Simplifying the anthropic impossibility result

[Stuart_Armstrong]

So. I previously demonstrated an anthropic impossibility theorem, showing that in Duplicates Sleeping Beauty, there was no possible probability theory that obeyed both the martingale condition and “simple Bayes” in non-anthropic situations.

This post will clarify and simplify the result, replacing the martingale with the law of total probability. Let’s start by fully defining both Simple Bayes and what a non-anthropic situation is.

Duplicates Sleeping Beauty. She is put to sleep on Sunday, and a coin is tossed; if Tails, she is duplicated into each room before being awoken on Monday. If Heads, she is awoken in Room 1. On Tuesday, she observes her room number.

Definitions

I’ll present the definition, illustrating with Duplicates Sleeping Beauty in quoted blocks.

Let’s start with our list of worlds and our prior over them; we’ll assume that every world is at least theoretically possible, ; if they aren’t, just removed them from the list. Then an agent who has seen history is in a non-anthropic situation if every world has at most one possible agent with history . Thus the agent may not know which world they are in, but they do know where they are within each world.

In Duplicate Sleeping Beauty, the worlds are , the heads world and the tails world, with the prior being .

She is non-anthropic on seeing the history - pre-duplication, so only one agent present in each world.

She is also non-anthropic on both Tuesday histories:

• (one possible agent with that history in , and one possible agent with that history in )

• (one possible agent with that history

These non-anthropic situations are outlined in green in the image below.

Simple Bayes is the condition that, if an agent finds themselves with a history in a non-anthropic situation, then they should update in the usual Bayesian way: world goes from to , and then the weighted sum is renormalised. Note that comes from the definition of the world itself.

In Duplicate Sleeping Beauty, both worlds are deterministic, so is either or . Between the three non-anthropic histories, , , and the two worlds, and , only one combination is impossible: . The others have .

For , the updates for the probabilities of worlds to goes as ; no need to even renormalise. Since it has the same conditional probabilities, the updates of world are the same. For , the updates go after renormalisation.

These posterior probabilities are given in green below.

The inconsistency

I’ll now show that there is no probability distribution for Sleeping Beauty that she can use to predict her observations and the probabilities of the worlds, that can obey the following conditions:

1. Simple Bayes in non-anthropic situations.

2. Doesn’t give to a unless is impossible.

3. If and are exhaustive and mutually exclusive, then and .

I’m not going to add assumptions about , instead just defining it to be .

The last degree of freedom

In fact, simple Bayes and the prior will fix almost entirely; the only real degree of freedom left will be determining (or ) which isn’t ultimately enough.

The problem description imposes (since room 2 is not occupied in the Heads world).

Simple Bayes on Sunday imposes that must reproduce the prior on the worlds themselves - , since is fully exhaustive itself, so implies implies ; and the same for .

Then simple Bayes on Tuesday gives . Hence . Similarly, so . I’m now going to make two key assumptions from the problem setup:

1. Since every history ends with Sleeping Beauty observing her room, and are exhaustive.

2. Since every Sleeping Beauty will only observe her own room, and are mutually exclusive^[1].

Anyway, the mutually exclusive and exhaustive condition gives . Similarly and must be mutually exclusive, and since and are exhaustive, . Inserting the values computed above gives , or, upon multiplying by :

• .

This implies , which is impossible since is possible and hence .

Where the different theories ere

So, where do the different anthropic theories go wrong?

• SIA breaks the law of total probability^[2] and the martingale from Sunday to Monday.

• SSA typically (SSA is multipley defined) breaks simple Bayes on Tuesday (it is fully martingale consistent).

• SIA over full future histories breaks simple Bayes on Sunday, giving probability to Heads for a fair coin that is not yet tossed (thus breaking the Principal Principle). Unlike standard SIA, it is fully martingale consistent.

In a sense, SSA and SIA over full future histories are similar; SSA takes the initial simple Bayes probabilities as correct, and pushes them forwards consistently (diverging from simple Bayes on the final probabilities); SIA over full future histories takes the final simple Bayes probabilities as correct, and pulls them back consistently (diverging from simple Bayes on the initial probabilities).

Standard SIA takes both the initial and the final simple Bayes probabilities as correct, and so can’t be consistent between them^[3].

