Say I have $X \sim \text{CUnif}(a, b)$ and $Y \sim \text{CUnif}(c, d)$. The parameters of $X$ and $Y$ overlap i.e., $a < c < b < d$.
How can I find a decision boundary in such case?
I am thinking of taking an arbitrary point (say $x_o$) as a decision boundary in between $b$ and $c$ and then find the probability of miss classification and then minimizing it.
Best Answer
If your datum $\alpha$ is $X$ or $Y$ with probabilities $\pi_X$ and $\pi_Y = 1 - \pi_X$ respectively, and the probability of mis-classification involves $\pi_X$ and $\pi_Y$, then you are a Bayesian and all you need to do is compare the likelihood ratio $\displaystyle \frac{p_X(\alpha)}{p_Y(\alpha)}$ (which takes on values $0, \frac{d-c}{b-a}, \infty$ depending on the value of $\alpha \in [a,d]$ to an appropriate threshold (which I will leave you to determine), and this can be reduced to a threshold test on the value of the datum $\alpha$.