The Principle of Indifference

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Here is a short primer on how to avoid mangling the principle of indifference: never use it to justify assigning probability 1/n for any small integer n (like 2, for instance, yielding probability 0.5) to an untestable hypothesis, because you're probably fooling yourself.

Now here's a longer primer!

First let's check the font of human wisdom, wikipedia.

Suppose that there are n > 1 mutually exclusive and collectively exhaustive possibilities. The principle of indifference states that if the n possibilities are indistinguishable except for their names, then each possibility should be assigned a probability equal to 1/n.

So, for example, suppose you have a standard 6-sided die. You have a hypothesis

H: This die will land on a 1 when rolled, given that it lands on one of its faces like normal, rolled like normal.

If I said to you "Due solely to the principle of indifference, you should assign probability 1/6 to H.", I would be wrong. Why is this? Because each of the possibilities, 1-6, are not indistinguishable except for their names. There is a different amount of mass removed due to indentations, different amounts of black ink and white ink on each side, etc. But due to the principle of indifference + our knowledge of physics, we can say "the differences between the sides are almost certainly insignificant and I don't know which way they'd bias the die anyway" and conclude P(H)~=1/6.

Seems easy, right? But it's easy to slip up and misapply. Take this example:

I have to ask: Do you think there's a better option? And in that case, what is it? Should we just assume that everything is equally likely?

That's the part that's driving me nuts! How many competing theories are there to explain reality? Infinite! We can't assign them all an agnostic value. Example:

God exists and he created the world 5 minutes ago and gave us false memories.
God exists and he created the world 4 minutes ago and gave us false memories.
I can literally create billions of competing theories with this alone by dividing time into smaller and smaller units. But should we be agnostic about each one? But that doesn't make sense. If you roll a fair die, you can't be agnostic about each number. You have to assign a .16666 value to each possible outcome.

Now this is a provocative point! Suppose for the sake of argument that

G: God exists and he created the world some time ago and gave us false memories.

Call G_n the hypothesis that he did this n minutes ago. What's wrong with saying "Due to the principle of indifference, P(G₁|G) = P(G₂|G) = P(G₃|G) = P(G₄|G) = ..."? Aren't all of these hypotheses indistinguishable except for their names? Don't they exhaustively cover the possibility space given G? Aren't they mutually exclusive?

It's clear that something is wrong because if we try to use the principle of indifference like this, then for any positive probability p, P(G₁|G) < p. Proof: sum(P(G_k|G),k=1..1+1/p) < 1 ==> P(G₁|G) < 1/(1+1/p) < p. So then P(G_n|G) = 0 for all n, but that doesn't work, because the laws of probability say limit(sum(P(G_k|G),k=1..n),n->inf) = 1.

Easiest first. They're definitely mutually exclusive, as God can only create the world once that we care about. We're talking about the world we think we know of, so no pedantic other universes or whatever. And they're exhaustive, because we're abstracting "one-and-a-half minutes ago" into G₂. So they must be distinguishable - but how? There is no possible test we can do to tell the difference between any of the two hypotheses!

Do you see what I did there? It's tricky. I just equivocated the word "indistinguishable". See, in mathematics, indistinguishable means something very specific, and it's not "inability to tell the difference via experiment" or whatever. Instead, look at the possibilities themselves. If the only difference between them is the names we give them, then they are indistinguishable. Note: not the "only detectable difference", the only difference. And in this case there is a very real difference between G₁ and G₂. In one the world has existed for longer and God's original fake-out was slightly less of a fake than in the other, regardless of whether we can ever tell which is which.

So they're distinguishable. The principle of indifference does not apply. What can we do instead, anything? It turns out yes, we can, but that's the subject of another post. The main result is that for sufficiently loose definitions of "smaller", the smaller n is, the larger G_n probably is. Smaller numbers are in a very real sense "less complex" than larger numbers.

How about this one:
1. We die and cease to exist
2. We die and go to heaven
3. We die and reincarnate
4. We die and merge with the one-mind
5. We die and Baal takes us to the 8th circle of Hell

Without any evidence, does that mean there's a low chance we cease to exist when we die because there are 4 other competing theories? Or should theories 2-5 be combined into one all-ecompassing theory: we die and don't cease to exist. That would get you back to at least an agnostic value between exisence after death and non-existence. But can you just lump a bunch of different theories together?

I can't answer your question. It's bugged me for a long time.

