Aggregation property of the Dirichlet distribution for layman

The aggregation property of the Dirichlet distribution has tons of practical examples.

It is useful when you are calculating posterior distributions with Dirichlet priors, which is probably the case where I've used it the most.

In terms of a layman explanation, I think the concept is a bit abstract. But I'll do my best.

Imagine you have a Dirichlet Distribution over flowers, samples from the Dir will be species of flowers, say you have 4 species, so you have k=4.

You tell a gardener to plant 100 flowers according to random samples from that Dirichlet distribution, so if a random sample gave you 0.2,0.6,0.1,0.1 (they have to sum to one) you'll plant 20, 60, 10 and 10 flowers of each variety respectively.

Now, suppose that someone comes and tells you that flowers k=3 and k=4 are actually the same variety, just that they have different colors. (Yeah, our gardener was very crappy, as well as our knowledge of botanic)

So you tell to your gardener, keep planting the flowers in such a way that they still use the same Dirichlet Distribution, but taking into account you have one flower less that got "aggregated" to the other one.

Because he is really proficient with Dirichlet Distributions, he knows that he can create a "new" Dirichlet by basically summing the parameters for k=3 and k=4, and the sample will follow an aggregate. Thus he'll end up planting the same number of flowers following the same distribution.