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 inclusion-exclusion principle for probability and 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

Instead of using the martingale, lets use the simpler inclusion-exclusion principle: that for any events and , . It's obvious from Venn Diagrams, and its harder to think of a simpler principle than that.

We'll also use the definition of conditional probability, namely that . The inclusion-exclusion principle will also give us that, if and are exhaustive and mutually exclusive, .

Chasing down the inconsistency

So assume for the moment that there was an anthropic probability distribution for Sleeping Beauty - thus means the probability for the agent of observing history . And assume that obeyed Simple Bayes in non-anthropic situations and the inclusion-exclusion principle (and the definition of conditional probability). We're going to show that this gives a contradiction; this part of the proof is not that illuminating, it's just a question of chasing down probabilities, term after term, until we find an issue.

For , the definition of conditional probability gives . But simple Bayes has already given us the first factor, , so this becomes .

For , .

We can apply inclusion-exclusion to and . Now . Since one of the histories or must happen on Tuesday, they are exhaustive, so can be dropped: .

Similarly, . For any Sleeping Beauty herself, these histories are mutually exclusive: only one can happen. So .

Then inclusion-exclusion gives , which is . We already know , so .

Then add the extra assumption that the possible history must have so mutual exclusivity^[1] of and gives a contradiction: .

Where the different theories ere

So, where do the different anthropic theories go wrong?

• SIA breaks inclusion-exclusion 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 (going wrong on the final probabilities); SIA over full future histories takes the final simple Bayes probabilities as correct, and pulls them back consistently (going wrong on the initial probabilities).

Standard SIA takes both the initial and the final probabilities as correct, and so can't be consistent between them.

1. ^{^}

   Notice that an outside observer can have a well-functioning over this situation, just not Sleeping Beauty herself. The outside observer can set perfectly consistent probabilities and , and never encounter a contradiction; that's because observing in a world, for an outsider, is not mutually exclusive with observing (they both happen in world ).

   From a given Sleeping Beauty's perspective, they are mutually exclusive, so she has to deal with the "excess" probability in . SIA deals with it by renormalising the whole sum; SSA deals with it by renormalising the part of the sum in world only.

   This illustrates why, fundamentally, there has to be a problem with a for Sleeping Beauty herself, and that we will find the problem if we search long enough. Because simple Bayes means that, on Sunday and Tuesday, the probabilities for Sleeping Beauty observing correspond to those for the outside observer knowing that that was observed. But the outside observer has a perfectly consistent over the whole of the experiment. Sleeping Beauty cannot take a consistent , add "also, and are now mutually exclusive", and expect the result to remain consistent.