University AI - Bayes Decision Rule

Bayes Decision Rule

Bayes decision rules relies on the maximum of the joint-probability

$$p(\underline{x},w_i)=p(\underline{x}|w_i)\cdot P(w_i)$$

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Bayes Decision Rule with Discriminant Functions:

$$\begin{array}{rl}& D(\underline{x})=w_i\text{iff}{g}_i(\underline{x})\ge g_j(\underline{x})\\ & \to g_i(\underline{x})=p(\underline{x}|{\omega }_i)\cdot P({\omega }_i)\\ & \to g_i(\underline{x})=\log (p(\underline{x}|{\omega }_i))+\log (P({\omega }_i))\end{array}$$

We define the Maximum Likelihood Decision as:

$$g_i(\underline{x})=\log (p(\underline{x}|w_i))+\log (P({\omega }_i))$$

In case of Gaussian assumptions, the MLD will become:

$$g_i(\underline{x})=-\frac{1}{2}(\underline{x}-\underline{{\mu }_i}{)}^T{\Sigma }_i^{-1}(\underline{x}-\underline{{\mu }_i})-\frac{1}{2}\log \left|{\Sigma }_i^{-1}\right|+\log P(w_i)$$

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Original Files

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