The words “falsifiable” and “testable” are sometimes used interchangeably, which imprecision is the price of speaking in English. There are two different probability-theoretic qualities I wish to discuss here, and I will refer to one as “falsifiable” and the other as “testable” because it seems like the best fit.

As for the math, it begins, as so many things do, with:

P (A_{i} | B) = \frac{P (B | A_{i}) P (A_{i})}{Σ_{j} P (B | A_{j}) P (A_{j})} .

This is Bayes’s Theorem. I own at least two distinct items of clothing printed with this theorem, so it must be important.

To review quickly, B here refers to an item of evidence, $A_{i}$ is some hypothesis under consideration, and the $A_{j}$ are competing, mutually exclusive hypotheses. The expression $P (B | A_{i})$ means “the probability of seeing B, if hypothesis Ai is true” and $P (A_{i} | B)$ means “the probability hypothesis $A_{i}$ is true, if we see $B$ .”

The mathematical phenomenon that I will call “falsifiability” is the scientifically desirable property of a hypothesis that it should concentrate its probability mass into preferred outcomes, which implies that it must also assign low probability to some un-preferred outcomes; probabilities must sum to 1 and there is only so much probability to go around. Ideally there should be possible observations which would drive down the hypothesis’s probability to nearly zero: There should be things the hypothesis cannot explain, conceivable experimental results with which the theory is not compatible. A theory that can explain everything prohibits nothing, and so gives us no advice about what to expect.

P (A_{i} | B) = \frac{P (B | A_{i}) P (A_{i})}{Σ_{j} P (B | A_{j}) P (A_{j})} .

In terms of Bayes’s Theorem, if there is at least some observation B that the hypothesis Ai can’t explain, i.e., P(B|Ai) is tiny, then the numerator P(B|Ai)P(Ai) will also be tiny, and likewise the posterior probability P(Ai|B). Updating on having seen the impossible result B has driven the probability of Ai down to nearly zero. A theory that refuses to make itself vulnerable in this way will need to spread its probability widely, so that it has no holes; it will not be able to strongly concentrate probability into a few preferred outcomes; it will not be able to offer precise advice.

Thus is the rule of science derived in probability theory.

As depicted here, “falsifiability” is something you evaluate by looking at a singlehypothesis, asking, “How narrowly does it concentrate its probability distribution over pos... ,,,,,,,,,,,,,,,,,,,,,,