Consider 3 populations with the same variables but with different likelihoods of getting drawn:
$$f(\alpha, \beta, \gamma) \qquad g(\alpha, \beta, \gamma)\qquad h(\alpha, \beta, \gamma) $$
$$P_f(\alpha) =0.75, \quad P_f(\beta) = 0.25, \quad P_f(\gamma) = 0$$ $$P_g(\alpha) =0.25, \quad P_g(\beta) = \frac{1}{12}, \quad P_g(\gamma) = \frac{7}{12}$$ $$P_h(\alpha) =0, \quad P_g(\beta) = 0, \quad P_g(\gamma) = 1$$
If we extract sample data from an unknown population say $\left( 2 \, \alpha, 1 \beta\right)$, then it is fairly easy to intuitively guess where the population likely came from (here most likely f). How do you mathematically calculate this though?
I attempted to use multinomial distributions. With it I can calculate "if we took this population (and therefore its proportions) then this configuration would happen $x\%$ of the time". By comparing the calculated result I can tell which one is the most likely but not by how much.
How can I determine which population I sampled from with a confidence interval?
A friend told me he suspected the bayensian theorem might help me for this but I'm not that well versed in mathematics so I'm not sure how that would work...
Your help would greatly appreciated!
-Donuts