"Normalization" is an arithmetical procedure carried out to make sure that a set of probabilities sums to exactly 1, in cases where we believe that exactly one of those possibilities is true.
For example, suppose that the odds of Alexander Hamilton winning a presidential election are 3 : 2. But Alexander Hamilton must either win or not win, so the probabilities of him winning or not winning should sum to 1. If we add 3 and 2, however, we get 5, which is an unreasonably large probability. If we rewrite the odds as 0.6 : 0.4, we've preserved the same proportions, but made the terms sum to 1. It therefore makes sense to say that Hamilton has a 60% probability of winning the election.
We could figure out how to normalize the odds by dividing each of the terms by the sum of terms, i.e., go from 3 : 2 to
More generally, if we have an enormous relative-odds function where has many components, and we want to convert this to a probability function on that sums to 1, we could just divide every element of by the sum of all elements in
%knows-requisite(Math2)That is:
Invalid LaTeX $\forall i: \mathbb{P}(H_i) = \frac{\mathbb{O}(H_i)}{\Sum_i \mathbb{O}(H_i)}.: TeX parse error: Undefined control sequence \Sum
Analogously, if is a continuous distribution on , we would normalize it (create a proportional probability function whose integral is equal to 1) by dividing by its own integral:
Whenever a probability function is proportional to something, we can obtain that probability function by normalizing that thing.
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