.. _DefDecisionBasedSA: Definition of Term: Decision-based sensitivity analysis ======================================================= In decision-based :ref:`sensitivity analysis` we consider the effect on a decision which will be based on the output :math:`f(X)` of a :ref:`simulator` as we vary the inputs :math:`X`, when the variation of those inputs is described by a (joint) probability distribution. This probability distribution can be interpreted as describing uncertainty about the best or true values for the inputs. We measure the sensitivity to an individual input :math:`X_i` by the extent to which we would be able to make a better decision if we could remove the uncertainty in that input. A decision problem is characterised by a set of possible decisions and a utility function that gives a value :math:`U(d,f(x))` if we take decision :math:`d` and the true value for the input vector is :math:`x`. The optimal decision, given the uncertainty in :math:`X`, is the one which maximises the expected utility. Let :math:`U^*` be the resulting maximised expected utility based on the current uncertainty in the inputs. If we were to remove the uncertainty in the i-th input by learning that its true value is :math:`X_i = x_i`, then we might make a different decision. We would now take the expected utility with respect to the *conditional* distribution of :math:`X` given that :math:`X_i = x_i`, and then maximise this with respect to the decision :math:`d`. Let :math:`U^*_i(x_i)` be the resulting maximised expected utility. This of course depends on the true value :math:`x_i` of :math:`X_i`, which we do not know. The decision-based sensitivity measure for the i-th input is then the value of learning the true value of :math:`X_i` in terms of improved expected utility, i.e. :math:`V_i = \text{E}[U^*_i(X_i)] - U^*`, where the expectation in the first term is with respect to the marginal distribution of :math:`X_i`. We can similarly define the sensitivity measure for two or more inputs as being the value of learning the true values of all of these. :ref:`Variance-based` sensitivity analysis is a special case of decision-based analysis, when the decision is simply to estimate the true output :math:`f(X)` and the utility function is negative squared error. In practice, though, variance-based sensitivity analysis provides natural measures of sensitivity when there is no specific decision problem.