Conditioning on Observers

Suppose two weeks' worth of coins are tossed ahead of time.

Then with probability 1/4, you will wake up twice in the evening. With probability 1/2, you will wake up 6 times in the morning and once in the evening. And with probability 1/4 you will wake up 12 times in the morning.

Then by your logic, you ought to say that your probability of waking in the morning is (1/4)x(0/2) + (1/2)x(6/7) + (1/4)x(12/12) = 3/7 + 1/4 = 19/28, rather than 1/2 if the coins are tossed 'just in time'.

How can whether the coins are tossed in advance or not change the subjective probability?

Could be. I figured not meaning to vote was more likely than a mind-change in the span of like 10 ten seconds on a math problem we've been talking about for days.

You're right; that's distinct from what happened to me. Neither of my voting links were bold after I voted the first time and refreshed.

Yet one more variant. On my view it's structurally and hence statistically equivalent to Iterated Sleeping Beauty, and I present an argument that it is. This one has the advantage that it does not rely on any science fictional technology. I'm interested to see if anyone can find good reasons why it's not equivalent.

The Iterated Sleeping Beaty problem (ISB) is the original Standard Sleeping Beauty (SSB) problem repeated a large number N of times. People always seem to want to do this anyway with all the variations, to use the Law of Large Numbers to gain insight to what they should do in the single shot case.

The Setup

• As before, Sleeping Beauty is fully apprised of all the details ahead of time.
• The experiment is run for N consecutive days (N is a large number).
• At midnight 24 hours prior to the start of the experiment, a fair coin is tossed.
• On every subsequent night, if the coin shows Heads, it is tossed again; if it shows Tails, it is turned over to show Heads.

(This process is illustrated by a discrete-time Markov chain with transition matrix:

[1/2 1/2] = P
[ 1   0 ]

and the state vector is the row

x = [ Heads Tails ], 

with consecutive state transitions computed as x * P^k

Each morning when Sleeping Beauty awakes, she is asked each of the following questions:

1. "What is your credence that the most recent coin toss landed Heads?"
2. "What is your credence that the coin was tossed last night?"
3. "What is your credence that the coin is showing Heads now?"

The first question is the equivalent of the question that is asked in the Standard Sleeping Beauty problem. The second question corresponds to the question "what is your credence that today is Monday?" (which should also be asked and analyzed in any treatment of the Standard Sleeping Beauty problem.)

Note: in this setup, 3) is different than 1) only because of the operation of turning the coin over instead of tossing it. This is just a perhaps too clever mechanism to count down the days (awakenings, actually) to the point when the coin should be tossed again. It may very well make a better example if we never touch the coin except to toss it, and use some other deterministic countdown mechanism to count repeated awakenings per coin toss. That allows easier generalization to the case where the number of days to awaken when Tails is greater than 2. It also makes 3) directly equivalent to the standard SB question, and also 1) and 3) have the same answers. You decide which mechanism is easier to grasp from a didactic point of view, and analyze that one.

• After that, Beauty goes on about her daily routine, takes no amnesia drugs, sedulously avoids all matter duplicators and transhuman uploaders, and otherwise lives a normal life, on one condition: she is not allowed to examine the coin or discover its state (or the countdown timer) until the experiment is over.

Analysis

• Q1: How should Beauty answer?
• Q2: How is this scenario similar in key respects to the SSB/ISB scenario?
• Q3: How does this scenario differ in key respects from the SSB/ISB scenario?
• Q4: How would those differences if any make a difference to how Beauty should answer?

My answers:

Q1: Her credence that the most recent coin toss landed Heads should be 1/3. Her credence that the coin was tossed last night should be 1/3. Her credence that the coin shows Heads should be 2/3. (Her credence that the coin shows Heads should be 1/3 if we never turn it over, only toss, and 1/K if the countdown timer counts K awakenings per Tail toss.)

