Methods for modelling distributions? As predictor X I have particle size distributions

and I would like to run a model y ~ X.
I.e. each trial has a response y of size N and a predictor X of size N*M.
The easy way would be to get some quartiles from each distribution and run a model y ~ X[q10] + X[q50] +X[q90].
However, I would prefer to do something with the whole distribution and not with summaries. So I was wondering what modelling methods I could try,

*

*Functional Data Analyses ("easy" and hopefully useful model for this problem)

*A Bayesian model. But in this case I don't know how to fit the distributions as data in a model.

To recap my question is,

*

*What methods can I use to model predictors which are distributions?

*In a Bayesian framework how can I feed distributions as data?

Thanks
 A: A possible option might be to consider each of the vectors corresponding to a single observation of a distribution as independent observations coming from the same distribution f and then model, not just the mean as usual in a regression setting, but all the parameters required to characterize the assumed distributional form using something like brms, that estimates distributional models: https://cran.r-project.org/web/packages/brms/vignettes/brms_distreg.html. I guess you could have a hierarchical model to account for the fact that you are liteally estimating as many distributions as sets of observations, and you could have the parameters of a "parent" distribution being themselves characterized by distributions and you just observe the "sons" of this "parent" distribution, say. Note this could get quite complicated very quickly though! But maybe there are methods to do exactly what you want that I am not aware, since this is really not my area of expertize, what I have suggested here is to think about the problem in a different way. And... I'm not really sure whether this is cheating a bit because note it assumes you treat the data from each vector as independent observations of some distribution f while in the case of particle size distributions the data, being proportions of a total, are not really independent observations from any given family. In that case, an interesting distribution to consider might be a  Dirichelet, which has exactly that property of having K classes and the sum of the probabilities in each class adding to 1: https://builtin.com/data-science/dirichlet-distribution. An interesting plot with visualizations of a K=3 class Dirichelet distribution is here. I guess in your case you want a considerably higher K. Any way, more that a definitive answer, food for thought.
