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In Bayesian decision theory, There is an analytical form of error rate, which is $$P(e)=\int P(e|\bf{x})p(\bf{x})d\bf{x}$$. For binary classification, we can compute the type I error probability with: $$P_1(e)=\int_{R_2} p(\bf{x}|\omega_1)d\bf{x}$$. Where $R_2$ is the region in which $\bf{x}$ is classified as $\omega_2$. But, this is computation-complex, for that we should do a multivariable calculus.

Then a single-variable calculus is introduced by log-likelihood ratio $h(x)=\ln\frac{ p(\bf{x}|\omega_2)}{p(\bf{x|\omega_1})}$. Now, it could be computed by: $$P_1(e)=\int_t^\infty p(h|\omega_1) dh$$, where $t=\ln\frac{P(\omega_1)}{P(\omega_2)}$.

I want to know how is the equation below established? $$P_1(e)=\int_{R_2} p(\mathbf{x}|\omega_1)d\mathbf{x} = \int_t^{\infty}p(h|\omega_1)dh$$

It would be better if you can offer the concrete deducing process.

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  • $\begingroup$ Can you share where you saw the equation so that we can verify the notation better? $\endgroup$ – gunes Dec 27 '19 at 21:05
  • $\begingroup$ Emmmmmmmmmmm, a Chinese textbook about pattern recognition. You can see the book here:book.douban.com/subject/5250752. And this part is from p24 and p34. $\endgroup$ – Whisht Dec 31 '19 at 2:07