Q2: Note that Beauty's epistemic state regarding the state of the coin, or whether it was tossed the previous midnight, is exactly the same on every morning, but without the use of drugs or other alien technology. She awakens and is asked the questions once every time the coin toss lands Heads, and twice every time it lands tails. In Standard Sleeping Beauty, her epistemic state is reset by the amnesia drugs. In this setup, her epistemic state never needs to be reset because it never changes, simply because she never receives any new information that could change it, including the knowledge of when the coin has been tossed to start a new cycle.

Q3: In ISB, a new experimental cycle is initiated at fixed times--Monday (or Sunday midnight). Here the start of a new "cycle" occurs with random timing. The question arises, does the difference in the speed of time passing make any difference to the moments of awakening when the question is asked? Changing labels from "Monday" and "Tuesday" to "First Day After Coin Toss" and "Second Day After Coin Toss" respectively makes no structural change to the operation of the process. Discrete-time Markov chains have no timing, they have only sequence.

In the standard ISB, there seems to be a natural unit of replication: the coin toss on Sunday night followed by whatever happens through the rest of the week. Here, that unit doesn't seem so prominent, though it still exists as a renewal point of the chain. In a recurrent Markov chain, the natural unit of replication seems to be the state transition. Picking a renewal point is also an option, but only as a matter of convenience of calculation; it doesn't change the analysis.

Q4: I don't see how. The events, and the processes which drive their occurence haven't changed that I can see, just our perspective in looking at them. What am I overlooking?

Iteration

I didn't tell you yet how N is determined and how the experiment is terminated. Frankly, I don't think it matters all that much as N gets large, but let's remove all ambiguity.

Case A: N is a fixed large number. The experiment is terminated on the first night on which the coin shows Heads, after the Nth night.

Case B: N is not fixed in advance, but is guaranteed to be larger than some other large fixed number N', such that the coin has been tossed at least N' times. Once N' tosses have been counted, the experiment is terminated on any following night on which the coin shows Heads, at the whim of the Lab Director.

Q5: If N (or N') is large enough, does the difference between Case A and B make a difference to Beauty's credence? (To help sharpen your answer, consider Case C: Beauty dies of natural causes before the experiment terminates.)

Note that in view of the discussion under Q3 above, we are picking some particular state in the transition diagram and thinking about recurrence to and from that state. We could pick any other state too, and the analysis wouldn't change in any significant way. It seems more informative (to me at any rate) to think of this as an ongoing prcess that converges to stable behavior at equilibrium.

Extra Credit:

This gets right to the heart of what a probability could mean, what things can count as probabilities, and why we care about Sleeping Beauty's credence.

Suppose Beauty is sent daily reports showing cumulative counts of the nightly heads/tails observations. The reports are sufficiently old as not to give any information about the current state of the coin or when it was last tossed. (E.g., the data in the report are from at least two coin tosses ago.) Therefore Beauty's epistemic state about the current state of the coin always remains in its initial/reset state, with the following exception. Discuss how Beauty could use this data to--

• corroborate that the coin is in fact fair as she has been told.
• update her credences, in case she accrues evidence that shows the coin is not fair.

For me this is the main attraction of this particular model of the Sleeping Beauty setup, so I'm very interested in any possible reasons why it's not equivalent.

[  0  1  0  0 ]
[ 1/2 0 1/2 0 ]
[  0  0  0  1 ]
[ 1/2 0 1/2 0 ]

0.5 * ( H + T ) + 0.5 * ( T + T ) = 0.5 * H + 1.5 * T.

HH: 1/4
HT: 1/4
TT: 1/2

[  0  1/2  0  1/2 ]
[ 1/2  0  1/2  0  ]
[  0   0   0   1  ]
[ 1/2  0  1/2  0  ]

0.25 * ( H + H ) + 0.25 * ( H + T ) + 0.5 * ( T + T ) = 0.75 * H + 1.25 * T

0.5 * ( H ) + 0.5 * ( T + T ) = 0.5 * H + 1.0 * T

[1/2, 1/2, 0]
[0,   0,   1]
[1/2, 1/2, 0]

Two ways to iterate the experiment:

1. Replicate the entire experiment 1000 times. That is, there will be 1000 independent tosses of the coin. This will lead between 1000 and 2000 awakenings, with expected value of 1500 awakenings.

and

1. Replicate her awakening-state 1000 times. Because her epistemic state is always the same on an awakening, from her perspective, it could be Monday or Tuesday, it could be heads or tails.

