Bentham’s Bulldog Ruins my Weekend
With the Self Indication Assumption
Recently, I had the pleasure of chatting with Bentham’s Bulldog, who has a great Substack you should all visit. During our conversation, we found ourselves debating the Self-Indication Assumption (SIA), a topic that I was introduce to recently and was quite confused about.
I had sometimes seen the SIA presented in the following way:
The Self-Indication Assumption states: “All other things being equal, an observer should reason as if they are randomly selected from the set of all possible observers.”
This framing puzzled me because I thought there were clear counterexamples.
However, during our discussion, Matthew (Bentham’s Bulldog’s real name) explained to me that the SIA is simply an application of Bayesian reasoning. This surprised me—it seems odd to give Bayesian reasoning a separate name—but I now believe I understand what he meant.
Conceptualizing the SIA as Bayesian reasoning done right is a powerful perspective—or at least it helped me fully appreciate the strength of the SIA. The key takeaway for me was that this perspective wants to justify the SIA, showing that it isn’t some arbitrary principle we’re adding into the mix; rather, it represents the proper application of probability theory to specific scenarios. If the SIA is simply reasoning correctly using probability theory, that would be a strong endorsement indeed.
Realizing the full strength of an argument only after a discussion isn’t exactly ideal! But hey, better late than never. However, there’s still a lingering doubt I can’t quite shake when thinking about these issues, one that is better explained in writing.
1. Reasoning According to the SIA
Suppose there is a situation where God flips a fair coin, and based on whether it lands on tails or heads, He decides to create a universe with either 10 people or 100 people. Alternatively, we could frame this scenario as having equal priors for two different population sizes. More formally, consider the two hypotheses:
H1: The universe contains 10 people.
H2: The universe contains 100 people.
Where
We are assigning equal probability to these two hypotheses, but—as we will see—the underlying point of this example remains unchanged as long as we assign any non-zero probability to each of them.
Suppose we then consider whether the evidence of our own existence gives us reason to place greater credence in one hypothesis over the other. In other words we are interested in
where the evidence E is given by our existance.
Now, let’s think about the probability of the evidence—namely, our existance—given the two hypotheses. Considering that there are multiple possible universes where people like us could exist, the SIA suggests we reason as if we are randomly selected from the set of all possible observers. As a result, the probability of our existence becomes proportional to the number of observers in the universe. Mathematically:
Let’s say for simplicity that
Then we have all we need to apply Bayes’ Theorem obtaining
And of course
We notice that by reasoning in this way—given the evidence of our own existence—we update our probability estimates in favor of the more populous universe. This update is proportional to the number of people in that universe.
For instance, if we hold even a very small prior credence that a universe with a googolplex of people exists, this framework implies we should update the likelihood of that universe being the one we exist in by a factor of a googolplex. As Matthew was delicately trying to explain during our conversation, the specific priors we assign to these hypotheses ultimately don’t matter much (as long as they are non-zero) because we can arbitrarily increase the likelihood of a hypothesis simply by increasing the number of people it posits.
In this framework we have treated our existence as a random sampling process from the set of all possible observers across all possible universes. However, it seems to me that there’s an alternative way to proceed.
2. The Naive Statistician Objection
When we arrive at the point of considering the probability of our existence given the two hypotheses, one might say:
“Hey! I exist. I know I exist—after all, I am experiencing my existence right now.”
No matter the population of the two universes, I exist. Therefore, the probability of my existence, in either of the two universes, is 1. It must be 1 because I am now currently existing. So,
This, to me, seems like a very natural way to proceed. And under this reasoning, of course, Bayes’s rule gives us no update.
Shocking.
3. What Now?
The question now becomes which is the correct way of proceeding.
Initially, I thought the tension arises when one attempts to conduct a sort of meta-analysis from an outside perspective: “Given that some observer with my exact memories, experiences, etc. exists, is that more or less likely under the big-universe or small-universe hypotheses?” This line of reasoning might lead one to weigh the hypotheses differently, favoring the one with more observers as the SIA suggests.
On the other hand, from a first-person perspective, the probability of the evidence that I exist—not as seen from a third-person view but as a direct fact of my own experience—is 1 in either universe.
As a result, I concluded that both approaches are valid, but they address different questions (I’ll let you guess which one I thought was more pertinent when pondering the existence of God).
At this point, I was ready to give myself a pat on the back and get back to doing actual work, confident that I had untangled one of those “hard” problems philosophers spend their lives muddling through.
But then a lingering suspicion crept in: my line of thinking seemed to suggest (though I’m not entirely sure) that I was a halfer in the Sleeping Beauty problem. So, for about a day, I happily embraced my newfound identity as a halfer in the Sleeping Beauty problem, feeling quite content about it until…I read another post by Matthew.
Now, I’m once again extremely confused and have decided I never want to talk about these matters again.
I had the same naive statician intuition, but i had no way to put it into words. nice post
Nice essay! If you want some reasons to prefer the SIA approach over the one where you don't update in favor of theories that predict more people exist, I give some in section 3 here https://benthams.substack.com/p/the-ultimate-guide-to-the-anthropic. In short, alternatives imply utter absurdity.