The theorem acquires its application to statistical inference when we
think of as a hypothesis which is being tested by measuring some data
. In real life, with noisy and incomplete data, we never have the
luxury of measuring directly, but only something depending on it in a
nonunique fashion. If we understand this dependence, i.e understand our
experiment, we know
. If only, (and this is a big IF!), someone gave us , then
we
would be able to compute the dependence of on from Bayes
theorem.
Going from to may not seem to be a big step for a man, but it is a giant step for mankind. It now tells us the probability of different hypotheses being true based on the given data . Remember, this is the real world. More than one hypothesis is consistent with a given set of data, so the best we can do is narrow down the possibilities. (If ``hypothesis'' seems too abstract, think of it as a set of numbers which occur as parameters in a given model of the real world)