The distinction between 1 and 2 is that, in 2, we are trying to repeatedly sample from the joint probability distributions that she should have on an awakening. In 1, we are replicating the entire experiment, with the double counting on tails.

This seems a distinction without a difference. The longer the iterated SB process continues, the less important is the distinction between counting tosses versus counting awakenings. This distinction is only about a stopping criterion, not about the convergent behavior of observations or coin tosses to expected values as it's ongoing. Considered as an ongoing process of indefinite duration, the expected number of tosses and of observations of each type are well-defined, easily computed, and well-behaved with respect to each other. Over the long run, #awakenings accumulates 1.5 times more frequently than #tosses. Beauty is never more than two awakenings away from starting a new coin toss, so whether you choose to stop as soon as an awakening has completed or until you finish a coin-toss cycle, the relative perturbation in the statistics collected so far goes to zero. Briefly, there is no "natural" unit of replication independent of observer interest.

She knows that it was a fair coin. She knows that if she's awake it's definitely Monday if heads, and could be either Monday or Tuesday if tails. She knows that 50% of coin tosses would end up heads, so we assign 0.5 to Monday&heads.

This would be an error. You are assigning a 50% probability to an observation (that it is Heads&Monday) without taking into account the bias that's built in to the process for Beauty to make observations. Alternatively, if you are uncertain whether Monday is true or not--you know it might be Tuesday--then you should be uncertain that P(Heads)=P(Heads&Monday).

You the outside observer know the chance of observing that the coin lands Heads is 50%. You presumably know this because you have corroborated it through an unbiased observation process: look at the coin exactly once per toss. Once Beauty is put to sleep and awoken, she is no longer an outside observer, she is a particpant in a biased observation process, so she should update her expectation about what her observation process will show. Different observation process, different observations, different likelhoods of what she can expect to see.

Of course, as a card-carrying thirder, I'm assuming that the question about credence is about what Beauty is likely to see upon awakening. That's what the carefully constructed wording of the question suggests to me.

She knows that 50% of coin tosses would end up tails,

except that as we agreed, she's not observing coin tosses, she's observing biased samples of coin tosses. The connection between what she observes and the objective behavior of the coin is just what's at issue here, so you can't beg the question.

In 1, people are using these ratios of expected counts to get the 1/3 answer. 1/3 is the correct answer to the question about the long-run frequencies of awakenings preceded by heads to awakenings preceded by tails. But I do not think it is the answer to the question about her credence of heads on an awakening.

Agreed, but for this: it all depends on what you want credence to mean, and what it's good for; see discussion below.

In 2, the joint probabilities are determined ahead of time based on what we know about the experiment.

Let n2 and n3 are counts, in repeated trials, of tails&Monday and tails&Tuesday, respectively. You will of course see that n2=n3. They are the same random variable. tails&Monday and tails&Tuesday are the same.

Let me uphold a distinction that's continually skated over, but which is crucial point of disagreement here. I think you're confusing your evidence for the thing evidenced. And you are selectively filtering your evidence, which amounts to throwing away information. Tails&Monday and Tails&Tuesday are not the same; they are distinct observations of the same state of the coin, thus they are perfectly correlated in that regard. Aside from the coin, they observe distinct days of the week, and thus different states of affairs. By a state of affairs I mean the conjunction of all the observable properties of interest at the moment of observation.

