# Bayesian classification using uniform distribution

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?

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 $$\pi_2>0, \pi_3>0$$.