# Predicting Y based on distribution of X

Suppose I have two random variables Y and X, where Y is given as one point while X is given as a distribution. I am trying to predict Y based on X, however I cannot put the whole distribution of X in a column as I do not have one value. I could estimate some statistic(s) for the whole distribution and use only that, however I loose too much information this way. Are there some options for such cases, to be able to include more (ideally all) information about the distribution in a standard tabular form to be used in standard statistical modeling?

As an example, suppose I am trying to predict the weather tomorrow (Y), and my X is a distribution of possible values obtained through simulations. How could I include as much information as possible about X, while still keeping the column numbers to a minimum, so as to avoid high-dimensional data?

Imagine that a simulation is run each day and produces a fixed number of samples for X, let's say 1000, which depicts a probabilistic outcome of Y for tomorrow. I am able to run such a simulation because I partly understand the process which generates Y. The simulations results - the whole distribution of X - is important, since it carries a lot of information about the range, shape, modes, etc. of the possible outcomes, and I postulate that a better prediction of Y can be obtained using information about the whole distribution, rather than using just one estimate.

Note1: the distributions of X are arbitrary and do not follow any standard ones.
Note2: the values change over time and are not stationary.

• Interesting question. So-called functional-data looks at it the other way around, predicting entire (density or other) functions. One wonders whether the simplest way would be to include different functionals of $X$ in a model for $Y$, e.g., the mean, higher moments, quantiles etc. Including all information for arbitrary distributions of $X$ is probably not realistic. Dec 15, 2021 at 12:38

We really could need some more information, for instance, what does $$Y$$ represent, and how do you do the simulations producing $$X$$? The answer by @Nuclear Hoagie: assumes the simulations give a true predictive distribution for $$Y$$, if that is so (and you assumed squared error for the prediction) the mean of the $$X$$'s is the correct answer.
But if not (and how can you know?), maybe try something else. One idea is to bin $$X$$ and make histogram counts, then use the bin counts as predictors. Another, probably better, is to compute various descriptors from $$X$$, maybe mean, median, some quantiles, ... and use those as predictors. The problem looks somewhat similar to abc (approximate bayesian computation) so maybe have a look ABC. How can it avoid the likelihood function? or ABC with Lotka-Volterra (or any dynamical system).