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I would like to ask you if my thought first and my answer then to the following problem is right.

Suppose that I have a 3-class 1-dim classification problem where the classes $\omega_1, \omega_2, \omega_3 $ are modelled by the following uniform distribution:

$p(x|\omega_1)= \left\{\begin{matrix} 1/5, & x\epsilon (0, 2)\cup (5, 8)\\ 0, & otherwise \end{matrix}\right.$

$p(x|\omega_2)=\left\{\begin{matrix} 1/9, & x\epsilon (0, 9)\\ 0, & otherwise \end{matrix}\right.$ and

$p(x|\omega_3)=\left\{\begin{matrix} 1, & x\epsilon (3, 4)\\ 0, & otherwise \end{matrix}\right.$

If the 3 classes are not equiprobable. Is there any combination of the priori probabilities that guarantees that x' = 3.5 will be assigned to $\omega_1$

My answer is No even when $p(\omega_2) = p(\omega_3) = 0 $ in this case the posterior probability $p(\omega_1|x)$ is also zero Is it right?

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You are right. Since $p(3.5 \ \omega_1)=0$, the posterior probability will also be 0, irrespective of the values of the priors $\pi_1$. And, since $p(3.5 \mid \omega_2)>0, p(3.5 \mid \omega_3>0$, those posteriors will be positive also, only assuming priors $\pi_2>0, \pi_3>0$.

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