.. _DiscUncertaintyAnalysis:

Discussion: Uncertainty analysis
================================

In uncertainty analysis, we wish to quantify uncertainty about simulator
outputs due to uncertainty about simulator inputs. We define :math:`X`
to be the uncertain true inputs, and :math:`f(X)` to be the
corresponding simulator output(s). In the emulator framework, :math:`f(.)`
is also treated as an uncertain function, and it is important to
consider both uncertainty in :math:`X` and uncertainty in :math:`f(.)`
when investigating uncertainty about :math:`f(X)`. In particular, it is
important to distinguish between the unconditional distribution of
:math:`f(X)`, and the distribution of :math:`f(X)` conditional on :math:`f(.)`.
For example:

#. :math:`\textrm{E}[f(X)]` is the expected value of :math:`f(X)`,
   where the expectation is taken with respect to both :math:`f(.)` and
   :math:`X`. The value of this expectation can, in principle, be obtained
   for any emulator and input distribution.
#. :math:`\textrm{E}[f(X)|f(.)]` is the expected value of :math:`f(X)`,
   where the expectation is taken with respect to :math:`X` only as
   :math:`f(.)` is given. If :math:`f(.)` is a computationally cheap function,
   we could, for example, obtain the value of this expectation using
   Monte Carlo, up to an arbitrary level of precision. However, when
   :math:`f(.)` is computationally expensive such that we require an emulator
   for :math:`f(.)`, **this expectation is an uncertain quantity**. We are
   uncertain about the value of :math:`\textrm{E}[f(X)|f(.)]`, because
   we are are uncertain about :math:`f(.)`.

There is no sense in which :math:`\textrm{E}[f(X)]` can be 'wrong': it
is simply a probability statement resulting from a choice of emulator
(good or bad) and input distribution. But an estimate of
:math:`\textrm{E}[f(X)|f(.)]` obtained using an emulator can be poor if we
have a poor emulator (in the :ref:`validation<DefValidation>` sense)
for :math:`f(.)`. Alternatively, we may be very uncertain about
:math:`\textrm{E}[f(X)|f(.)]` if we don't have sufficient training data for
the emulator of :math:`f(.)`. Hence in practice, the distinction is
important for considering *whether we have enough simulator runs for our
analysis of interest*.