Work in progress... any new arguments appreciated.

Comparing cosmological theories

To make progress in cosmology we really need to understand the fundamental issue of how to compare different theories and work out the posterior credences of different models in the light of what we know and observe.

Questions we would like to be able to answer:

Are theories with more observers favored? [SSA-A, SSA-C versus SIA-U below]
Are outcomes of correct theories giving more observers favored? [SSA-A versus SIA-C below]
Are theories that give no observers with high probability disfavored? [Yes, unless SIA-U]
How do you compare multiverse and universe theories?
Does the interpretation of QM matter?
Is the Doomsday argument correct? If not, why? [No - SIA or restricted reference class, or both]
When we talk about observers do we mean civilizations, individual observers, possible observers, observer-moments, etc?
What about if one (or more) theories contains an infinite number of observers?

Reasoning principles

SSA : Self Sampling Assumption (Nick Bostrom - anthropic-principle.com)

"Every observer should reason as if she were a random sample from the set of all observers in their reference class."

There are I think basically four ways of applying SSA:

SSA-A : Actual observers

"SSA should be applied only to observers in the reference class that actually exist". All that matters when comparing theories is if one or more observers exist (assuming you are a typical observer).

Proponents: Nick Bostrom, John Leslie (but see -MW below)

SIA-C : Self Indicating Assumption (Chance version)

"SSA should be applied to all observers in the reference class that are possible, but do not necessarily actually exist". You are more likely to exist in universe with lots of observers, so your existence favors a realization of the universe with lots of observers.

Proponent: AL + (probably) others

SSA-MW : SSA-A but Many Worlds observers are treated as actual

"SIA-C applied only if the Many Worlds interpretation is correct, otherwise SSA-A"

Proponent: Nick Bostrom, John Leslie

SIA-U : Self Indicating Assumption (Uncertainty version)

"SSA should be applied to all observers that we think might exist or could have existed, but do not necessarily actually exist." Subjective uncertainties about which theories are correct are treated in the same way as probabilistic outcomes of a correct theory.

Proponent: Ken Olum [though most of his paper is really arguing for SIA-C]

One could possibly split SIA-U into two positions depending on whether the uncertainties are about whether the universe is logically possible (e.g. depends critically on whether the 10^500th digit in the decimal expansion of pi is 0, whether the twin primes conjecture is true, etc), or physically possible (e.g. depends on M-theory being correct). [Ken Olum thinks dependence on the value of a digit of pi is equivalent to tossing a coin.] Tegmark might say that all logically possible universes actually exist, in which case SSA-A is equivalent to the latter case. Possible fusion arguments are

Diplomatic: The objective chance is given by SSA-A, the subjective credence by SIA-C. Both are different and it depends what we mean
Meta-Bayesian: We don't know which method of reasoning is correct, so apply meta-parameters to the reasoning methods depending on the credence of the method being correct
Many Worlds is true: SSA-MW and SSA-C then agree. As long as SIA-U is false and one rules out SSA-A being applied only within in the universe we observe (both sensible), we are then consistent.

One could perhaps argue that we shouldn't have fundamental ambiguities in our reasoning, and hence Many Worlds must be true.

Are theories with more observers favored?

Some useful though experiments are described in Ken Olum's paper and at anthropic-principle.com. Two arguments in favor of SIA in Olum's paper apply differently to SIA-U and SIA-C. Repeatability: if you make many universes the frequentist probability agrees with the result of SIA-C, but not with SIA-U. Interpretation of quantum mechanics: SIA-C considers only possibilities that would exist if M-W were true; SIA-U also considers possibilities that would not be in the M-W superposition.

The main argument against SIA seems to be Bostrom's "Presumptuous Philosopher": SIA favors universes with more observers, and hence any theory (possibly preposterous [with a very low prior]) with a large enough number of observers is very likely to be true compared to a theory with far fewer observers. With SIA-C this argument only applies within different probabilistic outcomes of a correct theory. With SIA-U even wrong theories with a larger number of observers will be favored.

I propose an additional thought experiment to clarify the distinction between SIA-U and SIA-C.

God's coin toss: Chance outcomes versus subjective uncertainty

We know that God makes universes in accordance with the rule

if R_1 make 10 observers
if R_2 make 1000 observers
if R_3 make 10^10^100 observers

R_1,R_2,R_3 are one of H,T or X, but we don't know which. So our
subjective uncertainties are P(R_1=H)=1/3, P(R_1=X)=1/3, P(R_2=H)=1/3,
etc.

