Dirichlet-multinomial distribution

Imagine you have a big bag full of different colored balls. Each time you reach in and grab a handful of balls, the number of balls of each individual color you grab might be different. The Dirichlet-Multinomial Distribution is like a tool we can use to figure out how likely it is that we'll get a certain combination of colors of balls in each handful.

To understand how this works, let's first talk about the Dirichlet Distribution. This is like another tool we use as part of the Dirichlet-Multinomial Distribution. The Dirichlet Distribution helps us figure out the probability of getting a certain proportion of each color of ball in our handful. For example, if we want to know how likely it is we'll get 50% red balls, 25% blue balls, and 25% green balls in our handful, we can use the Dirichlet Distribution to figure that out.

Once we know the probability of each proportion of colors in a handful, we can use the Multinomial Distribution to figure out how likely it is we'll actually get that combination of colors. The Multinomial Distribution takes into account the fact that we might have a different number of balls in each color, and multiplies the probability of getting each color together to get the overall probability of getting that specific combination of colors.

So, in summary, the Dirichlet-Multinomial Distribution helps us figure out the probability of getting a certain combination of colors of balls in each handful. We first use the Dirichlet Distribution to figure out the probability of getting each proportion of colors, and then we use the Multinomial Distribution to combine those probabilities and figure out the overall probability of getting a particular combination of colors.