The purpose of this post is to invite discussion on potential solutions to a mathematical problem.

Consider a stochastic simulation for which a set G of parameters determine the dynamics. From a single realization of the simulation, a value Y is measured. Since the simulation is stochastic, Y is a random variable whose probability distribution depends on the parameters G. (Y=Y_G)

The problem: I am interested in finding a parameterized probability distribution estimate of Y, and express the parameters as functions of the simulation parameters in G. So my question is, how could one go about doing this?

What I’m currently thinking of doing is the following procedure:

- Sample parameter values of G with a grid search
- For each set G, perform N simulations to obtain an empirical distribution of Y_G
- Visualize the empirical distributions for some parameter sets G, and choose an appropriate common parametric distribution (e.g. normal, exponential, etc.)
- For each Y_G, find the MLE’s of the distribution parameters
- Visualize the MLE parameters M against the simulation parameters G, and try to construct an analytical expression M = f(G) which fits the observations.

There are many scenarios where my proposed approach breaks down. It might be difficult to find a common parametric distribution which can be fit adequately to all empirical distributions of different Y_G. Even if such a distribution is found, it may be difficult to construct the sought after expression f(G).

Please feel free to share experiences with similar problem, or to just brainstorm solution ideas. Thanks.