I suggest reading Radford Neal.

Anthropic reasoning is what leads people to believe in miracles. Rare events have a high probability of occurring if the number of observations is large enough. But whoever that rare event happens to will feel like it couldn't have just happened by chance, because the odds of it happening to them was so large.

If you wait until the event occurs, and then start treating it as a random event from a single trial, forming your hypothesis after seeing the data, you'll make inferential errors.

Imagine that there are balls in an urn, labeled with numbers 1, 2,...,n. Suppose we don't know n. A ball is selected. We look at it. We see that it's number x.

non-anthropic reasoning: all numbers between 1 and n were equally likely. I was guaranteed to observe some number, and the probability that it was close to n was the same as the probability that it was far from n. So all I know is that n is greater than or equal to x.

anthropic reasoning: A number as small as x is much less likely if n is large. Therefore, hypotheses with n close to x are more likely than hypotheses where n is much larger than x.

Consider the case of Sleeping Beauty with an absent-minded experimenter.

If the coin comes up Heads, there is a tiny but non-zero chance that the experimenter mixes up Monday and Tuesday.

If the coin comes up Tails, there is a tiny but non-zero chance that the experimenter mixes up Tails and Heads.

The resulting scenario is represented in a new sheet, Fuzzy two-day, of my spreadsheet document.

Under these assumptions, Beauty may no longer rule out Tuesday & Heads. She has no justification to assign all of the Heads probability mass to Monday & Heads. She is therefore constrained to conditioning on being woken in the way that the usual two-day variant suggests she should, and ends up with a credence arbitrarily close to 1/3 if we make the "absent-minded" probability tiny enough.

Why should we get a discontinuous jump to 1/2 as this becomes zero?

Your argument is, I take it, that these counts of observations are irrelevant, or at best biased.

No, I was just saying that this, lim N-> infinity n1/(n1+n2+n3), is not actually a probability in the sleeping beauty case.>

I maintain that it is. I can guarantee you that it is. What obstacle do you see to accepting that? You've made noises that this is because the counts are correlated, but I haven't seen any argument for this beyond bare assertion. Do you want to claim it is impossible for some reason, or are you just saying you haven't seen a persuasive argument yet?

The disagreement seems to center on the denominator; it should count not awakenings, but coin-tosses.

No, I wouldn't say that. My argument is that you should use probability laws to get the answer. If you take ratios of expected counts, well, you have to show that what you get as actually a probability.

What would you require for proof? If I could show you a Markov chain whose behavior is isomorphic to iterated Sleeping Beauty, would that convince you?

I also am not sure what you mean when you say "use probability laws". Is there a failure to comport with the Kolmogorov axioms? Is there a problem with the definition of the events? Do you mean Bayes' Theorem, or some other law(s)? I also am deeply suspicious of the phrase "get the answer". I will have no idea what this could mean until we can eliminate ambiguity about what the question is (there seems to be a lot of that going around), or what class of questions you'll admit as legitimate.

defining feature of Sleeping Beauty--all Sleeping Beauty's awakenings are epistemically indistinguishable. She has no choice but to treat them all identically.

Hm, I think that is what I'm saying. She does have to treat them all identically. They are the same variable. That's why she has to say the same thing on Monday and Tuesday.

Up to this point, I see we are actually in strenuous agreement on this aspect, so I can stop belaboring it.

That's why an awakening contains no new info. If she had new evidence at an awakening, she'd give different answers under heads and tails.

I don't mean to claim that as soon as Beauty awakes, new evidence comes to light that she can add to her store of bits in additive fashion, and thereby update her credence from 1/2 to 1/3 along the way. If this is the only kind of evidence that your theory of Bayesian updating will acknowledge, then it is too restrictive. Since Beauty is apprised of all the relevant details of the experimental process on Sunday evening, she can (and should) use the fact that the predicted frequency of awakenings into a reset epistemic state is dependent on the state of the coin toss to change the credence she reports on such awakenings from 1/2 to 1/3. She can tell you this on Sunday night, just as I can tell you now, before any of us enter into any such experimental procedure. So her prediction about what she should answer on an awakening does not change from Sunday evening to Monday morning.

