Comparison of Likelihood Ratio between Max Likelihood and Bayes Decisions

Likelihood Ratio for Binary Classification

$$\boxed{LR(\mathbf{x})=\frac{p(\mathbf{x}|s=1)}{p(\mathbf{x}|s=2)}>\theta \Rightarrow \text{Decide for }s=1,\text{ else for }s=2.}$$

Variants of the threshold:

• Maximum Likelihood Decision

$$\theta =1$$

Here if the probability $p(\textbf x|s=1)$ 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. 

• Bayes Decision

$$\theta =\frac{P(s=2)}{P(s=1)}$$

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 $\theta$. 

*Note: Since we define $LR$ to be greater than $\theta$, you want it to be smaller for a preferable classification, hence the ratio is flipped.*