I have constructed a Neural Network to predict a function of i inputs. ie:
\begin{align*} y = f(x_1, x_2 .. x_i) \end{align*}
The network is working really well and gives me a good approximation of the mean at each point and extrapolates perfectly to the points where I have no data.
Now I believe the true y to be normally distributed around the output y (mean) for each input X. I'm trying to think of a sane way to get the standard deviation here. If I calculate the standard deviation from all matching X inputs and then feed it into a new neural network will that do the trick? I'm worried that leaves me with a severely reduced dataset and the calculation is not as reliable.
With all that dealing with errors in the back propogation step it seems like there should be some way to extract a form of variance...or is that a pipe dream? :D