* 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
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- 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
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* 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
$$$\frac{P(x|\omega_1)}{P(x|\omega_2)} \;\; \frac{\text{if } > \text{ then classifier chooses } \omega_1}{\text{if } < \text{ then classifier chooses } \omega_2} \;\; \frac{P[\omega_2]}{P[\omega_1]}$$$
$$$\frac{P(x|\omega_1)\times P[\omega_1]}{P(x|\omega_2)\times P[\omega_2]} \;\; \frac{\text{if } > \text{ then classifier chooses } \omega_1}{\text{if } < \text{ then classifier chooses } \omega_2} \;\; 1$$$
$$$\frac{P(\omega_1|x)}{P(\omega_2|x)} \;\; \frac{\text{if } > \text{ then classifier chooses } \omega_1}{\text{if } < \text{ then classifier chooses } \omega_2} \;\; 1$$$
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* 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
$$$\frac{P(x|\omega_1)}{P(x|\omega_2)} \;\; \frac{\text{if } > \text{ then classifier chooses } \omega_1}{\text{if } < \text{ then classifier chooses } \omega_2} \;\; 1$$$