Full Non-Indexical Conditioning Also Assumes A Self Sampling Process.

In a previous post, I argued for the distinction between the first-person and physical person. I purpose the root cause of paradoxes in anthropics is considering it as an Observation Selection Effect. I.E. treating the first-person perspective as a sample. SSA and SIA disagree on the correct selection process. But in my opinion, indexicals such as “I” and “now” shouldn’t be treated as a random sample in the first place.

I didn’t touch on another camp, Full Non-Indexical Conditioning (FNC). Unlike SSA and SIA, FNC does not state anything in the like of “treat oneself as randomly selected from such and such”. Nevertheless, I am against FNC for the exact same reason. Because in my opinion, FNC still implicitly assumes a particular sampling process and considers indexicals the outcome of said process.

A Simplified Use of FNC

For the Sleeping Beauty Problem, FNC’s approach is as follows. Once waking up in the experiment, there is no need to assume anything like “today is randomly selected from all days/all awakened days.” Just make observations. For example, say I find the weather is cloudy. Then without using any indexicals, it can be stated “Beauty is awake on a cloudy day”.

Now, “Beauty is awake on a cloudy day” is more likely to be true if there are two awakenings instead of just one. Given two awakenings (tails), as long as there is one cloudy day during the experiment the statement is true. If there is only one awakening (heads) then it has to be cloudy on that exact day. So this seemingly independent information about the weather would bump up the probability of Tails.

That alone would not make the Tails twice likely as Heads. Because there is a chance the two days during the experiment are both cloudy. In which case the observation would not favour either head or tail. However, if I consider ALL my experiences after waking up (the feeling of airflow on the skin, the sequence of things I see, etc) then it is practically impossible for the two awakenings to be identical. So the probability of Tails should be nearly (equals to in any practical sense) twice that of Heads. Numerically, FNC would give the same answer as SIA.

I think FNC still implicitly assumes a self-sampling process. Because a non-indexical statement such as “Beauty is awake on a cloudy day” alone is not enough for the purposed probability update. This is best shown with an example.

Boy or Girl

Say couple A has two children. We randomly sample a boy from them. Maybe by telling the parents to bring a boy tomorrow if they have one. The next day a boy is presented to us. Now, what is the probability that both children are boys? This is a very easy question. The answer is ¹⁄₃. (BB, BG, GB, GG are equal-probable, successfully sampling a boy rejects GG). Now we ask the boy, on which day of the week are you born on? Say he answers “Wednesday”. Learning this does not change the probability. Since the day of the week is independent of the sex of the sibling. All seven possible answers are symmetrical to the problem.

In contrast, say couple B also has two children. But we specifically sample a boy born on Wednesday. i.e. telling the parents to bring us a boy born on Wednesday if they have one. The next day a boy is presented. Now, what is the probability that both children are boys? In this case, the probability is higher. Because it is much more likely to have a boy born on Wednesday if they have two boys. The answer, in this case, is ¹³⁄₂₇ (the probability of BB is not exactly twice of BG or GB due to the chance both boys are born on Wednesday). The seven days of the week are not symmetrical to the problem due to the specific sampling process involved.

The Assumed Sampling Process

In both cases, “the couple have a boy born on Wednesday” is a true statement. But the probabilities are updated differently. Because the same information is learned through different sampling processes.

Back to the sleeping beauty problem. When waking up in the experiment, subsequently discovering the sky is cloudy and all other experiences, Beauty does not perform any sort of sampling whatsoever. She just opens her eyes and sees. While we can ignore her first-person perspective in all of this and make a truthful, non-indexical statement such as “Beauty is awake on a cloudy day”, that statement alone is not enough to update the probability as suggested by FNC. It has implicitly adopted a self-sampling process similar to couple B’s case, instead of alternatives such as couple A’s case.

Put it straightly, FNC act as if someone with all the experience of me is being sampled among all observers, and successfully found. It treats oneself as the said sample. Whether this couple-B-type assumption is more reasonable than couple-A-type is debatable, just like the relative merit of FNC vs SSA/SIA is. But the bottom line is, even though more subtle, FNC is still an assumption. It also treats the first-person perspective as a selection outcome. It is an Observation Selection Effect.