PHYS771 Lecture 17: Fun with the Anthropic Principle (2006) | Hacker News Reader

PHYS771 Lecture 17: Fun with the Anthropic Principle (2006)

scottaaronson.com

60 pointsTomte15 comments

The problem here arises in thinking this has anything to do with bayes rule, probability or any formalism of this kind.

The rule `P(B|A) = P(A|B)P(B)/P(A)` is only a model of ratios the follow a certain sort of set logic (and measure to [0, 1]).

It says nothing about what this model refers to. Does `B` refer to a proposition? A belief? An event? Bayesians say "belief", but then what do these beliefs model?

Do I believe "the sky is blue" ? What would it mean to have this as a belief? Do I have some model of the sky, of blue? etc. Why should we suppose that beliefs should operate according to bayes rule?

Yes, for any given class of beliefs, relative to some model of the world, you can construct an argument that bayes rule must apply. But these arent beliefs, they're truth-apt propositions which model a world. Beliefs are just mental states, and little constrains their consistency.

Since we do not know what the world is like. We do not know that believing, "frozen water could fall in a glass of water in ordinary conditions" contradicts, "water is largely comprised of h2o" -- so we can believe both at the same time.

So what's the issue here? The issue is that the formalism is useless for generating descriptions of the world (events, beliefs, propositions, ... whatever you like). And it's only when we have a set of such descriptions which actually model features of reality, and which therefore have some consistency between them... that we can actually construct any kind of formal model of reasoning. At this point, most of the work is done.

Thus much of the suppose normative force, and epistemic weight, of all this machinery is an illusion. Since we have no idea what beliefs affect the probabilities of what others; of what beliefs correspond to features of the world; and so on... no problems are resolved.

I'm always irritated by professors who will stand in front of a room and demonstrate people's irrationality against some experiment they perform, asserting that the model of the experiment people have in their heads is "the one on the slide", and then conclude that they are wrong. The issue is, in my experience without exception, that the audience is not so dumb as the professor. And has not made the very bizarre assumptions scrawled on the board; and hence the entire formal model is invalid.

I have a hard time grokking your wording, but I'd like to point out that Bayes' Rule is typically "typed" to events (i.e. B is of type "event").

That being said, I don't think this needs to be the case. The syntax could "compile" differently depending on context.

From my understanding, Bayes' Rule is more about updates to pre-existing probabilities, which we label with the suggestive name 'beliefs', which aren't necessarily tied to a human's mental state.

I do agree with your last paragraph, but I also accept that those profs likely do so because some majority or plurality of students do hols that belief. That being said, there's definitely room for different wording.

in Bayesian epistemology, the arguments to probability "functions" are beliefs.

To simplify my phrasing: to model any aspect of reality requires inventing a formalism; that formalism itself largely fails to correspond to reality. insofar as it does, the formalism is useless.

Bayesian epistemology pretends to greater insights than it has, because it assumes that the modelling relation is simple; whereas, really its where the whole part of epistemology lies.

The dice room puzzle described here is a little frustrating, because no-one seems to get it right. In the finite population case the anthropic argument for "I'm likely to be the in the last round" is wrong. In the countably infinite case it can't be applied (no discrete uniform distribution on a countably infinite set). I wrote some ramblings about it here, though I'm not sure it's super-clear: https://harrybraviner.github.io/posts/2024-01-28-anthropic_d...

I thought the writeup was convincing and addresses the heart of the paradox. The only nitpick: you can have uniform probability distributions on infinite sets, like [0,1]: https://en.wikipedia.org/wiki/Continuous_uniform_distributio... There p(x) = 0 for any x, but for fixed e, p(x +/- e) is the same for all x.

But you can't have such a distribution on an unbounded set, which is where the paradox fails. If we had a uniform distribution on an unbounded set, p(x +/- e) has to be the same for all x and therefore nonzero, but

  p(1 +/- e) + p(2 +/-e) + ...

has to sum to <= 1. It is an infinite sum of nonzero terms so this is a contradiction. (The same argument works if you drop the epsilon for thinking of a distribution on the integers).

