Incorrect prior distribution

An incorrect a priori distribution is a situation when, in the Bayes theorem, the sum (integral) of a priori probabilities does not yield 1 or is not limited at all.

Justification

If Bayes's theorem is written as follows:

$P(A_{i}\mid B)={\frac {P(B\mid A_{i})P(A_{i})}{\sum _{j}P(B\mid A_{j})P(A_{j})}}\,,$ ${\ displaystyle P (A_ {i} \ mid B) = {\ frac {P (B \ mid A_ {i}) P (A_ {i})} {\ sum _ {j} P (B \ mid A_ { j}) P (A_ {j})}} \ ,,}$

it becomes clear that it will remain true if all a priori probabilities $P(A_{i})$ ${\ displaystyle P (A_ {i})}$ and $P(A_{j})$ ${\ displaystyle P (A_ {j})}$ multiply by a constant.

The posterior probabilities will still in total (or upon integration) give 1, regardless of the absolute values of the a priori probabilities.

Incorrect prior distribution

Justification

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