p(t|f)

Recently, while talking about books I have said

“..not everything that is not technical is fiction ..and not everything that is not fiction is technical”

In a moment later, I recollected that these two statements are logically equivalent. Yet, I sensed urge to add second statement to conversation, as if it was different. I have got interested in mechanism behind it and under what conditions it is useful.

Let’s formalise it a bit with t (technical) and f (fiction). Logically both statements are the same
~t && ~f = ~f && ~t

How about conditional probability?

‘probability of not technical if it is not fiction’
p(~t|~f) = p(~t && ~f) / p(~f)

‘probability of not fiction if it is not technical’
p(~f|~t) = p(~f && ~t) / p(~t)

They are not equivalent anymore.

It seems that I had strong inclination to treat both statements as conditional probabilities. Naturally, my next question was “is it only me”? Apparently, people often think of conditional probability when they use material implication in natural language (paper). I expect there are all sorts of gaps between formal logic and our intuition.

When is this useful?

If we posses knowledge about all the books out there then we can check if there are any books that are not technical and not fiction. However, in more realistic setting we don’t know everything at hand, instead we learn gradually about the world via experiments. Moreover, these experiments are costly. For example, if each experiment takes 1h then only 24 can be done daily. With opportunity cost it becomes much worse, each experiment takes slot from all other possible experiments that could have been conducted.

Simple frequentist approach for such tests would be count number of items that are both not technical and not fiction. We check book one by one and see if it is fiction or technical. If we have found at least some, then good — we have proven that such cases exists. If we did not found it, then we can not say anything for sure.

Now, if we know that some pool of books has higher chance of being technical then by knowing conditional probabilities we can prioritise accordingly and find a book of 3rd kind faster, simply because our test are now more effective. If our resources for tests are capped, then it becomes a question whether we can find an answer at all.

Our brain is very nuanced, but there are reasons to believe that it is Bayesian(ish), at least sometimes.