Tristan Cook comments on Tristan Cook’s Quick takes

[Tristan Cook]

Anthropics: my understanding/​explanation and takes

In this first comment, I stick with the explanations. In sub-comments, I’ll give my own takes

Setup

We need the following ingredients

• A non-anthropic prior over worlds $W_1,...,W_n$ where $\sum P\left(W_i\right)=1$ ^[1]

• A set $O_i$ of all the observers in $W_i$ for each $i$.

• A subset of observers $Y_i\subseteq O_i$ of observers in each world $W_i$ for each $i$ that contain your exact current observer moment

  • Note it’s possible to have some $Y_i$ empty—worlds in your non-anthropic prior where there are zero instances of your current observer moment

• A reference class (you can choose!) $R=(R_1,...,R_n)$ where $R_i\subseteq O_i$. I call $R_i$ the reference set for world $i$ which is a subset of the observers in that world.

  • Generally one picks a ‘rule’ to generate the $R_i$ systematically

    • For example “human observers” or “human observers not in simulations”.

  • All anthropic theories technically use a reference class (even the self-indication assumption)

  • Generally one chooses the reference class to contain all of the you-observer moments (i.e.$Y_i\subseteq R_i$ for all $i$)

And finally, we need a choice of anthropic theory. On observation $E$ (that you are your exact current observer moment), the anthropic differ by the likelihood $P\left(E|W\right)$.

Self-indication assumption (SIA)

Bostrom gives the definition of something like

All other things equal, one should reason as if they are randomly selected from the set of all possible observer moments [in your reference class].

This can be formalised as

$P_{SIA,R}\left(E|W_i\right)=\frac{|R_i\cap Y_i|}{{\sum }_i|R_i|}\propto |R_i\cap Y_i|$

Note that

• The denominator - which is the total number of observers in all possible worlds that are in our reference class—is independent of $i$, so we can ignore it when doing the update since it cancels out when normalising.

• The standard approach of SIA is to take the reference class equal to be all observers, that is $R_i=O_i$ for all $i$. In this case, the the update is proportional to $|Y_i|$

  • This is true for any reference class with $Y_i\subseteq R_i$ (i.e. our reference class does not exclude any instances of our exact current observer moment).

Heuristically we can describe SIA as updating towards worlds in direct proportion to the number of instances of “you” that there are.

Updating with SIA has the effect of

• Updating towards worlds that are multiverses

• Probably thinking there are many simulated copies of you

• Probably thinking there are aliens (supposing that the process that leads to humans and aliens is the same, we update towards this process being easier).

• (The standard definition) breaking when you have a world with infinite instances of your exact current observer moment.

Self-sampling assumption (SSA) ^[2]

Bostrom again,

All other things equal, one should reason as if they are randomly selected from the set of all actually existent observer moments in their reference class

This can be formalised as

$P_{SSA,R}\left(E|W_i\right)=\frac{|R_i\cap Y_i|}{|R_i|}$

Note that

• The denominator—which is the total number of observers in the reference set in world $i$ - is not independent of i so does not cancel out when normalising our update

• In the case that one takes a reference class to include all exact copies of ones’ current observer moment (i.e. $Y_i\subseteq R_i$), we have the expression simplify to $\frac{Y_i}{R_i}$.^[3]

Heuristically we can describe SSA as updating towards worlds in direct proportion to how relatively common instances of “you” are in the reference class in the world.

Minimal reference class SSA

This is a special case of SSA, where one takes the minimal reference class that contains the exact copies of your exact observer moment. That is, take $R_i=Y_i$ for every $i$. This means for any evidence you receive, ruling out worlds that do not contain a copy of you that has this same evidence.

Note that the formula for this update becomes $|Y_i|/|Y_i|$ - for worlds where there is at least one copy of us, the likelihood is 1 and is otherwise 0.^[4]

Updating with SSA has the effect of

• Updating towards worlds that are small but just contain copies of you (in the reference class)

• For non-minimal reference classes, believing in a ‘Doomsday’ (i.e. updating to worlds with fewer (future) observers in your reference class, since in those worlds your observations are more common in the reference set).

• Having generally weaker updates than SIA—the likelihoods are always in [0,1].

Both SIA and SSA

• Update towards worlds where you are more ‘special’,^[5] for example updating towards “now” being an interesting time for simulations to be run.

• Can be extended to worlds containing infinite observers

• (In my mind) are valid ‘interpretations’ of what we want ‘probability’ to mean, but not how I think we should do things.

• Can be Dutch-booked if paired with the wrong decision theory^[6]

1. ^

   One could also easily write this in a continuous case

2. ^

   Strictly, this is the strong self-sampling assumption in Bostrom’s original terminology (the difference being the use of observer moments, rather than observers)

3. ^

   One may choose not to do this, for example, by excluding simulated copies of oneself or Boltzmann brain copies of oneself.

4. ^

   The formal definition I gave for SSA is only for cases where the reference set is non-empty, so there’s not something weird going on where we’re deciding ⁰⁄₀ to equal 0.

5. ^

   In SIA, being ‘special’ is being common and appearing often. In SSA, being ‘special’ is appearing often relative to other observers

6. ^

   Spoilers: using SIA with a decision theory that supposes you can ‘control’ all instances of you (e.g. evidential like theories, or functional-like theories) is Dutch-bookable. This is also the case for non-minimal reference class SSA with a decision theory that supposes you only control a single instance of you (e.g. causal decision theory).

[Tristan Cook]

How I think we should do anthropics

I think we should reason in terms of decisions and not use anthropic updates or probabilities at all. This is what is argued for in Armstrong’s Anthropic Decision Theory, which itself is a form of updateless decision theory.

In my mind, this resolves a lot of confusion around anthropic problems when they’re reframed as decision problems.

If I had to pick a traditional anthropic theory...

I’d pick, in this order,

1. Minimal reference class SSA

2. SIA

3. Non-minimal reference class SSA

I choose this ordering because both minimal reference class SSA and SIA can give the ‘best’ decisions (ex-ante optimal ones) in anthropic problems,^[1] when paired with the right decision theory.

Minimal reference class SSA needs pairing with an evidential-like decision theory, or one that supposes you are making choices for all your copies. SIA needs pairing with a causal-like decision theory (or one that does not suppose your actions give evidence for, or directly control, the actions of your copies). Since I prefer the former set of decision theories, I prefer minimal reference class SSA to SIA.

Non-minimal reference class SSA, meanwhile, cannot be paired with any (standard) decision theory to get ex-ante optimal decisions in anthropic problems.

For more on this, I highly recommend Oesterheld & Conitzer’s Can de se choice be ex ante reasonable in games of imperfect recall?

1. ^

   For example, the sleeping beauty problem or the absent-minded driver problem