I’ve already written about this before but I’ve thought of another way of explaining this.
As I wrote in that post that I linked to above, probabilities aren’t facts about objects or phenomena that we look at or experience. If you flip a coin and it lands heads twice, the probability of it landing tails on the third flip is the same as the probability of it landing heads on that third flip.
But people who think that probability is an aspect of the coin similar to its weight or its color will think that 50% probability is physically tied to the coin, so it *must* account for the lack of landing tails on the next flip. As though there is a god of coin flips who has to make sure that the books are accounted for.
Again, this is wrong. And this next scenario I think explains why.
In a standard deck of cards, there’s a 1/52 chance of pulling any specific card, right? What if we have two people, Alice and Bob, who want to pull from the deck. Except, Alice has memorized the order of the cards in the deck and Bob hasn’t.
What is the probability of Bob drawing an Ace of Spades on the first draw? For us and Bob, it’s 1/52. But for Alice — because she’s memorized the order of the cards — it’s virtually certain (e.g., 99.99% or 0.000…1%) to her which card Bob will draw.
If 1/52 was some intrinsic aspect of the deck of cards, then how can there be two different probabilities? Obviously, because probability is a description of our uncertainty. It only exists in our minds. The reader of that thought experiment and Bob are operating under uncertainty. Alice, on the other hand, is not because she’s memorized the order of the cards.
Furthermore, Bayes is all about updating on new evidence. What if there was some third actor, Chad, who mixed up the deck of cards outside of Alice’s knowledge? Now, Alice may think that the next card’s probability is either 100% or 0%, but this is not true either. Now Chad has the certainty.
If Bob draws a card that Alice doesn’t think he should draw, how can she possibly do a Bayesian update on either 0% or 100%? She has to do the equivalent of moving faster than the speed of light in order to update; it literally takes infinite bits of data in order to update from 0% or 100% to some other number. Try it:
P(H | E) = P(E | H) * P(H) / P(E)
50% = ??? * 0% / 1.9% or
50% = ??? * 100% / 1.9%
This situation can be repeated over and over again, introducing new characters manipulating the deck outside of other people’s knowledge. And this demonstrates that not only is probability subjective and in your head, but that a Bayesian probability of 0% or 100% is not a probability at all because those numbers cannot be updated.