A short example to show how increasing input uncertainty can decrease output uncertainty.
In variance-based sensitivity analysis, the aim is to understand how individual uncertain inputs \(X_1,\ldots, X_d\) (or groups of inputs) contribute to the variance of \(Y=f(X_1,\ldots, X_d)\). The choice of input distribution (apart from whether the inputs are dependent or not) is outside the scope of many papers on the subject, and the default, in the case of independent inputs is to assume each input has the \(U[0,1]\) distribution (and we can always transform to \(U[0,1]\)). But, of course, the results of the analysis won’t mean much if we haven’t thought about the input distributions properly.
An example I like to show this is the following. Suppose our (one-input) model is given by the function \(f(x) = \exp(-x)\), and consider the variance of \(Y=\exp(-X)\), where \(X\sim U[0,b]\). (So the aim here would be simply to quantify uncertainty in \(Y\) rather than apportion it to different model inputs, but it’s easy to consider extending this example to multiple inputs.)
After a little algebra, we have
and so increasing \(b\) increases the variance of \(X\) but decreases the variance of \(Y\). This is illustrated in the plot below. The solid line shows \(f(x)\), and the two dashed lines show different uniform distributions: one with \(b=1\), and one with \(b=10\). As \(b\) increases, we put more probability on the region of input space where \(f(x)\) is relatively flat, and less where the ‘interesting stuff’ happens.
- being ‘conservative’ in our specification of input uncertainty doesn’t necessarily result in a conservative specification of output uncertainty;
- simply specifying a range for a model input, and then assuming a uniform distribution over that range may not be sensible: elicit your input distributions carefully!
I will write a separate post on eliciting probability distributions later on, and for now will just plug some software and papers described here.