the sun (because they feel safer) which then increases the chance of cancer (modelled as below):

Now imagine that the two effects perfectly balanced so that sunscreen made no difference as to whether you got sunscreen. The simplest model that would capture the related probabality distribution is far simpler than that above. It looks like this:

The only problem is, this doesn’t represent the causal system. So the causal model cannot be the simplest one which captures the probability distribution.

**Augmented Causal Simplicity**

Which leads to a more complicated conjecture: A causal model is the one with the least arrows that still manages to capture the relevant fully augmented probability distribution. In a basic sense, the fully augmented probability distribution simply means captures the probabilities regardless of how we intervene in the model. So say we intervene in the above model to set the amount of time spent in the sun. In the first graph, this means sunscreen no longer changes the amount of time spent in the sun but it does change the chance of getting cancer. In the second model, changing the time spent in the sum will not capture this probabalistic relationship.

So by demanding that a model capture the probability distribution under all possible interventions, the causal model can still be said to be the simplest that captures the distribution. This then implies that Bayes Nets representation of causality are fundamental.

**The next post in the sequence is Modeling type and token causal relevance**