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I often see in the books that the Bayes' theorem is used without the normalisation denominator as: $$ P(A\mid B)\propto P(A)\cdot P(B\mid A) $$ While I understand the reasoning behind it (the normalisation constant does not directly influence the inference) I was wondering about the implementation of this relationship in real-life examples.

I have some classifier that produces posterior probabilities $P(A\mid B)$ for 3 classes. I have the information about the prior $P(A)$ probability, but the normalisation value is not so straightforward to compute. I would like to compute the likelihood $P(B\mid A)$ to subsequently use it in a HMM as an emission probability. One way I see is to compute all 3 likelihoods and then scale them such that they add to 1. Would this be correct?

Thanks

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Yes. If you get three likelihood values $L_1$,$L_2$,$L_3$. The probabilities can be calcluated as

$$P_i=\frac {L_i} {\sum_{k=1}^3 L_k}$$

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