It's like what Jack said about types and tokens. It's like Vladimir_Nesov said:

The distinction between types and tokens is only relevant when you want to interepret your tokens as being about something else, their types, rather than about themselves. But types are carved out of observers' interests in their significance, which are non-objective, observer-dependent if anything is. Their variety and fineness of distinction is potentially infinite. As I mentioned above, a state of affairs is a conjunction of observable properties of interest. This Boolean lattice has exactly one top: Everything, and unknown atoms if any at bottom. Where you choose to carve out a distinction between type and token is a matter of observer interest.

Two subsequent states of a given dynamical system make for poor distinct elements of a sample space: when we've observed that the first moment of a given dynamical trajectory is not the second, what are we going to do when we encounter the second one? It's already ruled "impossible"! Thus, Monday and Tuesday under the same circumstances shouldn't be modeled as two different elements of a sample space.

I'll certainly agree it isn't desirable, but oughtn't isn't the same as isn't, and in the Sleeping Beauty problem we have no choice. Monday and Tuesday just are different elements in a sample space, by construction.

if she starts out believing that heads has probability 1/2, but learns something about the coin toss, her probability might go up a little if heads and down a little if tails.

What you seem to be talking about is using evidence that observations provide to corroborate or update Beauty's belief that the coin is in fact fair. Is that a reasonable take? But due to the epistemic reset between awakenings, there is never any usable input to this updating procedure. I've already stipulated this is impossible. This is precisely what the epistemic reset assumption is for. I thought we were getting off this merry-go-round.

Suppose, for example, she is informed of a variable X. If P(heads|X)=P(tails|X), then why is she updating at all? Meaning, why is P(heads)=/=P(heads|X)? This would be unusual. It seems to me that the only reason she changes is because she knows she'd be essentially 'betting' twice of tails, but that really is distinct from credence for tails.

Ok, I guess it depends on what you want the word "credence" to mean, and what you're going to use it for. If you're only interested in some updating process that digests incoming information-theoretic quanta, like you would get if you were trying to corroborate that the coin was inded a fair one to within a certain standard error, you don't have it here. That's not Sleeping Beauty, that's her faithful but silent, non-memory-impaired lab partner with the log book. If Beauty herself is to have any meaningful notion of credence in Heads, it's pointless for it be about whether the coin is indeed fair. That's a separate question, which in this context is a boring thing to ask her about, because it's trivially obvious: she's already accepted the information going in that it is fair and she will never get new information from anywhere regarding that belief. And, while she's undergoing the process of being awoken inside the experimental setup, a value of credence that's not connected to her observations is not useful for any purpose that I can see, other than perhaps to maintain her membership in good standing in the Guild of Rational Bayesian Epistomologists. It doesn't connect to her experience, it doesn't predict frequencies of anything she has any access to, it's gone completely metaphysical. Ok, what else is there to talk about? On my view, the only thing left is Sleeping Beauty's phenomenology when awakened. On Bishop Berkeley's view, that's all you ever have.

Beauty gets usable, useful information (I guess it depends on what you want "information" to mean, too) once, on Sunday evening, and she never forgets it thereafter. This information is separate from, in addition to the information that the coin itself is fair. This other information allows her to make a more accurate prediction about the likelihood that, each time she is awoken, the coin is showing heads. Or whether it's Monday or Tuesday. The information she receives is the details of the sampling process, which has been specifically constructed to give results that are biased with respect to the coin toss itself, and the day of the week. Directly after being informed of the structure of the sampling process, she knows it is biased and therefore ought to update her prediction about what relative frequencies per observation will be of each observable aspect of the possible state of affairs she's awoken into-- Heads vs. Tails, Monday vs. Tuesday.

I think I might understand the interpretation that a halfer puts on the question. I'm just doubtful of its interest or relevance. Do you see any validity (I mean logical coherence, as opposed to wrong-headedness) to this interpretation? Is this just a turf war over who gets to define a coveted word for their purposes?

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?

I don't agree, because the question is about her subjective probability at an awakening. The betting question you described is a different one.

