What Are Probabilities, Anyway?

In order answer questions like "What are X, anyway?", we can (phenomenologically) turn the question into something like "What can we do with X?" or "What consequences does X have?"

For example, consider the question "What are ordered pairs, anyway?". Sometimes you see "definitions" of ordered pairs in terms of set theory. Wikipedia says that the standard definition of ordered pairs is:

(a, b) := {{a}, {a, b}}

Many mathematicians find this "definition" unsatisfactory, and view it not as a definition, but an encoding or translation. The category-theoretic notion of a product might be more satisfactory. It pins down the properties that the ordered pair already had before the "definition" was proposed and in what sense ANY construction with those properties could be used. Lambda calculus has a couple constructions that look superficially quite different from the set-theory ones, but satisfy the category-theoretic requirements.

I guess this is a response at the meta level, recommending this sort of "phenomenological" lens as the way to resolve these sort of questions.

This word "possible" carries a LOT of hidden baggage. If math tells us anything its that LOTS of things SEEM possible to us because we aren't logically omniscient but aren't really possible.

While we're at it, how about we drop "worlds" from the mix. I don't think it adds anything. If we replace it with "information flows" do things work better?

Lumping probabilities in with utilities sounds pretty close to Vladimir Nesov's Representing Preference by Probability Measures.

Copied from a chat where I tried to explain interpretations 3 and 4 a bit more:

I'm not sure what it means for a world to be more real either, but to the extent the idea makes sense in the many worlds interpretation of quantum mechanics (where some Everett branches are somehow "more real" or "exist more") it seems reasonable to extend that to other mathematical structures. One intuition pump is to imagine that the multiverse literally consists of an infinite collection of universal Turing machines, each initialized with a random input tape. So that's #3 in my post. I see #4 as sort of a fallback position, where if there is no fact of the matter about which worlds have more "measure" or "reality fluid" that can be discovered through philosophical reasoning or other kinds of investigation, you can still have UDT "add up to normality" (e.g., make a UDT-based AI behave in a way that a typical human would find reasonable) by hard coding a "prior" into its utility function, which can be interpreted as it simply caring more about some worlds more than others.

Your getting yourself in trouble because you assume that puzzling questions must have deep answers when usually the question itself is flawed or misleading. In this case there just seems to be a need for any explanation of the kind you offer nor would be of any use anyway.

These 'explanations' you offer of probability aren't really explaining anything. Certainly we do succesfully use probability to reason about systems that behave in a deterministic classical fashion (rolling dice probably counts). No matter what sort of probability you believe in you have to explain that application. So introducing 'objective' probability merely adds things we need to explain (possible worlds etc..).

The correct approach is to step back and ask what is it that needs explaining. Well probability is really nothing but a fancy way of counting up outcomes. So once we justify describing the world in a probabilistic fashion (even when it's deterministic in some sense) the application of mathematical inference to reformulate that description in more useful ways is untroubling. In other words if it's reasonable to model rolling two six sided dice as being independent uniformly random variables on 1...6 counting up the combinations and saying there is a 1/6 chance of getting a 7 doesn't raise any new difficulties.

So the question just comes down to is it reasonable of us to model the world using random variables?. I mean one might worry that some worlds were deeply 'tricky' in that almost always when it appeared two objects behaved like independent random variables in reality there was some hidden correlation that would eventually pop out to bite you in the ass and then once you'd taken that correlation into account another one would bite you and so on and so on.

But if you think about it for awhile this isn't really so much a question about the nature of the world as it is a purely mathematical question. If we keep factoring out by our best predictions will the remaining unaccounted for variation in outcomes appear to be random, i.e., make modeling it as random variables an accurate way to make predictions? Well that's actually kinda complicated, I have a theorem (well tiny tweak of someone else's theorem plus interpratation) which I believe says that yes indeed it must work this way. I won't go into it here but let me just say one thing to convince you of it's plausibility.

Basically the argument is that things only fail to look random because we notice a more accurate way of predicting their behavior. The only evidence for a sequence of observations failing to be random according to the supposed distribution would be a pattern in the observations not captured by R so would in turn yield a more accurate distribution. So basically the claim is that we can always simply divide up any observable into the part we can predict (i.e. a distribution of outcomes) and the part we can't. Once you mod out by the part you can predict by defintion anything left is totally unpredictable to you (e.g. computable machines) and thus can't detectably fail to look random according to it's distribution since that would be a better prediction.

This isn't rigorous (it's complicatd) but the point is that Randomness is nothing but our inability to make any better predictions

All possible worlds are real, and probabilities represent how much I care about each world.

Right, so maybe we need to rethink this whole rationality thing, then? I mean, since there are possible worlds where god exists, under this view, the only difference between a creationist and a rational atheist is one of taste?