The answer here is that in the absence of any evidence, each possible theory must be evaluated per its complexity and then we get a situation like the G_ns above: a ton (infinity) of distinguishable theories, some more complex and thus intrinsically less likely than others. But the mere fact that you cannot in practice tell the difference between two theories does not make them indistinguishable from a principle of indifference standpoint, and if you try to assign probability 0.5 to one of them,

you're probably fooling yourself

 

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... Do you see what I did there? It's tricky. I just equivocated the word "indistinguishable". See, in mathematics, indistinguishable means something very specific, and it's not "inability to tell the difference via experiment" or whatever. Instead, look at the possibilities themselves. If the only difference between them is the names we give them, then they are indistinguishable. Note: not the "only detectable difference", the only difference. And in this case there is a very real difference between G₁ and G₂. In one the world has existed for longer and God's original fake-out was slightly less of a fake than in the other, regardless of whether we can ever tell which is which.

So they're distinguishable. The principle of indifference does not apply. What can we do instead, anything? It turns out yes, we can, but that's the subject of another post. The main result is that for sufficiently loose definitions of "smaller", the smaller n is, the larger G_n probably is. Smaller numbers are in a very real sense "less complex" than larger numbers.


The answer here is that in the absence of any evidence, each possible theory must be evaluated per its complexity and then we get a situation like the G_ns above: a ton (infinity) of distinguishable theories, some more complex and thus intrinsically less likely than others. But the mere fact that you cannot in practice tell the difference between two theories does not make them indistinguishable from a principle of indifference standpoint, and if you try to assign probability 0.5 to one of them,

Nice OP, GA. Looks like poor Malerin has been barking up the wrong tree; and now we must coax him down, eh?

You raise an interesting point about distinguishing between G1 and G2: G2 is less of a "fake" because earthlings in G2 have 2 minutes worth of "real" memory, vs 1 minute in G1. (I'm not sure, though, how one could distinguish between "real" and "fake" memory, however many minutes of real vs fake memory are involved, if God's fake-out is perfect, whenever it occurs.)

I have to ask: Do you think there's a better option? And in that case, what is it? Should we just assume that everything is equally likely?

That's the part that's driving me nuts! How many competing theories are there to explain reality? Infinite! We can't assign them all an agnostic value. Example:

God exists and he created the world 5 minutes ago and gave us false memories.
God exists and he created the world 4 minutes ago and gave us false memories.
I can literally create billions of competing theories with this alone by dividing time into smaller and smaller units. But should we be agnostic about each one? But that doesn't make sense. If you roll a fair die, you can't be agnostic about each number. You have to assign a .16666 value to each possible outcome.

However, from the above quoted, I think Malerin's confusion may hinge on something more basic (though I haven't been following the sourced thread): a failure to first distinguish, then evaluate, each hypothesis in a chain of hypotheses, which Malerin's example is.

Assume there's no way to research any and all hypotheses in the chain. You use the PoI to evaluate the probability of a chain of hypotheses by applying it to each link sequentially. Thus:

"[link 1] God exists and [2] he created the world [specifier] n minutes ago and [3] gave us false memories..."

L1: P (God exists).. by PoI = 50%; (or not) = 50%

L2: P (he created the world | L1) = 50% of 50% = 25%

L3: P (gave us false memories | L1 & L2) = 50% of 25% = 12.5%
& specifier
n minutes ago: P (n=x | L1 & L2 & L3) = .125 / age of world in minutes; for x = 0, 1, 2, ... age. Even given an age of eternity, this reintegrates to .125.

How about this one:
1. We die and cease to exist
2. We die and go to heaven
3. We die and reincarnate
4. We die and merge with the one-mind
5. We die and Baal takes us to the 8th circle of Hell

Without any evidence, does that mean there's a low chance we cease to exist when we die because there are 4 other competing theories? Or should theories 2-5 be combined into one all-ecompassing theory: we die and don't cease to exist. That would get you back to at least an agnostic value between exisence after death and non-existence. But can you just lump a bunch of different theories together?

I can't answer your question. It's bugged me for a long time.

Another chain of disguised as level, unlinked hypotheses.

[1] P(we die) = 100%. (no PoI yet given abundant evidence for mortality)

[2] P(and cease to exist | [1]) = 50% (assuming PoI); not [2] = 50%

[3] P(z | [1] & not [2]), where z = {heaven, reincarnate, merge, Baal} -- assuming that list is mutually exclusive and exhaustive -- 1 / possibilities: ie, 1/4 x 50% = 12.5%.

As I say, I haven't really been following whatever thread this came from, but I'm guessing something like the above may help rescue Malerin (how does a dog get up a tree anyhow? Shouldn't that be a cat? But cats don't have a bark. But trees do. Stupid metaphors -- now I'm confused!)

+ apologies in advance if this was already covered in-thread...