1. ^

   It’s the third one that separates Sleeping Beauty’s internal probabilities from an observer looking in from the outside. For these probabilities are not hard to estimate from the outside: and . Here is estimating the probability that the observation exists, while is Sleeping Beauty trying to estimate the probability that she observes that history.

2. ^

   The real result is that any anthropic probability distribution used by Sleeping Beauty to predict observations must break in some way. The law of total probability is currently the simplest property of probability that I can show must break (given simple Bayes); there are probably simpler ones.

3. ^

   I noted that inserting inert duplications into the problem description—duplicates that are created in the heads world and immediately deleted—allowed probability theories to get around the restrictions here. This “works” (in as much as redefining the problem to be a different problem can be considered to work) because they are not attacking the mutual exclusivity of and , but the exhaustiveness.

   The duplicate created and destroyed in the heads world will not see any more history that , the original history on Sunday. That means that , with some of the probability going to the dead duplicate. Then, the only reasonable probability assignments are , , and the probability of being a dead duplicate is .

[jessicata]

SIA breaks inclusion-exclusion

Skeptical; couldn’t we have SSSA+SIA which assigns:

• ¹⁄₈ to Heads, Sunday

• ¹⁄₈ to Tails, Sunday

• ¹⁄₈ to Heads, Monday, Room 1

• ¹⁄₈ to Heads, Tuesday, Room 1

• ¹⁄₈ to Tails, Monday, Room 1

• ¹⁄₈ to Tails, Tuesday, Room 1

• ¹⁄₈ to Tails, Monday, Room 2

• ¹⁄₈ to Tails, Tuesday, Room 2

This is a distribution over centered worlds. Since it is a probability distribution it doesn’t break inclusion-exclusion.

You might be thinking “SIA but not SSSA” in which case I’m not sure how you’re computing the probabilities and how you’re getting that they don’t obey inclusion/​exclusion?

[Stuart_Armstrong]

What does this give as P(Monday|Sunday)? Seems it has to be zero.

Now obviously P(Monday|Sunday)=0 in the sense that “If it’s Sunday, it’s not Monday”, but how does SSSA+SIA encode “If it’s Sunday, what’s the probability that it will be Monday?” or “If I’ve observed Sunday, what’s the probability I will observe Monday?” It feels that it’s given up causality.

Unless I’ve made a mistake (very possible), my result shows that if SSSA+SIA can be turned into a personal probability distribution for Sleeping Beauty to use for predicting her experiences as well as world-facts, it will then break ^[1] in that form.

EDIT: I’ve replaced inclusion-exclusion with the law of total probability and simplified the proof further.

1. ↩︎

   The fundamental reason it will break is that h_1 and h_2 are mutually exclusive and exhaustive full histories for Sleeping Beauty, but any sensible probability system wants to put P(h_1)=1 and P(h_2)=1/​2 (which are the correct probabilities for the existence of these histories, since they are not mutually exclusive from the outside).

   The system has too few degrees of freedom; you can’t take the outside-observer probability distribution (which is what simple Bayes forces on Sunday and Tuesday), substitute in mutual exclusivity, and not have something break. Inclusion-exclusion is just the simplest thing that shows the break, currently.

[jessicata]

Yes P(Monday|Sunday)=0, that’s straightforward from assigning probabilities to centered worlds.

How does SSSA+SIA encode “If it’s Sunday, what’s the probability that it will be Monday?”

Given a centered world it is easy to tell whether it will be Sunday in an objective sense (of future light-cone of the present moment containing Monday) but that’s probably not what you’re asking about.

“If I’ve observed Sunday, what’s the probability I will observe Monday?”

Baseline for physicalists should be either “100%” (because some duplicate will observe Monday, and “I” spans all of them) or “NaN” (because once you know your centered world, there is no more to know).

People manifestly disagree on what probabilities to assign to centered worlds, so it’s not like “probabilities for future experience” are required for making non-trivial anthropic claims.

Re: causality, I refer to the physicalist baseline, which adds no graph edge from Tails + Sunday Beauty to either “Tails + Monday + Room 1” Beauty or “Tails + Tuesday + Room 2″ Beauty. There is no argument-from-causality for adding such an edge to the physicalist baseline.

my result shows that if SSSA+SIA can be turned into a personal probability distribution for Sleeping Beauty to use for predicting her experiences as well as world-facts, it will then break in that form.

