# Forward propagation of uncertainty with stochastic response variable

I have a forward uncertainty quantification (UQ) problem, where my solution of a physical system depends on a number of inputs and I want to find how known uncertainties in the inputs propagate into the solution of the physical system. The response variable in the simple case is deterministic -- given a set of inputs, the output is always the same. This one isn't a problem to work out.

But now I have a more complicated physical system. The inputs are the same, but now the solution of the system has additional randomness. So the outputs of the system are stochastic in nature -- every set of inputs will generate a distribution of responses rather than a single, deterministic response. For these cases, the stochastic solution is the result of a sequence of independent random events that take place at independent random locations. These events have no relation to the inputs, but the solution may change based on the interaction of these events with the inputs. There may be thousands of events so I don't think I can just treat them as additional dimensions in the stochastic space.

What I cannot figure out is how to find the final uncertainty when the response is a pdf rather than a deterministic response. I could do something like a maximum likelihood estimation on each simulation result given a set of inputs and treat the distribution parameters as response variables for the forward model, but then I have to presume a distribution for the stochastic response of each simulation. Is this the only way? Or is there a way to build a pdf of pdfs? I'd also like to be able to isolate the impact of the uncertain inputs from the impact of the stochastic solution.

I'm sure others have done this in the past, but I'm not able to use the right keywords to find the papers that demonstrate it.