She would also bet at an awakening. If you ask her to bet when she just broke up, it would seem weird that she would say "my subjective probability for heads is 1/2, but I'll only willing to bet up to $20 - 1/3 of the winnings if it's heads."

It seems even weirder in the Xtreme Sleeping Beauty, where she's awakened a thousand times : "my subjective probability for heads is 1/2, but I'm only willing to bet up to 6 cents".

Yes, you get a different result if you change the betting rules where only one bet per "branch" counts, but I don't see why that's closer to the problem as originally stated.

Suppose we write out SB as a world-program:

SleepingBeauty(S(I)) =
{
    coin = rnd({"H","T"})
    S("starting the experiment now")
    if(coin=="H"):
        S("you just woke up")
        S("you just woke up")
    else:
        S("you just woke up")
    S("the experiment's over now")
    return 0
}

This notation is from decision theory; S is sleeping beauty's chosen strategy, a function which takes as arguments all the observations, including memories, which sleeping beauty has access to at that point, and returns the value of any decision SB makes. (In this case, the scenario doesn't actually do anything with SB's answers, so the program ignores them.)

An observer-moment is a complete state of the program at a point where S is executed, including the arguments to S. Now, take all the possible observer-moments, weighted by the expected number of times that a given run of SleepingBeauty contains that observer moment. To condition on some information, take the subset of those observer-moments which match that information. So, P(coin=heads|I="you just woke up") means, of all the calls to S where I="you just woke up", weighted by probability of occurance, what fraction of them are on the heads branch? This is 1/3. On the other hand, P(coin=heads|I="the experiment's over now")=1/2.

Suppose we write out SB as a world-program:

SleepingBeauty(S(I)) =
{
    coin = rnd({"H","T"})
    S("starting the experiment now")
    if(coin=="H"):
        S("you just woke up")
        S("you just woke up")
    else:
        S("you just woke up")
    S("the experiment's over now")
    return 0
}

This notation is from decision theory; S is sleeping beauty's chosen strategy, a function which takes as arguments all the observations, including memories, which sleeping beauty has access to at that point.

An observer-moment is a complete state of the program at a point where S is executed, including the arguments to S. Now, take all the possible observer-moments, weighted by the probability that a given run of SleepingBeauty contains that observer moment. To condition on some information, take the subset of those observer-moments which match that information. So, P(coin=heads|I="you just woke up") means, of all the calls to S where I="you just woke up", weighted by probability of occurance, what fraction of them are on the heads branch? This is 1/3. On the other hand, P(coin=heads|I="the experiment's over now")=1/2.

This appears to be where you are getting confused. Your probability tree in your post was incorrect. It should look like this:

If you think about writing a program to simulate the experiment this should be obvious.

No. I assert P(H|W) = 1/21 in this case.

Two ways of seeing this: Either calculate the expected number of wakings conditional on the coin flip (m/20 and m for H and T). [As in SB]

Alternatively consider this as m copies of the single constant game, with uncertainty on each waking as to which one you're playing. All m single constant games are equally likely, and all have P(H|W) = 1/21. [The hoped for PSB intuition-pump]

No; between sedation and amnesia you know nothing but the fact that you've been woken up, and that 20 runs of this experiment are to be performed.

Why would an earlier independent trial have any impact on you or your credences, when you can neither remember it nor be influenced by it?

As I see it, initially (as a prior, before considering that I've been woken up), both Heads and Tails are equally likely, and it is equally likely to be either day. Since I've been woken up, I know that it's not (Tuesday ∩ Heads), but I gain no further information.

Hence the 3 remaining probabilities are renormalised to 1/3.

Alternatively: I wake up; I know from the setup that I will be in this subjective state once under Heads and twice under Tails, and they are a priori equally likely. I have no data that can distinguish between the three states of identical subjective state, so my posterior is uniform over them.

If she knows it's Tuesday then it's Tails. If she knows it's Monday then she learns nothing of the coin flip. If she knows the flip was Tails then she is indifferent to Monday and Tuesday. 1/3 drops out as the only consistent answer at that point.