Likelihood Ratio

Use-case

The Likelihood ratio is another way of writing the Bayes rule. It conveniently allows the use of a threshold which dictates which classification or decision is to be made.

Mathematical Definition

For a two class case,

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

Variants of the threshold:

• Maximum Likelihood Decision

$$\theta =1$$

• Bayes Decision

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

• Bayes Risk Decision

$$\theta =\frac{{\lambda }_{12}-{\lambda }_{22}}{{\lambda }_{21}-{\lambda }_{11}}\cdot \frac{P(s=2)}{P(s=1)}$$

Note: ${\lambda }_{12}$ is the cost of deciding state $1$ given state $2$ is true, so an and so forth.

![[Pasted image 20240423154225.png|]]

Note: The threshold has same units as the $y-$axis.

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References

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