In Probability Space & Aumann Agreement, I wrote that probabilities can be thought of as weights that we assign to possible world-histories. But what are these weights supposed to mean? Here I’ll give a few interpretations that I've considered and held at one point or another, and their problems. (Note that in the previous post, I implicitly used the first interpretation in the following list, since that seems to be the mainstream view.)

- Only one possible world is real, and probabilities represent beliefs about which one is real.
- Which world gets to be real seems arbitrary.
- Most possible worlds are lifeless, so we’d have to be really lucky to be alive.
- We have no information about the process that determines which world gets to be real, so how can we decide what the probability mass function p should be?

- All possible worlds are real, and probabilities represent beliefs about which one I’m in.
- 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.

- Not all possible worlds are equally real, and probabilities represent “how real” each world is. (This is also sometimes called the “measure” or “reality fluid” view.)
- Which worlds get to be “more real” seems arbitrary.
- Before we observe anything, we don't have any information about the process that determines the amount of “reality fluid” in each world, so how can we decide what the probability mass function p should be?

- All possible worlds are real, and probabilities represent how much I care about each world. (To make sense of this, recall that these probabilities are ultimately multiplied with utilities to form expected utilities in standard decision theories.)
- Which worlds I care more or less about seems arbitrary. But perhaps this is less of a problem because I’m “allowed” to have arbitrary values.
- Or, from another perspective, this drops another another hard problem on top of the pile of problems called “values”, where it may never be solved.

As you can see, I think the main problem with all of these interpretations is arbitrariness. The unconditioned probability mass function is supposed to represent my beliefs before I have observed anything in the world, so it must represent a state of total ignorance. But there seems to be no way to specify such a function without introducing *some* information, which anyone could infer by looking at the function.

For example, suppose we use a universal distribution, where we believe that the world-history is the output of a universal Turing machine given a uniformly random input tape. But then the distribution contains the information of which UTM we used. Where did that information come from?

One could argue that we do have some information even before we observe anything, because we're products of evolution, which would have built some useful information into our genes. But to the extent that we can trust the prior specified by our genes, it must be that evolution approximates a Bayesian updating process, and our prior distribution approximates the posterior distribution of such a process. The "prior of evolution" still has to represent a state of total ignorance.

These considerations lead me to lean toward the last interpretation, which is the most tolerant of arbitrariness. This interpretation also fits well with the idea that expected utility maximization with Bayesian updating is just an approximation of UDT that works in most situations. I and others have already motivated UDT by considering situations where Bayesian updating doesn't work, but it seems to me that even if we set those aside, there is still reason to consider a UDT-like interpretation of probability where the weights on possible worlds represent how much we care about those worlds.

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?

Do you mean something precise by "information flows"?

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:

... (read more)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 hav... (read more)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?

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 . . .

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

realeven mean, by the way? Interpretation 1 withrealtaken to mean ‘of or pertaining to the world I'm in’ (as I would) is equivalent to Interpretation 2 withrealtaken to mean ‘possible’ (as Tegmark would, IIUC) and to Interpretation 3 withrealtaken to mean ‘likely’ and to Interpretation 4 withrealtaken 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.... (read more)

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... (read more)

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

meananything? 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.