alex_zag_al comments on Putting in the Numbers - Less Wrong

8 Post author: Manfred 30 January 2014 06:41AM

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Comment author: alex_zag_al 02 February 2014 04:02:24AM *  0 points [-]

Whoa, you think the only correct interpretation of "there's a die that returns 1, 2, or 3" is to be absolutely certain that it's fair? Or what do you think a delta function in the distribution space means?

I haven't been able to follow this whole thread of conversation, but I think it's pretty clear you're talking about different things here.

  • Obviously, the long-run frequency distribution of the die can be many different things. One of them, (1/3, 1/3, 1/3), represents fairness, and is just one among many possibilities.
  • Equally obviously, the probability distribution that represents rational expectations about the first flip is only one thing. Manfred claims that it's (1/3, 1/3, 1/3), which doesn't represent fairness. It could equally well represent being certain that it's biased to land on only one side every time, but you have no idea which side.
Comment author: Vaniver 02 February 2014 07:58:19AM 0 points [-]

I think it's pretty clear you're talking about different things here.

I thought so too, which is why I asked him what he thought a delta function in the distribution space meant.

One of them, (1/3, 1/3, 1/3), represents fairness, and is just one among many possibilities.

Right; but putting a delta function there means you're infinitely certain that's what it is, because you give probability 0 to all other possibilities.

It could equally well represent being certain that it's biased to land on only one side every time, but you have no idea which side.

Knowing that the die is completely biased, but not which side it is biased towards, would be represented by three delta functions, at (1,0,0), (0,1,0), and (0,0,1), each with a coefficient of (1/3). This is very different from the uniform case and the delta at (1/3,1/3,1/3) case, as you can see by calculating the posterior distribution for observing that the die rolled a 1.

Comment author: alex_zag_al 03 February 2014 05:24:31AM 0 points [-]

okay, and you were just trying to make sure that Manfred knows that all this probability-of-distributions speech you're speaking isn't, as he seems to think, about the degree-of-belief-in-my-current-state-of-ignorance distribution for the first roll. Gotcha.

Comment author: alex_zag_al 03 February 2014 05:09:49AM *  0 points [-]

Okay... but do we agree that the degree-of-belief distribution for the first roll is (1/3, 1/3, 1/3), whether it's a fair die or a completely biased in an unknown way die?

Because I'm pretty sure that's what Manfred's talking about when he says

There is a single correct distribution for our starting information, which is (1/3,1/3,1/3),

and I think him going on to say

the "distribution across possible distributions" is just a delta function there.

was a mistake, because you were talking about different things.

EDIT:

I thought so too, which is why I asked him what he thought a delta function in the distribution space meant.

Ah. Yes. Okay. I am literally saying only things that you know, aren't I. My bad.

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