* 3 variants from LRT decision rule. * Likelihood is probability density function * LRT means if you know 2 probability density functions, you can utilize them for the decision ================================================================================ - Bayes Criterion * It uses cost terms as well as LRT $$$\dfrac{P(x|\omega_1)}{P(x|\omega_2)} \;\; \dfrac{\text{if } > \text{ then classifier chooses } \omega_1}{\text{if } < \text{ then classifier chooses } \omega_2} \;\; \dfrac{(C_{12}-C_{22})P[\omega_2]}{(C_{21}-C_{11})P[\omega_1]}$$$ $$$C{12}$$$: cost when classifier classifies 1 to 2 ================================================================================ * Maximum A Posteriori criterion: it has simple cost term * Zero-one cost function: * if classifier classifies correctly, then, cost is 0 * if classifier classifies incorrectly, then, cost is 1 $$$\frac{P(x|\omega_1)}{P(x|\omega_2)} \;\; \frac{\text{if } > \text{ then classifier chooses } \omega_1}{\text{if } < \text{ then classifier chooses } \omega_2} \;\; \frac{P[\omega_2]}{P[\omega_1]}$$$ $$$\frac{P(x|\omega_1)\times P[\omega_1]}{P(x|\omega_2)\times P[\omega_2]} \;\; \frac{\text{if } > \text{ then classifier chooses } \omega_1}{\text{if } < \text{ then classifier chooses } \omega_2} \;\; 1$$$ $$$\frac{P(\omega_1|x)}{P(\omega_2|x)} \;\; \frac{\text{if } > \text{ then classifier chooses } \omega_1}{\text{if } < \text{ then classifier chooses } \omega_2} \;\; 1$$$ ================================================================================ * Maximum Likelihood criterion: * it has same prior probability, $$$\frac{P[\omega_1]}{P[\omega_2]}=1$$$ * it has zero-one cost function * Then, ML can be written as following $$$\frac{P(x|\omega_1)}{P(x|\omega_2)} \;\; \frac{\text{if } > \text{ then classifier chooses } \omega_1}{\text{if } < \text{ then classifier chooses } \omega_2} \;\; 1$$$