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.

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 .