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Still interesting anyway. Thoughts:

1. how do we know that the variables are not a subset of possible variables, selected for their general interest to humans and so the simplicity simply due to the hidden confounder 'interesting enough to humans to collect'?
2. could this be an artifact of representation? Something like (although that seems to have meaningful explanations)
3. the general observation reminds me of Cohen's paper 'The Earth is Round p<0.05' where he notes on pg 4 that 'the nil hypothesis' is always false and quotes a psychology saying that "Everything is related to everything else".

Eric Schwitzgebel

Gwern: Thanks for the thoughtful comment!

On 1: I agree that's possible! It was my intent to acknowledge that possibility by nodding to the fact that I'm not generalizing to aliens and looking at the kinds of variables we experience in day-to-day life.

On 2: Benford's law is a cool regularity. It's also a pattern that holds across a wide range of variables -- so it's another way into the type of project above. There are representational issues in the data, e.g., some data that it seemed natural to put in logarithmic form, which impacted the Wildness tests (harder to get 10x or 1/10 but easier to flip positive to negative. Here's one place where I tried to just be "stupid" about things.

On 3: I didn't know Cohen's paper. (Thanks for the link!) Paul Meehl also said some things about everything being related, as I recall. But relatedness and complexity seem to me to be somewhat separable. Relations can be simple or complex, and arguably nothing is simpler than a zero correlation....


> Relations can be simple or complex, and arguably nothing is simpler than a zero correlation....

That gets you into 'whose definition of simplicity'; if you look at it from an entropy/information theory/computation complexity point of view, 2 variables which are uncorrelated are as complex as it is possible to be, since knowing one variable does not predict in the slightest bit the other variable and the 2 variables have no shorter encoding than the variables separately.

Also consider the opposite case: if 0 correlation is the simplest possible relationship, then surely 1 correlation is the most complex possible relationship. Yet you can get a correlation of 1 with any variable x just by looking at the correlation of x and... x.

Eric Schwitzgebel

Indeed, simplicity is a complex issue! I agree that encoding each value of each variable is information intensive if they are uncorrelated. But encoding *their relationship* is arguably informationally cheap! I'd be inclined to think that both zero and one correlations are simple relationships, compared, say, to the relationship y = (sin(a + bx)+c)^(f+gx^2) -- which is still pretty simple compared to the entire universe of possible functions. (I might get into this a bit in my follow-up post.)

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3QD Prize 2012: Wesley Buckwalter