Suppose that I have to draw from a Dirichlet distribution a multinomial distribution over three events: $$\theta \sim Dir(\alpha_1, \alpha_2, \alpha_3)$$
If $$\alpha_1 = \alpha_2 = \alpha_3 \land \alpha_1 + \alpha_2 + \alpha_3 >> 3$$ $\theta$ distribution is likely to be an uniform distributioin over the three events.
On the contrary, if: $$\alpha_1 = \alpha_2 = \alpha_3 \land \alpha_1 + \alpha_2 + \alpha_3 < 3$$ $\theta$ is likely to be a multinomial distribution that puts the majority of its mass on one of those three events.
My question is: how should I choose the $\alpha_1, \alpha_2, \alpha_3$ if I want $\theta$ to be a distribution that assigns high probability to two events and low probability to the third one?
In other words, I would like that the three most probable values for $\theta$ are:
- $\theta_1 = 0.5, \theta_2=0.5, \theta_3=0$
- $\theta_1 = 0.5, \theta_2=0, \theta_3=0.5$
- $\theta_1 = 0, \theta_2=0.5, \theta_3=0.5$
Furthermore, is that easy to generalize this to the case in which we have $N>3$ events?