Here’s an attempt. It seems what you want is a distribution over personal histories. We can filter histories for some sort of finality (death, or end of experiment). Here the final histories are . SSSA+SIA over centered worlds conditional on finality says . And .

Your proof uses however SSSA+SIA conditional on finality gives . If we consider that assigns probabilities over centered worlds, it is not ridiculous to assign or . (SIA in general has a prior over centered worlds that differs from the baseline “distribution over universes” that is motivated by physics.)

To be clear I don’t think this “conditioning on finality” is a good way to assign probabilities (I assume it breaks in other cases) however it might provide a counterexample to the claim there is no way to assign personal probabilities compatible with SSSA+SIA in this example.

SIA breaks inclusion-exclusion

You are still making this claim, I don’t know what you think SIA is as an algorithm, what probabilities you think it applies here, and why you think it breaks inclusion /​ exclusion.

[Stuart_Armstrong]

That design is SIA over full histories (see the end of the post), which breaks simple Bayes on Sunday.

The simplest version of SIA (C-SIA, counting SIA) upweighs worlds by the number of copies of you there is in them at any one time and is otherwise fully Bayesian. This is consistent with deaths, divergence of copies, and the initial creation of you(s). The impossibility result shows that it isn’t consistent with duplication, though.

A potential criticism of SSSA+SIA: if it doesn’t intrinsically assign subjective probabilities to agents inside the problem, what is it for? These problems aren’t difficult to analyse probabilistically from the outside; it’s from the inside that we need an effective tool.

[jessicata]

Ah I agree this breaks simple Bayes on Sunday, and is not really compatible with SSSA+SIA.

I’m still not sure how you’re getting the SSA /​ SIA asymmetry on inclusion /​ exclusion. It seems you could take the SSA probabilities and re-scale them by population (maybe here it should be population of Beauty-days?). In which case the result would still be probabilities so would be compatible with inclusion /​ exclusion.

A potential criticism of SSSA+SIA: if it doesn’t intrinsically assign subjective probabilities to agents inside the problem, what is it for?

It assign probabilities for the agents inside to know their centered world (that is, all physical facts plus “what is here + now”), that’s enough for things like “estimating size of universe” or “figuring out if today is Tuesday” or “figuring out if coin is heads”, which people disagree on.

[Stuart_Armstrong]

SIA is well defined, so you can calculate with it, and notice the probability of Heads go from ¹⁄₂ on Sunday to ¹⁄₃ on Monday to 0 or ¹⁄₂ on Tuesday. The Monday → Tuesday transition is consistent with probability rules, the Sunday → Monday is not. Inclusion-exclusion (now replaced with the law of total probability) is one of the easiest ways to pinpoint the inconsistency.

SSA has multiple definitions, but in it’s “halfer” format, with reference class being Sleeping Beauty copies currently in that universe, it assigns ¹⁄₂ to Heads on Sunday, ¹⁄₂ on Monday, and ²⁄₃ or 0 on Tuesday. It is Bayes consistent, but violates simple Bayes on Tuesday. Other SSAs may behave differently.

It assign probabilities for the agents inside to know their centered world (that is, all physical facts plus “what is here + now”), that’s enough for things like “estimating size of universe” or “figuring out if today is Tuesday” or “figuring out if coin is heads”, which people disagree on.

I feel that it probably has timeline inconsistencies, such as assigning a different size to the universe at different moments, without any observations to update on. The source of this feeling is that if it were consistent on timelines, then it would assign a reasonable internal agent distribution to future observations.

After all, if you’ve assigned probabilities to the day, the coin, how many agents there are, and the room, this should be now constraining your expected observations.

[jessicata]

I agree SIA is non-martingale (common thirder property). I think one thing you are saying is that if we agree with “simple Bayes” we can’t have probabilities over personal histories, and a big motivation for SIA is simple Bayes, so SIA-ers will tend to not assign consistent probabilities over personal histories; I’d agree there.

I feel that it probably has timeline inconsistencies, such as assigning a different size to the universe at different moments, without any observations to update on.

