Tuesday, July 16, 2013

Trees and limits of probabilistic reasoning

Suppose you're one of the nodes of this infinite tree, cut off for the purposes of the diagram, but you have no information whatsoever on which node you are.  Region A is exactly like region B.  And region B is exactly like region C.

So that you're in D must be at least as likely as that you're in B or C, but that you're in B is just as likely as that you're in A, and ditto for being in C.  Hence that you're in D is at least twice as likely as that you're in A.  But that you're in D obviously has the same probability as that you're in A.

Thus, P(A)=P(D)2P(A).  Hence P(A)=0.  But of course the whole tree is equal to three copies of A, plus the point 0.  So if you can assign probabilities, then you're certain to be at 0.  Which is absurd, especially since you can recenter the graph at another point and run the argument again.

Philosophically, this is a nice illustration of the severe limits to probabilistic reasoning.  My eight-year-old son is looking at what I'm posting and says: "It's just probabilities and I'm not going to use probabilities on this tree.  It's obvious."  He wonders why I am posting such obvious things.

Mathematically, all that this displays is that of course there is a paradoxical decomposition of a regular tree, and hence that there is no finitely-additive symmetry-invariant probability measure on a regular tree.


James Bejon said...

This may be a really dumb question/observation...

But suppose the past is infinite. Presumably, this means there are an infinite number of past points in time (nodes?) at which things could have turned out differently to the way in which they have turned out.

So, how come there's a non-zero probability of today's world arising?

Alexander R Pruss said...

Do we have good reason to think it's non-zero?

Justin said...
This comment has been removed by the author.
Justin said...

Because it exists?

Alexander R Pruss said...

If space is continuous, all sorts of zero probability things happen.

James Bejon said...

Presumably, then, all sorts of things of probability 1 don't happen?

I think I'd take that as a reason to deny that space is continuous.

Alexander R Pruss said...

Sure. You can get zero probability events happening in at least four scenarios:
- continuous space (dart throws)
- continuous time (particle decay times)
- infinite past
- infinite future
- infinite universe/multiverse

Let me explain the last three. If there are infinitely many independent coin tosses, then for any infinite sequence of heads/tails, the probability of getting precisely that sequence is zero.

Alexander R Pruss said...

That's five, I guess.

James Bejon said...

Hmmm...my instant reaction, then, would be to deny that space is continuous, that time is continuous, etc.,...Does that sound a bit extreme?

Suppose I've been tossing a coin for an infinite amount of time. Would you say that P(heads every time) = P(tails every time) = P( (neither heads nor tails) every time)?