There is a connection somewhere

Feb 22, 2017

A note on Bayes' theorem

Thomas Kuhn discussed the notion of paradigms as things that every expert in a field assumes to be true. He then proceeded to study the overturning of these paradigms, scientific revolutions. An example of a paradigm would be the believe of 19th century physicists that every wave needs a medium, which was the key assumption of the ether theory. At the time this was a reasonable believe, every wave studied in detail so far had a medium, air for sound, water for tidal waves and perhaps ropes for demonstration experiments, but light was different. Or perhaps not, it does not take too much redefinition to claim that an unexcited electromagnetic field constitutes a medium. It would certainly not be the medium 19th century physicists had in mind, but with that redefinition we change the claim "19th century physicists were mistaken about the existence of a medium." to "19th century physicists were mistaken about the nature of a medium."

Interestingly a similar thing happens with Bayes' theorem: Let $(X, \mu)$ be a measurable space, $G \subset X$ a subset with finite measure and $A, B \subset G$. Then we can interpret $P(A)=\frac{\mu(A)}{\mu(G)}$ as a probability measure on $G$ and specifically

$$ P(A)=\frac{\mu(A)}{\mu(G)} =\frac{\mu(A\cap G)}{\mu(G)} = P(A|G). $$

Let now $F\subset X$, $\mu(F)$ finite and $A\subset F$. Then

$$ P(B|A)=\frac{P(A\cap B)}{P(A)}\frac{P(B)}{P(B)}\\ =\frac{\mu(A\cap B)}{\mu(A)} \frac{\mu(B)}{\mu(G)} \frac{\mu(G)}{\mu(B)} \left(\frac{\mu(F)}{\mu(F)}\right)^2\\ =P(A|B)\frac{\mu(A)}{\mu(B)} \frac{\mu(F)}{\mu(F)}\\ = P(A|B)\frac{\tilde{P}(B)}{P(A|F)} $$ with $\tilde{P}(B)=\mu(B)/\mu(F)$, for which $\tilde{P}(B)=P(B)$ if $B\subset F$.

This is concerning for two reasons, first there is no relationship between $B$ and $F$. So in case of 'obviously' $F$, one assumes that $B=B\cap F$ and therefore assumes the standard form of Bayes' theorem. On the other hand, even if $B\subset F$, we can not choose between $F$ and $G$. This is the case with the difference between existence and nature of a medium above, it seems that there are interpretive systems that are too strong to be empirically accessible.

The Persian King of King Xerxes had the Hellespont whipped and cursed, after a bridge collapsed. Obviously ridiculous, but the second bridge did hold.

I am not entirely sure, what to make of these. Probably the most conservative interpretation is, a model does not necessarily hold at points were it was not tested. A bit more speculative, we can add believes as long as they do not interfere (too much) with the actual workings of reality.