I think your writeup was basically clear on this in terms of the math, just some of the language was a bit confused.

Yeh, U[0, 1] is different because it assigns non-zero probabilities to intervals, not points. In this case we're assuming that we live in an uncountable population (each real in [0, 1] is a person), so you can't do things like assign a unique number to each person. There, even if the maniac goes on kidnapping forever, he will only kidnap a countable subset of the population. Thinking about this honestly makes my brain hurt a little.

Thanks for writing this.

Is this concept (dice room puzzle, doomsday argument) at all related to the st Petersburg paradox? https://en.m.wikipedia.org/wiki/St._Petersburg_paradox

I think it's a little different. It's like asking "what is the probability that a given coin is in the round that the player wins?" But the St Petersburg paradox isn't about that, it's purely about how many coins the player wins. I suppose it runs into similar problems when you ask about prizes that are so large that the bank runs out of coins, but I think it remains interesting even if you cap it at some finite number of coin flips. It still has the small-probability-of-huge-payout property.

If you've ever looked at the Kelly Criterion, that seems related (and in fact is one of the articles linked to from that Wikipedia page). There you maximise expected log return at each round, and I think that tames the infinity in this case (though I have _not_ checked that).

Reality is not a chess board. Thing is, we don’t know what we don’t know and we can't put a probability on that.

We don't even know how much we don't know, but I would intuitively say very much.

> We don't even know how much we don't know, but I would intuitively say very much.

I choose to believe that there is, and always will be, much more that we do not know than what we know. That is truly exciting because it means our curiosity can never be satiated.

> I would intuitively say very much

Potential knowledge is probably infinite. How much is a finite fraction of infinite? Likely not much.

To be fair, when we say "we know much" we don't compare with all that potential knowledge, but with the knowledge of the generations that preceded us, or compared to the uneducated goat herder on some steppe, or some such thing. But this is also tricky, because they knew/know thing that we don't and so we find ourselves in a muddy morass, even when guessing how much we know (compared to someone else), let alone trying to accurately assess this. Unless we are very specific of course, like narrowing it down to maths or nuclear science.

Come to think of it, that's probably what you meant.

OK, scratch this.

You're describing games of incomplete information (i.e. as opposed to a perfect game like chess). Interestingly, when I look this up, the first thing that pops up is "Bayesian Game".

A Bayesian game is a strategic decision-making model that assumes players have incomplete information. In such games, players are aware of the rules and the mechanics governing the game, but they lack full knowledge about certain aspects, like the strategies or payoffs of other players. Poker is a classic example where the game rules are clear to everyone, but players do not know each other's hands.

This contrasts with the real world, where we do not fully understand the "rules" of reality. While we have well-established theories like Newtonian mechanics, quantum mechanics, and general relativity, these frameworks do not provide a complete explanation of everything. In reality, we are continually searching for new rules and understandings.

Imagine playing a game with three known rules while being told there are additional unknown rules. Calculating the odds of these unknown rules based on the existing ones is simply not possible from my perspective.

Then we have concepts like the anthropic principle, which states that "conditions observed in the universe must allow the observer to exist." This statement is a tautology, meaning it is self-evidently true. It's similar to saying, "a game that is played must have conditions that support playing." While tautologies are logically true, they do not offer new information and thus don't allow for substantive new conclusions to be drawn. They simply restate the obvious without providing additional insights.

The argument of "we can't say because of all we don't know" needs constant updating.

Early on, you have Elihu in Job arguing that we can't understand creation because why it rains and where snow comes from is beyond human understanding.

Now we learn it as a nursery rhyme.

Maybe we should entertain the idea that we do have adequate information to assess our overall situation and see what we find.

The issue of Bayes rule (it’s not really theorem) is that you have to concretely define your set of possible hypotheses beforehand.

In the real world, there are unknown unknowns, and those can’t be put into the Bayes rule calculation.

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