Читать книгу Informatics and Machine Learning. From Martingales to Metaheuristics онлайн

63 страница из 101

For k > 0, P(|X − E(X)| > k) ≤ Var(X)/k²

So far we have counts and probabilities, but what of the probability of X when you know Y has occurred (where X is dependent on Y)? How to account for a greater state of knowledge? It turns out the answer to this was not put on a formal mathematical footing until half way thru the twentieth century, with the Cox derivation [101] .

The rules of probability, including those describing conditional probabilities, can be obtained using an elegant derivation by Cox [101] . The Cox derivation uses the rules of logic (Boolean algebra) and two simple assumptions. The first assumption is in terms of “b|a,” where b|a ≡ “likelihood” of proposition b when proposition a is known to be true. (The interpretation of “likelihood” as “probability” will fall out of the derivation.) The first assumption is that likelihood c‐and‐b|a is determined by a function of the likelihoods b|a and c|b‐and‐a: