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I was going through Probabilistic Reasoning In Intelligent Systems by Judea Pearl .In chapter 3 the author tries to motivate the need for qualitative representation of independence relations, that do not require numerical equality check - $P(A,B)$ = $P(A|B)P(B)$. This he proposes because, human beings do not always depend on numerical calculations/probability estimates to reach judgements about independencies. For example, a person can conclude that the chances of a burglary at his/her home tonight is independent of the chances of war in next 10 years, without assigning any probability estimates to the individual events.

My query is, whether this claim - that independence relations are qualitative in nature - is always true. Following is an example where numerical estimates become important.

Let us have a pair of 4-sided dice.

The independence of first die roll from second is quite intuitive; the independence in this case has more to do with the process-wise independence of the two events. However,

$A$ = first die shows 1
$B$ = sum of two dice rolls is 5
Here the independence is purely due to numerical values. Causal diagram of the scenario - Causal diagram of two die rolls

shows that they are dependent. Even though the qualitative nature of the dependence remains (by virtue of the arrow), it’s the numeric values involved that make the two events independent. This independence is of a quantitative nature that’s possible due to the particular numbers involved in this context. How can such independencies be inferred from qualitative aspects alone?

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    $\begingroup$ I think this is an instance of a map–territory fallacy. Whether you model (mentally or computationally) two events as independent and whether they physically are are not the same question. $\endgroup$
    – Arya McCarthy
    Commented Aug 7, 2022 at 2:06
  • $\begingroup$ @AryaMcCarthy your point does give a hint towards the fact that process-wise (causal) independencies are the ones that we have an intuition about. And those are the ones that are better represented by graphical models like Markov/Bayesian networks or causal diagrams. But maybe numerically induced independencies, which are not intuitive, cannot always be captured by graphical models. This I am guessing though. Would help if someone can shed some light on that. $\endgroup$
    – Anirban Chakraborty
    Commented Aug 7, 2022 at 2:23
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    $\begingroup$ @Scriddie please note that the pair of dice are 4 sided, possible values of a roll being {1,2,3,4}. Probability of getting a sum of 5 without any knowledge of first die outcome is $1/4$. Again, given first die outcome is 1, probability of sum being 5 is also $1/4$. Hence the independence. This is not intuitive, and the independence is peculiar to the numerical values chosen for the example. This forms the crux of the question. $\endgroup$
    – Anirban Chakraborty
    Commented Aug 18, 2022 at 12:46
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    $\begingroup$ @AnirbanChakraborty thanks! In this case, it is not clear to me why "the qualitative nature of the dependence remains (by virtue of the arrow)". One could rectify the diagram by removing the arrow and hence the contradiction. While humans may be able to make some statements about independence easily without checking the probabilities, this is clearly not one of them. $\endgroup$
    – Scriddie
    Commented Aug 18, 2022 at 13:00
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    $\begingroup$ The presence or absence of arrows is inferred by testing for independence using interventional data. In a graph with two variables and no unobserved ones, a lack of statistical dependency implies the absence of a causal dependency. There should thus not be an arrow. For more information, see this video. $\endgroup$
    – Scriddie
    Commented Aug 18, 2022 at 14:39

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In your example, the independence is not obvious other than by considering the probabilities at play. This shows that humans can not always make correct statements about independence without considering the probabilities numerically. However, the way one arrives at a conclusion does not affect the binary nature of dependence, which is either present or absent and can therefore be considered qualitative.

In many circumstances, human understanding of the world may be mechanistic, which allows us to make many statements about statistical dependencies without having explicitly calculated them (e.g. there is simply no plausible mechanism connecting my daily cereal consumption with the length of the monsoon season in India). In his subsequent works on causality, Pearl goes to great lengths to show that such mechanistic understanding can be modeled through a causal structure that encodes (conditional) independence relationships qualitatively.

In conclusion, the independencies you mention (ones that humans can easily reason about, as well as ones that we need to check numerically) are not different in a statistical sense and can therefore be equally represented in graphical models.

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  • $\begingroup$ The statement - 'In his subsequent works on causality, Pearl goes to great lengths to show that such mechanistic understanding can be modeled through a causal structure that encodes (conditional) independence relationships qualitatively.' - assures me that in the book Causality by Pearl I would then find this capacity of graphical models to encompass both types of independencies, intuitive and non-intuitive. Did I understand your answer correctly? $\endgroup$
    – Anirban Chakraborty
    Commented Aug 18, 2022 at 14:10
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    $\begingroup$ These independencies are not different in a statistical sense and thus equivalent for probabilistic modeling. Since different things may be intuitive to different people, a textbook would likely not frame it like that. However, you would find examples of causal graphical models encoding (conditional) independencies, some of which may be intuitive whilst others may not be. You will also find a lot on how independencies can encode the notions of intervention and counterfactuals. The fact that those ideas seem intuitive to humans may explain why some of the independencies seem "natural" to us. $\endgroup$
    – Scriddie
    Commented Aug 18, 2022 at 14:30
  • $\begingroup$ I have not yet gone through the quantitative aspects (relations that may stand behind the arrows) of a causal graphical structure; only the qualitative aspects (where d-separation indicates conditional independence). However, I am guessing that the relation behind the arrow joining the two nodes ("first die outcome" and "sum of dice pair outcome") can always be designed to indicate both dependence and independence, based on the node values. Is that the type of idea presented in Pearl's works? $\endgroup$
    – Anirban Chakraborty
    Commented Aug 18, 2022 at 14:39
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    $\begingroup$ An arrow does not indicate dependence and independence simultaneously. An arrow in a causal graphical model means that changing the value of the parent affects the value of the child, which is called causal dependency. The exact effect depends on the functional form of the causal dependency. $\endgroup$
    – Scriddie
    Commented Aug 18, 2022 at 14:49

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