Estabilishing a Statistically Based Causal Relationship

X89codered89X

Hi all,

I was curious about how i would go about showing that samples of a variable separated in time may have a causal relationship. This actually may be more stochastic processes than pure statistics becuase I'm assuming random variables [itex] X, Y [/itex] have distributions [itex] f(x; k), g(y;k) [/itex] where k is a discrete index representing time samples. How would I prove that X->Y in the traditional sense of logic that "Given X, then Y", where the truth of this statement ranges from -1 to 1.

Also just my thinking but "Given Y, then X" would not just be the negative of "Given X, then Y"

I don't have anything in my stat book about this, but maybe it's just too basic? Not Sure. Thanks for the help.

mfb

Where do you expect a causal relation? X->Y for some k?
You can find a correlation (if there is one), but that won't give you a causal relationship between both.

X->Y where X comes before Y? You cannot rule out a common influence on both just based on that correlation, but at least you can rule out Y->X.

Stephen Tashi

In the traditional sense of logic, the truth of "If X then Y" doesn't range from -1 to 1. In traditional logic, the truth of "If X then Y" is either true or false and it is a function of the truth or falsity of the propositions X,Y.. So you need to rephrase your question.

("Implication" is a topic of traditional logic. "Causation" is not. In fact, mathematics does have any standard definition for "causation". Discussions of causation are in the scope of Philosophy and Metaphysics.)

mathandpi

Judea Pearl has done quite a bit of work on causality, especially through Bayesian networks. Googling his name, you will find quite a few general-audience articles that might be interesting.

More mathematically, we may consider Bayesian Networks through graphical models and consider "interventions" in the model. In particular, see "Causal inference in statistics:
An overview" by Pearl at http://ftp.cs.ucla.edu/pub/stat_ser/r350.pdf

Everyone always like to say "Correlation does not imply Causation", so it is nice to be able to think about the other direction!

ImaLooser

If X and Y are correlated, then they may have a causal relationship. If not correlated, then no causal relationship. If it is stochastic processes, they might be correlated with some delay.

X89codered89X

Math and Pi: this is so close to what I was looking for it's not even funny. Thank you

Stephen tashi: yes I suppose you are right. I may need to revise my range of outcome to 0 to 1.

Mfb: I am talking about a metric in which you conclude some analog truth value to "x causes y" using both time series for all k.

Edit* ImaLooser: Based on MathandPi's Post (after actually starting to read the material from Pearl), Causation does not imply correlation since it's actually possible that the causation is nonlinear (from my understanding since correlation would imply, if anything at all, a linear causation between [itex] X [/itex] and [itex] Y [/itex]). There is no reason for causation to be an inherently linear operation in general.