How to best analyse distribution of distributions [closed]

Question

How can one go by analyzing a distribution of distributions ?

An example of such a dataset is as given in this question : What does "a distribution over distributions" mean?

Suppose there are boxes with chocolates, with some portion of dark and sweet chocolates. And you are interested in eating them (chocolates, not - boxes).

You pick at random one of the boxes. (Some kinds of boxes can be more common than others.) Then, you can pick at random one of the chocolates.

So you have a distribution (a collection of boxes) of distributions (chocolates in a box).

Assume minor changes to the above example : there are 10 different types of chocolates in a box (instead of 2) and each box is different (e.g., some from different manufacturers or different factories of the same company)

Then:

1. What statistical/data analysis techniques can we use to understand the distribution of dark chocolate across different boxes compared to white ones, or understand the weight distribution or weight to color relationship ?
2. More specifically, how can we best visualize the relationship between the chocolates across different boxes - without plotting a 2D grid of box plots/probability distributions etc. (not scalable)
3. Can we reduce the relationship to 1 number - for example instead of visualizing a frequency bar plot for each box, can we instead reduce the statistic to 1 number for each box.
4. How can we see the relationship between sequential chocolates across different boxes - e.g., in some boxes we notice 5th-10th chocolates are bigger than the 1st-4th. But how can we test this across 1000 boxes ?

Sorry for the long question and I hope it's not ambiguous.

I would like to know where/how to start analyzing a distribution of distributions (dist-of-dist).

If it's 1 distribution, we can first plot it, try to fit a known theoretical PDF, look for cross-correlation/pearsons correlation for different metrics within the chocolates of the same box, BUT when it's a dist-of-dist, I don't know which existing data analysis methods suit best, apart from exhaustive eye-balling.

Thanks

Answer 1

You have about a dozen questions, but to answer the one about distribution of candy types, the question you point to asks about the Dirichlet distribution. In particular, the Dirichlet is used as a prior in a Dirichlet-Multinomial model.

This models the number of each kind of candy in a particular box as a Multinomial. But the Multinomial has parameters which you'd have to learn (or guess), and the Dirichlet models these parameters. (The Dirichlet, in turn has parameters, which also must be learned or guessed.)