Assuming we have two classes $ C_1$ and $ C_2$ represented as two Gaussians with $ (2\mu_2, \sigma)$ and $ (\mu_2, \sigma)$ . We know further that $ \mu_2 > 0$ and $ p(C_1) = p(C_2)$ .

We want now calculate the decision boundary assuming that classifying $ x \in C_2$ as $ C_1$ is three times more expensive than the opposite. Classifying $ x$ correctly has no costs.

At first we note that we need some representation of the loss function. Let $ \alpha_1$ be the decision for $ C_1$ and $ \alpha_2$ the decision for $ C_2$ , then we get

$ \begin{align} R(\alpha_1|x) &= \lambda_{11}p(C_1|x) + \lambda_{12}p(C_2|x)\ R(\alpha_2|x) &= \lambda_{21}p(C_1|x) + \lambda_{22}p(C_2|x) \end{align}$

where $ \lambda_{ij} = \lambda(\alpha_i|C_j)$ is the cost for deciding for class $ i$ and being in fact class $ j$ .

Hence we have $ \lambda_{11} = \lambda_{22} = 0$ and with $ \lambda_{21} = c$ we get $ \lambda_{12} = 3c$ .

We are now searching the the decision boundary, where it holds $ R(\alpha_1|x) = R(\alpha_2|x)$ . We can use the previous equations and substitute the $ \lambda$ ‘s: \begin{align} R(\alpha_1|x) = 3c\cdot p(C_2|x) &= c \cdot p(C_1|x) = R(\alpha_2|x)\ \Longleftrightarrow 3c\frac{p(x|C_2)p(C_2)}{p(x)} &= c\frac{p(x|C_1)p(C_1)}{p(x)}\ \Longleftrightarrow 3p(x|C_2) &= p(x|C_1)\ \Longleftrightarrow 3 \exp(-\frac{(x – 2\mu_2)^2}{2\sigma^2}) &= \exp(-\frac{(x – \mu_2)^2}{2\sigma^2})\ \Longleftrightarrow \frac{\exp(-\frac{(x – \mu_2)^2}{2\sigma^2})}{\exp(-\frac{(x – 2\mu_2)^2}{2\sigma^2})} &= 3\ \Longleftrightarrow – \frac{(x – \mu_2)^2}{2\sigma^2} + \frac{(x – 2\mu_2)^2}{2\sigma^2} &= \ln(3)\ \Longleftrightarrow \frac{-x^2 + 2x\mu_2 – \mu_2^2 + x^2 – 4x\mu_2 + \mu_2^2}{2\sigma^2} &= \ln(3)\ \Longleftrightarrow \frac{-2x\mu_2}{2\sigma^2} &= \ln(3)\ \Longleftrightarrow x &= -\ln(3)\frac{\sigma^2}{\mu_2} \end{align}

Sorry for this long equivalent transformation. Is my basic approach correct? And is my result also correct?

Thank you for reading and helping!