God tosses a coin, which comes out H or T. X is impossible. We find
ourselves observing and wondering about the number of observers. What is
the probability of there being 10^10^100 observers?

Answers (very nearly exactly):

SSA-A: 1/3
SIA-C: 2/3
SIA-U: 1

SSA-MW always gives the answer for SSA-A if M-W is false, SIA-C is M-W is true. To be more careful, replace "God tosses a coin" by "God is a non-reference class observer who makes an observation of a coin in the superposition [H+T]/sqrt(2)". The subjective uncertainties could be about logical truths (e.g. we know that R_3=X iff the 10^500th digit in the ternary expansion of pi is a zero), or they could be physical uncertainties (e.g. about the correct GUT if there are several logical possibilities).

Weighing the arguments

As I understand it, the challenges to each model are

SSA-A
The frequentist argument about repeatability has to be wrong, so this is some sort of ultra-subjectivist Bayesianism. If you are a thirder on the sleeping beauty problem what exactly is the difference? If you do not subscribe to SSA-MW, why not? In assessing the prior God didn't yet know that you exist, but on making the argument you do, so the information in the prior is not the same as the information when you observe. Outcomes in which observers are common are not favored relative to outcomes where there is just one.
SIA-C
If you don't exist you can't sample yourself, so possible observers are not equivalent to actual observers {at least if M-W is false). You may be unhappy about having large probabilities to be in an outcome which was relatively unlikely but has a large number of observers [but this isn't really an argument]. Theories in which observers are common are not favored relative to theories where there is just one.
SIA-U
It is impossible to rule out wrong theories that predict enormous number of observers. Repeatability argument fails [universes with lots of observers predicted by a wrong theory are never instantiated]. Well know importance of counter-factual possibilities [e.g. bomb-testing], but subjective uncertainty about impossible theories is quite different.

In all cases the infinite case can be difficult.

Are theories that give no observers with high probability disfavored?

Consider three theories:

A: 99% chance of no observers, 1% chance of 1 observer
B: 99% chance of no observers, 1% chance of 1000 observers
C: Certainty of 1 observer

Other things being equal which is more likely? I think all can agree C has 100 times higher credence than A. With SSA-A and SIA-C presumably A and B are equally likely. With SIA-U B has 1000 times higher credence that A, and also 10 times higher credence than C.

Anthropic Conditioning

If you have a theory, how do you compare predictions with what we observe? Neil suggests starting with something very simple (e.g. carbon must form), gradually becoming more specific until you get agreement. The less you need to condition the greater the credence of the theory. This is consistent with SSA - if, after conditioning on carbon, everything agrees with observations then all other carbon-based life forms would get similar observations and we are a typical observer. On the other hand if we have to condition on something more specific (say formation of high-water planets), the subset of observers of which we are typical is smaller and the theory is disfavored relative to a theory which would give our observations for all observers [if there (possibly?) exist observers that didn't evolve on high-water planets].

Random thoughts

Consider the case where you are told that there are two GUTs, A and B, and that A is true only iff theorem T is true (the truth of T has prior credence 1/2 - for example T could be the statement that the 10^100th digit in the decimal expansion of pi is ). Does your credence of T depend on the number of observers predicted by A and B? It should it be possible to change your credence of T by observation, when A or B correlates with something observable, eg. in the case where one of A or B does not predict any observers one can be ruled out immediately. But if SIA-C is correct one's credence should be independent of the number of observers, as long as the number is greater than zero. Perhaps the existence of oneself as an observer is really a physical observation, and not strictly related to anthropic reasoning - an impirical measurement of which theory is correct, and hence the truth of T. The observation that I exist contains a great deal of empirical information about the composition and history of the universe. The observation that a plant exists would give much the same information. What we are interested in is the effect of the possible existence of other observers that we do not observe on our beliefs.

SSA-A does not agree with repeatability arguments, nor does SIA-U. For things to be repeatable they have to be possible.

The doomsday argument is explictly concerned with possible futures, and hence SIA-C negates it exactly with an arbitarily large definition of the reference class, without the need for SIA-U.

If A predicts that 99% of its observers observe O, but B predicts only 50%,given that we observe O how does this affect our credence of T? Just apply Bayes, P(T|O) = P(O|T)P(T)/P(O), so P(T|O)/P(~T|O) = 0.99/0.5, so P(T|O) = 2/3 (approx). If we observe ~O, then P(T|~O)=0.02 (approx). Now if B predicts 100 times more observers than A, what do we think?

Antony Lewis Sept 2001.