We’re still at work on the piranha theorems. But, in the meantime, I happened to show somebody this:

There can be

somelarge and predictable effects on behavior, but not a lot, because, if there were, then these different effects would interfere with each other, and as a result it would be hard to see any consistent effects of anything in observational data. The analogy is to a fish tank full of piranhas: it won’t take long before they eat each other.

And she said, wait, you better check to see if this is right. Are piranhas cannibals? That doesn’t seem right, if they’re cannibals they’ll just eat each other and die out. But if they’re not cannibals, the analogy doesn’t work.

So when I got home, I looked it up. I googled *are piranhas cannibals*. And this is the first thing that came up:

So my analogy is safe, and we’re good to go.

**P.S.** I guess I could’ve titled the above post, Are Piranhas Cannibals?, but that would’ve violated the anti-SEO principles of this blog. Our general rule is, make the titles as boring as possible, then anyone who clicks through to read the post will be pleasantly surprised by all the entertainment value we offer.

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Quote from the blog post: “There can be some large and predictable effects on behavior, but not a lot, because, if there were, then these different effects would interfere with each other, and as a result it would be hard to see any consistent effects of anything in observational data. The analogy is to a fish tank full of piranhas: it won’t take long before they eat each other.”

Perhaps (the gist, and/or implications, of) this reasoning allows me to point to yet another one of my annoyances concerning current day psychological science.

I thought the goal of psychological science is to explain, predict, understand behaviour and/or phenomena, etc. I reason a tremendousely important part of this might be to compare theories, compare variables, compare constructs, etc. to see which one best explains, predicts, and helps in understanding, things.

If this makes (any) sense, i reason this 1) might deserve some more acknowledgement, and 2) could be relevant (in some way or form) for the “piranha project” mentioned above.

Anon said,

“I thought the goal of psychological science is to explain, predict, understand behaviour and/or phenomena, etc. I reason a tremendousely important part of this might be to compare theories, compare variables, compare constructs, etc. to see which one best explains, predicts, and helps in understanding, things.”

I would hope that the goal is “… to see which one best explains, predicts, and helps in understanding things, under what circumstances,” but my impression is (sadly) that there is little of what Anon suggests, and even less of my amended version.

actually I clicked on this one because I thought “piranha theorems” sounded pretty dang interesting. Now I feel bamboozled. ;)

Shecky:

No, there really are some piranha theorems, formalizations of the idea described in my quoted paragraphs. A couple of mathematicians at Columbia are working on it.

Being cannibals does not necessarily mean they are able to catch their peers and die out. Piranhas in a four-dimensional fish tank cannot come across each other by randomly chasing no matter how eager they are, and this is when those math assumptions on p and n kick in.

I had no idea the piranha problem was about piranhas eating each other. I thought they were eating a cow.

But, does the cannibal interpretation really make sense? Wouldn’t a Brownian motion interpretation make more sense? Perhaps call it “Brownian Bowling Balls”. If people are bombarded by strong psych influences, they should bounce all over the place continually. This is what standard Brownian motion theory predicts, that an object bombarded by bowling balls will follow a wildly fluctuating random walk. The fact that we do not see this huge volatility means the effects of each individual influence must be small. We can then infer the effect of each individual influence from the volatility of the overall random walk. (You can run the math in either direction. The math is about a hundred years old.)

The cannibal interpretation implies they all eat each other and the effect dies out. I don’t think that is really correct. The individual effects don’t necessarily cancel, they cancel somewhat, but there is always some motion and there is some net drift in one direction or the other. That is standard Brownian motion theory too.

Terry:

Interesting point. To clarify: the piranha argument is a reductio ad absurdum:

ifthese large and persistent effects all existed,thenthey’d interfere with each other in a way that such large and persistent effects would not be observed,hencethey don’t exist.Your bowling ball analogy is better in that it works without the absurdum part being required.

Right. That is the other explanation for what we observe, that all the effects are negatively correlated in some weird way that they almost exactly cancel out.

Quote from above: “The cannibal interpretation implies they all eat each other and the effect dies out. I don’t think that is really correct.”

From a bigger picture perspective this may actually be correct in some way.

We’ve had decades of (ever increasingly silly?) “priming” research that (with a lot of other things) culminated to a point where a lot of this stuff is now viewed very, very critically (and perhaps even suspiciously).

“Priming” was hip and happening for about 10-20 (?) years, where everyone jumped on board the hype-train. But now, all the “seminal findings” are sort of laughed at, and “the big names” of this field are looked at with pity (if nothing else).

In this sense, perhaps “priming” research sort of ate itself and died.

https://www.youtube.com/watch?v=FJD-j2cXmS0. :~)

Andrew, I love the observation that there are ‘an infinite number of primes.’ Instead of calling this the Piranha Theorem, I think it should reference Euclid somehow. I’m not sure how, though: in this case the point is that there’s an infinite number of primes, but there can only be a small number of large primes.

To readers who don’t know: giving people some seemingly irrelevant experience or information before they do something is called ‘priming’ them.

Phil said,

“there’s an infinite number of primes, but there can only be a small number of large primes.”

Uhh … If we’re talking about prime numbers, the cardinality of the set of primes greater than the integer N is independent of N.

what’s the density of primes do for large x… let’s say the kernel density using a unit normal kernel of the indicator fuction of the primes for large x.

I don’t know how to answer your question as you have posed it after the …, but if you let pi(x) = # of primes less than or equal to x, then the Prime Number Theorem says pi(x) ~ x/log(x) –i.e., the limit as x goes to infinity of pi(x)/(x/log(x)) is 1.