Iterated Sleeping Beauty and Copied Minds

by lucidfox4 min read21st Dec 201034 comments


Sleeping Beauty Paradox
Personal Blog

Before I move on to a summation post listing the various raised thought experiments and paradoxes related to mind copying, I would like to cast attention to a particular moment regarding the notion of "subjective probability".

In my earlier discussion post on the subjective experience of a forked person, I compared the scenario where one copy is awakened in the future to the Sleeping Beauty thought experiment. And really, it describes any such process, because there will inevitably be a time gap, however short, between the time of fork and the copy's subjective awakening: no copy mechanism can be instant.

In the traditional Sleeping Beauty scenario, there are two parties: Beauty and the Experimenter. The Experimenter has access to a sleep-inducing drug that also resets Beauty's memory to the state at t=0. Suppose Beauty is put to sleep at t=0, and then a fair coin is tossed. If the coin comes heads, Beauty is woken up at t=1, permanently. If the coin comes tails, Beauty is woken up at t=1, questioned, memory-wiped, and then woken up again at t=2, this time permanently.

In this experiment, intuitively, Beauty's subjective anticipation of the coin coming tails, without access to any information other than the conditions of the experiment, should be 2/3. I won't be arguing here whether this particular answer is right or wrong: the discussion has been raised many times before, and on Less Wrong as well. I'd like to point out one property of the experiment that differentiates it from other probability-related tasks: erasure of information, which renders the whole experiment a non-experiment.

In Bayesian theory, the (prior) probability of an outcome is the measure of our anticipation of it to the best of our knowledge. Bayesians think of experiments as a way to get new information, and update their probabilities based on the information gained. However, in the Sleeping Beauty experiment, Beauty gains no new information from waking up at any time, in any outcome. She has the exact same mind-state at any point of awakening that she had at t=0, and is for all intents and purposes the exact same person at any such point. As such, we can ask Beauty, "If we perform the experiment, what is your anticipation of waking up in the branch where the coin landed tails?", and she can give the same answer without actually performing the experiment.

So how does it map to the mind-copying problem? In a very straightforward way.

Let's modify the experiment this way: at t=0, Beauty's state is backed up. Let's suppose that she is then allowed to live her normal life, but the time-slices are large enough that she dies within the course of a single round. (Say, she has a normal human lifespan and the time between successive iterations is 200 years.) However, at t=1, a copy of Beauty is created in the state at which the original was at t=0, a coin is tossed, and if and only if it comes tails, another copy is created at t=2.

If Beauty knows the condition of this experiment, no matter what answer she would give in the classic formulation of the problem, I don't expect it to change here. The two formulations are, as far as I can see, equivalent.

However, in both cases, from the Experimenter's point of view, the branching points are independent events, which allows us to construct scenarios that question the straightforward interpretation of "subjective probability". And for this, I refer to the last experiment in my earlier post.

Imagine you have an indestructible machine that restores one copy of you from backup every 200 years. In this scenario, it seems you should anticipate waking up with equal probability between now and the end of time. But it's inconsistent with the formulation of probability for discrete outcomes: we end up with a diverging series, and as the length of the experiment approaches infinity (ignoring real-world cosmology for the moment), the subjective probability of every individual outcome (finding yourself at t=1, finding yourself at t=2, etc.) approaches 0. The equivalent classic formulation is a setup where the Experimenter is programmed to wake Beauty after every time-slice and unconditionally put her back to sleep.

This is not the only possible "diverging Sleeping Beauty" problem. Suppose that at t=1, Beauty is put back to sleep with probability 1/2 (like in the classic experiment), at t=2 she is put back to sleep with probability 1/3, then 1/4, and so on. In this case, while it seems almost certain that she will eventually wake up permanently (in the same sense that it is "almost certain" that a fair random number generator will eventually output any given value), the expected value is still infinite.

In the case of a converging series of probabilities of remaining asleep - for example, if it's decided by a coin toss at each iteration whether Beauty is put back to sleep, in which case the series is 1/2 + 1/4 + 1/8 + ... = 1 -- Beauty can give a subjective expected value, or the average time at which she expects to be woken up permanently.

In a general case, let Ei be the event "the experiment continues at stage i" (that is, Beauty is not permanently awakened at stage i, or in the alternate formulation, more copies are created beyond that point). Then if we extrapolate the notion of "subjective probability" that leads us to the answer 2/3 in the classic formulation, then the definition is meaningful if and only if the series of objective probabilitiesi=1..∞ P(Ei) converges -- it doesn't have to converge to 1, we'll just need to renormalize the calculations otherwise. Which, given that the randomizing events are independent, simply doesn't have to happen.

Even if we reformulate the experiment in terms of decision theory, it's not clear how it will help us. If the bet is "win 1 utilon if you get your iteration number right", the probability of winning it in a divergent case is 0 at any given iteration. And yet, if all cases are perfectly symmetric information-wise so that you make the same decision over and over again, you'll eventually get the answer right, with exactly one of you winning the bet, even no matter what your "decision function" is - even if it's simply something like "return 42;". Even a stopped clock is right sometimes, in this case once.

It would be tempting, seeing this, to discard the notion of "subjective anticipation" altogether as ill-defined. But that seems to me like tossing out the Born probabilities just because we go from Copenhagen to MWI. If I'm forked, I expect to continue my experience as either the original or the copy with a probability of 1/2 -- whatever that means. If I'm asked to participate in the classic Sleeping Beauty experiment, and to observe the once-flipped coin at every point I wake up, I will expect to see tails with a probability of 2/3 -- again, whatever that means.

The situations described here have a very specific set of conditions. We're dealing with complete information erasure, which prevents any kind of Bayesian update and in fact makes the situation completely symmetric from the decision agent's perspective. We're also dealing with an anticipation all the way into infinity, which cannot occur in practice due to the finite lifespan of the universe. And yet, I'm not sure what to do with the apparent need to update my anticipations for times arbitrarily far into the future, for an arbitrarily large number of copies, for outcomes with an arbitrarily high degree of causal removal from my current state, which may fail to occur, before the sequence of events that can lead to them is even put into motion.