This is standard thirding. In standard Sleeping Beauty, between Sunday → Monday, Beauty gets no non-trivial observations. Yet her epistemic state changes, in the following sense: Usually, you should move your ‘current time’ beliefs forward according to a clock (assuming you have a good internal clock). However, from Sunday → Monday, Beauty should assign some probability to “it is Tuesday” which is not updating according to an internal clock. (Actually, halfers agree here, that upon awakening, she should consider it plausible that it’s Tuesday.). Thirders correspondingly assign more probability to Tails in a non-Martingale fashion (and this is not so counter-intuitive if you account for the internal clock issue). This is a probability for an objective event that happened in the past predictably changing. It violates Martingale, yet Dutch book arguments suggest that violating Martingale here is necessary to avoid sure losses (assuming CDT, which is the common background Dutch book assumption).

Analogous case for universe size: Number of times Beauty wakes up depends on how scientists measure size of universe on Monday (on Sunday no one knows what the size is).

[Stuart_Armstrong]

That’s standard Sleeping Beauty; I crafted duplicate Sleeping Beauty so that there was no memory-loss issue. Sleeping Beauty always knows what day it is.

Are you saying that SIA+SSSA suffers from these problems?

[jessicata]

SSSA+SIA does not assign consistent probabilities over full personal histories. (It assigns probabilites over centered worlds and some amount of personal history probability can be derived from that, yet won’t satisfy assumptions of OP for personal probabilities, e.g. won’t consider to be exhaustive.)

In standard Sleeping Beauty there is no memory loss from Sunday → Monday. This is part of David Lewis’s argument for single-halfing (using Martingale assumption). So violating Martingale even when your mind hasn’t been messed with in the intervening time is standard thirding.

With duplicate Sleeping Beauty we have analogous Dutch book arguments for thirding. If you assign to Heads on Sunday then you must assign to Heads on Monday if you are a CDT to avoid sure losses. Sure loss for ¹⁄₂ Heads on Sunday, ¹⁄₂ heads on Monday is: on Sunday, the bookie, who has no info advantage on Beauty, offers “-$6 if Heads, +$7 if Tails”, which Beauty accepts; then on Monday the bookie, who has also been split so he doesn’t know which room it is, offers “+$5 if Heads, -$4 if Tails”; the total shared account has -$1 if Heads, -$1 if Tails, since the Monday bet happens twice if Tails.

(One can appeal to utility function re: shared accounts, however, if it’s a shared account for an altruistic fund, which all copies of Beauty care about equally, we still have a problem)

Someone can hold “I don’t buy Dutch book arguments at all, I think Martingale can be supported on other assumptions”, however historically, diachronic Dutch book arguments have been a significant argument for Martingale.

[Stuart_Armstrong]

Yep!

I suppose it’s back to my old conclusion—decision theory is more fundamental that probability theory. https://​​arxiv.org/​​abs/​​1110.6437

[jessicata]

Just finished up a post on probabilities over centered worlds.

[JBlack]

This impossibility appears to hold in non-anthropic situations as well.

For example, in the “Tails False Memory” scenario I commented to earlier posts: a coin is flipped and on Tails, Beauty is given a false memory of being asked “what is your credence for heads” and having answered. Then unconditionally Beauty is asked “are you sure”. Beauty knows these rules.

In both possible worlds at every point in time, there is only one agent and therefore at most one agent with any given history. So this is a non-anthropic situation at all times by the given definition.

Let the relevant histories be h1 = “I remember being asked the first question only and not having answered yet”, and h2 = “I remember being asked both questions, having answered the first”. Let the worlds be wT = “the coin comes up Tails” and wH = “the coin comes up heads”.

Beauty knows that (wT, h1) is impossible, and so Q(wT | h1) = 0 and therefore Q(wH | h1) = 1. The subsequent state (wH, h2) is a non-anthropic situation and so Beauty must use the Simple Bayes rule to update, so Q(wH | h2) = 1 also. But Q(wH | h2) + Q(wT | h2) = 1, so Q(wT | h2) = 0 which requires that Q(wT, h2) = 0.

But (wT, h2) is a possible state and must not be assigned zero credence, concluding the impossibility proof.

[Stuart_Armstrong]

Yep, seems right.

Personally, I consider false memory to be a form of anthropic scenarios—you don’t know exactly who you are.