Subjective Subjective Probabilities - the Probabilities of Miraculous Healing

Bayes Rule

A refresher for the Bayes rule:

         P(B|A) * P(A)
P(A|B) = -------------
             P(B)

        P(B) * P(A|B)
P(A) = --------------
           P(B|A)

Now, suppose:

• A: "I" get cured of a supposedly incurable disease
• B: miracles exist

Also, assume that P(A) is mostly low, but empirically not zero. Sometimes people do suddenly get cured without a good reason.

Let us investigate the subjective probabilities of:

1. The devout believer,
2. the denialist, and
3. the "rational experimentalist"

The Devout Believer

Obviously, P(B) ~= 1

          1 * P(A|B) <- [Jesus factor, or am I worthy of salvation?]
P(A) ~= --------------
          P(B|A) ~= P(B) =~ 1

Basically the subjective probability of A depends on whether you think you will be saved. "Jesus" helps quite a bit in boosting the probability here. For those who object to the subjective probability here, don't fret -- a subjective probability of 99.999% still doesn't mean anything in the frequentist world. It just means they have messed up priors.

However, for those who believe not only in subjective truth but also in subjective probability, this might ring a bell.

The Denialist

The denialist just pretends B isn't a thing. So,

• P(B) is very low
• P(B|A) is very low (won't believe the evidence even if something apparently miraculous happened)
• P(A) ~= P(A|B) -- basically, B is not a real thing, P(A|B) is just P(A) because B is irrelevant.

The Rational Experimentalist

"I'll believe it when I see it" -- your local Rational Experimentalist

• P(B) is very low (denote it as 'e')
• P(B|A) is very high, i.e. "I will believe in miracles if I get cured"

So,

        e * P(A|B)       e
P(A) = -------------- < --- ~= 0
            0.99        .99

Here, the math is straightforward, but the interpretation is funny. On the surface, the rational experimentalist is just tautologically applying the formula, since in order to be able to say "I will believe in miracles if I get cured" given a low P(B) prior, P(A) must have had been very very low (i.e. virtually no possibility of a non-miraculous cure). Then the math is just as expected.

But if we take a subjective reality world view, that whatever we do not know is truly not yet settled, then "discovering" or "deciding" that P(B|A) is high is logically the same act as "discovering" or "deciding" that P(A) is low.

i.e. **If** you have a rational choice to decide that P(B|A) is high, in making that choice you have also decided that P(A) is very very low. (It gets even lower if you believe you're not worthy).

Conclusions

I wanted to say "that's why frequentist approaches to miracle finding doesn't work", but I suppose that doesn't really follow. I guess what really follows is that even if you think you "keep an open mind" and are "willing to change your stance" given extraordinary evidence, the fact that you are willing to change your stance implies you subjectively believe the chance of this extraordinary evidence is slim -- because (duh) it is extraordinary. In a subjective interpretation of reality, you'll probably discover the world being quite ordinary.

In a world with a low rate of miracles, the only way to increase your chances of seeing it is to believe it. Note that I didn't say that if you believe you will see.

Also, suppose we fixate a value to P(A) in a frequentist context, we can actually imply something about the ratio of devout believers and rational experimentalists. There's enough wriggle room to make everything look consistent especially with the "Jesus factor". I suppose we don't want to use this to deduce the number of "Jesus" equivalents we can have... But then, we could hypothetize that the math works out that in order to efficiently map each person's subjective probabilities to the frequentist rates, it seems that the universe could just assign miracles to those whose P(A) is high, and refrain from giving miracles to those whose P(A) is low. Then everybody's observations matches their probabilities.

Further Reading

https://github.com/hnfong/public-crap/blob/main/writings/2022/09-Subjective_Truth.md