Full Non-Indexical Conditioning (FNC) by Radford Neal (link to the original paper in pdf) is another anthropic camp distinct from SSA and SIA. It states:
“one should condition on all the evidence available, including all the details of one’s memory, but without considering “indexical” information regarding one’s place in the universe (as opposed to what the universe contains).”
What it means is that I shall use all information I have. However, instead of treating that information as something which “I” know, interpret it as “someone” with all that experience exists in the universe. Then update the probability basing on that particular someone’s existence.
This maneuver is basically switching from the first-person perspective to an objective (imaginary impartial observer’s) perspective. That alone means it is invalid according to the Perspective-Based Argument (PBA). It needs to explain why, among all the things the universe contains, a probability update is performed on that particular “someone”. Nonetheless, it might be interesting to compare it with other well-known camps such as SSA and SIA.
As stated in my core positions, the perspective switch puts indexicals such as “I” and “now” in a tough spot. On one hand, “I”/”now” are regarded as a regular observer/moment with no inherent logical significance. On the other hand, the analysis is specifically focused on “I”/”now” rather than any other observers/moments. To explain this attention, it is necessary to treat the perspective center as the outcome of an imaginary sampling process. FNC is no exception. Unlike SSA or SIA whose sampling process is explicitly stated in advance, FNC’s sampling process is implied after observations are made. This is best shown with a thought experiment.
Before I dive into the example, it should be pointed out that arguments resembling FNC have been proposed before. Titelbaum’s Technicolor Beauty, and Rosenthal’s Beauty with a Dime, both follow the same logic. They are essentially special cases of FNC where definitive differences between awakenings are available. The analysis is simpler in these cases. Here I will use the Technicolor Beauty problem as an example.
Technicolor Beauty
The problem is identical to the Sleeping Beauty Problem with one additional detail. The experimenter would randomly choose one day and paint Beauty’s room red while painting the room blue on the other day. This selection is independent of the coin toss. Say Beauty wakes up, after opening her eyes she found out the room is red. How should she reason about the probability of the coin toss?
According to FNC, the analysis should be as follows: after seeing the red room, do not interpret that as “the room is red today/now/during this awakening”. Discard that indexical view and switch to an outsider’s perspective. Interpret that as an objective statement of “there is an awakening on the day the room is red” and update the probability base on that. Beauty is guaranteed to be awake on the red day if the coin landed on Tails, with only a 50% chance for Heads. Therefore the probability of Heads shall decrease to 1/3.
By this argument, the seemingly irrelevant information of the room color, known to be independent of the coin toss, is critical to the calculation. It drives the probability update. Similarly, in the original sleeping beauty problem, every detail observed by Beauty after waking up is important. The shape of the cloud outside, the wrinkle on the blanket, the feeling of faint airflow on her skin, it shall be interpreted as “there is an awakening on the day with all these details”. That will cause the probability of Heads updates to 1/3. (Strictly speaking, it will cause the probability to be almost exactly 1/3. Because without a known definitive difference, there is a theoretical possibility for all observable details to be exactly the same on both days.)
The Imagined Sampling Process
As said earlier, FNC has to assume a sampling process to reconcile the problem caused by the perspective switch. It may not seem obvious in the example above, but that is indeed the case. Because factual statements alone, such as “the red day has an awakening”, does not justify the proposed probability update.
Imagine you know a friend Alice who has two children but you do not know their gender. One day you told her: “If you have a boy, bring one to see me tomorrow.” On the next day, she brought one to you. Here the probability of Alice having two boys is 1/3. Then you asked the boy: “are you the older or the younger child?”. “Older” he answered. The probability of both children being boys would remain at 1/3. (Assuming Alice does not have any preference in choosing which boy to bring if there are more than one.)
Now imagine you have another friend Bob who also has two children with their gender unknown to you. One day you told him: “If your older child is a boy, bring him to see me tomorrow.” On the next day, you saw the boy. Here the probability of Bob having two boys is 1/2.
In both problems, the statement “the older child is a boy” is factually correct. However, the seniority only affects the probability calculation in the latter case. Because in Bob’s case the older child is being specifically sampled. Whereas in Alice’s case the boy being the older child is merely a piece of independent information. It also explains why the probability is higher for Bob. Because while sampling for a boy, Bob’s condition is more restrictive.
Back at the Technicolor Beauty. While the statement “the red day has an awakening” is true, to update the probability as suggested by FNC implies a sampling process similar to Bob’s case. If an impartial outsider specifically samples the red day and found Beauty awake, then he shall perform the update as proposed by FNC. However, that is not the only way for an outsider to find “the red day has an awakening”. Like Alice’s case, an outsider could also randomly sample an awakening, then just happen to find the room red. Yet FNC disregards that possibility and bets on the former sampling process without providing any justification. So FNC at its core, like SSA and SIA, is an assumption. An assumption about treating the first-person center as the outcome of some imagined sampling process.
Of course, none of the sampling processes is correct. Beauty did not perform any sampling at all. She simply opened her eyes and discovered all the independent details. The sampling assumptions would be unnecessary if we reason from Beauty’s viewpoint, i.e. if we do not disregard the indexical information.