The key pieces of information she uses to arrive at this revised estimate are:

• That the questions will be asked in a reset epistemic state. This requires her to give the same answer on all awakenings.
• That the frequency of awakenings is dependent in a specific way on the result of the coin toss. This requires her to update the credence she'll report on awakenings from 1/2 to 1/3.

this is a probability tree corresponding to an arbitrary wake up day

Huh? If tails, then Beauty is (always) woken on Monday. Why do you have probability=1/2 there?

(likewise for Tuesday)

If you just apply Bayes rule, you get 1/2.

Apply it to what terms?

I'm not sure what more I can say without starting to repeat myself, too. All I can say at this point, having formalized my reasoning as both a Python program and an analytical table giving out the full joint distribution, is "Where did I make a mistake?"

Where's the bug in the Python code? How do I change my joint distribution?

So I think I figured this whole thing out. Are people familiar with the type-token distinction and resulting ambiguities? If I have five copies of the book Catcher in the Rye and you ask me how many books I have there is an ambiguity. I could say one or five. One refers to the type, "Catcher in the Rye is a coming of age novel" is a sentence about the type. Five refers to the number of tokens, "I tossed Catcher in the Rye onto the bookshelf" is a sentence about the token. The distinction is ubiquitous and leads to occasional confusion, enough that the subject is at the top of my Less Wrong to-do list. The type token distinction becomes an issue whenever we introduce identical copies and the distinction dominates my views on personal identity.

In the Sleeping Beauty case, the amnesia means the experience of waking up on Monday and the experience of waking up on Tuesday, while token-distinct are type-identical. If we decide the right thing to update on isn't the token experience but the type experience: well the calculations are really easy. The type experience "waking up" has P=1 for heads and tails. So the prior never changes. I think there are some really good reasons for worrying about types rather than tokens in this context but won't go into until I make sure the above makes sense to someone.

That is a difference, but it seems independent from the point I intended the example to make. Namely, that a relative frequency can still represent a probability even if its denominator includes duplicates - it will just be a different probability (hence why one can get 1/3 instead of 1/2 for SB).

In your example the experimenter has learned whether you have cancer. And she reflects that knowledge in the structure of the experiment: you are woken up 9 times if you have the disease.

Set aside the amnesia effects of the drug for a moment, and consider the experimental setup as a contorted way of imparting the information to the patient. Then you'd agree that with full memory, the patient would have something to update on? As soon as the second day. So there is, normally, an information flow in this setup.

What the amnesia does is selectively impair the patient's ability to condition on available information. it does that in a way which is clearly pathological, and results in the counter-intuitive reply to the question "conditioning on a) your having woken up and b) your inability to tell what day it is, what is your credence"? We have no everyday intuitions about the inferential consequences of amnesia.

Knowing about the amnesia, we can argue that Beauty "shouldn't" condition on being woken up. But if she does, she'll get that strange result. If she does have cancer, she is more likely to be woken up multiple times than once, and being woken up at all does have some evidential weight.

All this, though, being merely verbal aids as I try to wrap my head around the consequences of the math. And therefore to be taken more circumspectly than the math itself.

Basically, the 2 wakings on tails should be thought of as one waking. You're just counting the same thing twice.

This is true if we want the ratio of tails to wakings. However...

When you include counts of variables that have a correlation of 1 in your denominator, it's not clear what you are getting back. The thirders are using a relative frequency that doesn't converge to a probability

Despite the perfect correlation between some of the variables, one can still get a probability back out - but it won't be the probability one expects.

Maybe one day I decide I want to know the probability that a randomly selected household on my street has a TV. I print up a bunch of surveys and put them in people's mailboxes. However, it turns out that because I am very absent-minded (and unlucky), I accidentally put two surveys in the mailboxes of people with a TV, and only one in the mailboxes of people without TVs. My neighbors, because they enjoy filling out surveys so much, dutifully fill out every survey and send them all back to me. Now the proportion of surveys that say 'yes, I have a TV' is not the probability I expected (the probability of a household having a TV) - but it is nonetheless a probability, just a different one (the probability of any given survey saying, 'I have a TV').