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Decision Theory I [1* P]

Consider two non-negative numbers $ a$ and $ b$ , and show that, if $ a \le b$ , then $ a \le (ab)^{1/2}$ . Use this result to show that, if the decision regions of a two-class classification problem are chosen to minimize the probability of misclassification, this probability will satisfy

$\displaystyle p($mistake$\displaystyle ) \le \int \left[p({\bf x},C_1)p({\bf x},C_2) \right]^{1/2}d{\bf x}.$

Haeusler Stefan 2010-01-19