To me, the god world seems much easier to deal with and more pleasant. So why not shun rationality all together if probabilities are actually arbitrary - if thinking it really does make it so?

Before I’ve observed anything, there seems to be no reason to believe that I’m more likely to be in one world than another, but we can’t let all their weights be equal.

We can't? Why not? Estimating the probability of two heads on two coinflips as 25% is giving existence in worlds with heads-heads, heads-tails, tails-heads, and tails-tails equal weight. The same is true of a more complicated proposition like "There is a low probability that Bigfoot exists" - giving every possible arrangement of objects/atoms/information equal weight, and then ruling out the ones that don't result in the evidence we've observed, few of these worlds contain Bigfoot.

Hmmm - caring as a part of reality? Why not just flip things up, and consider that emotion is also part of reality. Random by any other name. Try to exclude it and you'll find you can't no matter how infinitely many worlds you suppose. There's also calculus to irrationality . . .

All possible worlds are real, and probabilities represent how much I care about each world. ... Which worlds I care more or less about seems arbitrary.

This view seems appealing to me, because 1) deciding that all possible worlds are real seems to follow from the Copernican principle, and 2) if all worlds are real from the perspective of their observers, as you said it seems arbitrary to say which worlds are more real.

But on this view, what do I do with the observed frequencies of past events? Whenever I've flipped a coin, heads has come up about half the time. If I accept option 4, am I giving up on the idea that these regularities mean anything?

What does real even mean, by the way? Interpretation 1 with real taken to mean ‘of or pertaining to the world I'm in’ (as I would) is equivalent to Interpretation 2 with real taken to mean ‘possible’ (as Tegmark would, IIUC) and to Interpretation 3 with real taken to mean ‘likely’ and to Interpretation 4 with real taken to mean ‘important to me’.

It depends. We use the term "probability" to cover a variety of different things, which can be handled by similar mathematics but are not the same.

For example, suppose that I'm playing blackjack. Given a certain disposition of cards, I can calculate a probability that asking for the next card will bust me. In this case the state of the world is fixed, and probability measures my ignorance. The fact that I don't know which card would be dealt to me doesn't change the fact that there's a specific card on the top of the deck waiting to be dealt. If I knew more about the situation (perhaps by counting cards) I might have a better idea of which cards could possibly be on top of the deck, but the same card would still be on top of the deck. In this situation, case 1 applies from the choices above.

Alternately consider photons going through a double slit in the classical quantum physics experiment. If the holes are of equal size and geometry, a photon has a 50% chance of passing through each slit (the probabilities can be adjusted, for example by changing the width of one slit). One of the basic results of quantum physics is that the profile of the light through both slits is not the same as the sum of the profiles of the light through each slit. In general, it is not possible to say which slit a given photon when through, and attempting to make that measurement changes the answer. In this situation, case 3 of the above post seems to apply.

My point is that the post's question can't be answered for probabilities in general. It depends.

2 and 4 are much the same if you only care about worlds you are in.

The post would be much better if a definition of "possible world" was given. When giving definitions, perhaps to define what does "real" precisely mean would be beneficial.

More or less, I interpret "reality" as all things which can be observed. "Possible", in my language", is something which I can imagine and which doesn't contradict facts that I already know. This is somewhat subjective definition, but possibility obviously depends subjective knowledge. I have flipped a coin. Before I have looked at the result, it was possible that it came up heads. After I have looked at it, it's clear that it came up tails, heads are impossible.

Needless to say, people rarely imagine whole worlds. Rather, they use the word "possible" when speculating about unknow parts of this world. Which may be confusing, since our intuitive understanding of the word doesn't match its use.

Even if defined somehow objectively (as e.g. possible world is any world isomorphic to a formal system with properties X), it seems almost obvious that real world(s) and possible worlds are different categories. If not, there is no need to have distinct names for them.

So before creating theories about what probability means, I suggest we unite the language. These things have been discussed here already several times, but I don't think there is a consensus in interpretation of "possible", "real", "world", "arbitrary". And, after all, I am not sure whether "probability" even should be interpreted using these terms. It almost feels like "probability" is a more fundamental term than "possible" or "arbitrary".

I must admit that I am biased against "possible worlds" and similar phrases, because they tend to appear mostly in theological and philosophical discussions, whose rather empty conclusions are dissatisfactory. I am afraid of lack of guidelines strong enough to keep thinking in limits of rationality.

All possible worlds are real, and probabilities represent how much I care about each world.

Could you elaborate on what it means to have a given amount of "care" about a world? For example, suppose that I assign (or ought to assign) probability 0.5 to a coin's coming up heads. How do you translate this probability assignment into language involving amounts of care for worlds?

Why should probabilities mean anything? How how would you behave differently if you decided (or learned) a given interpretation was correct?

As long as there's no difference, and your actions add up to normality under any of the interpretations, then I don't see why an interpretation is needed at all.