Bayesian Rule Forms

1. If $P({\omega }_i|\mathbf{x})={\max }_{j}P({\omega }_j|x)$, then $\mathbf{x}\in {\omega }_i$
2. If $p(\mathbf{x}|{\omega }_i)P({\omega }_i)={\max }_{j}p(\mathbf{x}|{\omega }_j)P({\omega }_j)$, then $x\in {\omega }_i$
3. If $l(\mathbf{x})=\frac{p(\mathbf{x}|{\omega }_1)}{p(\mathbf{x}|{\omega }_2)}>\frac{P({\omega }_2)}{P({\omega }_1)}$, then $\mathbf{x}\in {\omega }_1$, and vice versa.
4. Let $h(\mathbf{x})=-\ln [l(x)]=-\ln p(\mathbf{x}|{\omega }_1)+\ln p(\mathbf{x}|{\omega }_2)$.
   • If $h(\mathbf{x})<\ln \left(\frac{p({\omega }_1)}{p({\omega }_2)}\right)$, then $\mathbf{x}\in {\omega }_1$, and vice versa. We call $l(\mathbf{x})$ as the likelihood ratio, and $h(\mathbf{x})$ as the log-likelihood ratio.