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I wanted to pin down what the intuitive phrase:

r.v. X influences r.v. Y

as precisely and as rigorously as I could, and wanted to check if my interpretation was correct and complete with the community.

The way I interpret that phrase is as the following:

X influences Y if any observation (or realization) of the r.v X affects the probability distribution for the values of Y.

So for example, if we are wondering what the probability for a certain value of Y might be, we might go reference its distribution $p_Y(y)$ for an answer. However, if we believe that X could influence Y, then we believe that the distribution for Y is affect by a realization of X, yielding a different but valid distribution $p_{Y|X}(y|x)$.

Is this a correct way of interpreting what "r.v. X influences r.v. Y" means? Are there alternative interpretations of that sentences in probability, statistics, machine learning or inference?

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Your definition bears some similarity to Judea Pearl's definition, but is missing something very important. We need the concept of manipulation, as opposed to observation. Causal claims are claims about the results of manipulations, even if we can learn about them via observation.

The results of observations and manipulations can be very different. For example: if I were to observe that a patient has nicotine-stained fingers, then the probability that that patient has lung cancer would increase. But if I perform a manipulation, and paint the patient's hands yellow, then the probability of lung cancer does not change.*

Manipulation and observation have other different properties. Observation is symmetric (if learning the value of $X$ informs me about the value of $Y$, then the converse is also true), whereas manipulation is asymmetric (if manipulating $X$ leads to a change in $Y$, we can't assume that manipulating $Y$ will lead to a change in $X$).

Philosophers have attempted to define causation and manipulation in terms of observation alone, and every such attempt has run into some irresolvable difficulty. So the current approach in the causal literature is to take the idea of manipulation as a primitive, and define causation in terms of it. Pearl uses the "do" operator to represent manipulations: $do(X=x)$ means manipulating $X$ so that it takes the value $x$. Using this operator, we can define influence concisely:

$X$ causally influences $Y$ iff there exist at least two values of $X$, $x$ and $x'$, such that $P(Y\;|\;do(X=x)) \neq P(Y\;|\;do(X=x'))$

*Note: The phrase "controlling for X, Y and Z" actually means conditioning on the values of X, Y and Z – that is, observing them. This can mislead non-statisticians, because "control" sounds like a kind of manipulation.

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    $\begingroup$ Are you quite sure about this? "Observation is symmetric (if learning the value of X informs me about the value of Y, then the converse is also true)" I think association can be directional: if I observe that someone won a marathon, that tells me (with certainty) that that individual was born, however knowing that someone was born tells me nothing about whether they won a marathon. Another example would be the non-functional relationships comprising some of the examples in Reshef &Co.'s "Detecting novel associations in large data sets" (sciencemag.org/content/334/6062/1518 ) $\endgroup$ – Alexis Nov 10 '14 at 5:58
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    $\begingroup$ @Alexis Actually, learning that Joe was born does make it more likely that Joe won a marathon, compared to when I'm unsure about whether Joe was born at all! Probabilistic dependence/independence is symmetric: if A is dependent on B, then B is dependent on A. But I agree that the degree of dependence can be asymmetric (as in your example), for some measures of dependence. $\endgroup$ – Lizzie Silver Nov 10 '14 at 6:43

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