Likelihood Ratio for Binary Classification
Variants of the threshold:
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Maximum Likelihood Decision
Here if the probability greater than the denominator. The threshold has to be greater than one. This is same as choosing making a classification based on which classification has a higher probability.
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Bayes Decision
Here we make use of the Priors. Thus if we already know that a particular classification is more probable, we can use this knowledge to essentially weight the threshold .
Note: Since we define to be greater than , you want it to be smaller for a preferable classification, hence the ratio is flipped.