.. _ProcApproximateUpdateDynamicMeanandVariance:

Procedure: Use simulation to recursively update the dynamic emulator mean and variance in the approximation method
==================================================================================================================

Description and Background
--------------------------

This page is concerned with task of :ref:`emulating<DefEmulator>` a
:ref:`dynamic simulator<DefDynamic>`, as set out in the variant
thread on dynamic emulation
(:ref:`ThreadVariantDynamic<ThreadVariantDynamic>`).

The approximation procedure for iterating the single step emulator
(:ref:`ProcApproximateIterateSingleStepEmulator<ProcApproximateIterateSingleStepEmulator>`)
recursively defines

.. math::
   \mu_{t+1} &= \mathrm{E}[ m^*(w_t,a_{t+1},\phi)|f(D)], \\
   V_{t+1} &= \mathrm{E}[
   v^*\{(w_t,a_{t+1},\phi),(w_t,a_{t+1},\phi)\}|f(D)] +
   \mathrm{Var}[m^*(w_t,a_{t+1},\phi)|f(D)],

where the expectations and variances are taken with respect to :math:`w_{t}`,
with :math:`w_{t} \sim N_r(\mu_{t},V_{t})`. If the :ref:`single
step<DefSingleStepFunction>` emulator has a linear mean and a
separable Gaussian covariance function, then :math:`\mu_{t+1}` and
:math:`V_{t+1}` can be computed explicitly, as described in the procedure
page for recursively updating the dynamic emulator mean and variance
(:ref:`ProcUpdateDynamicMeanAndVariance<ProcUpdateDynamicMeanAndVariance>`).
Otherwise, simulation can be used, which we describe here.

Inputs
------

-  :math:`\mu_{t}` and :math:`V_{t}`
-  The single step emulator, conditioned on training inputs :math:`D ` and
   outputs :math:`f(D)`, and hyperparameters :math:`\theta`, with posterior
   mean and covariance functions :math:`m^*(\cdot)` and :math:`v^*(\cdot,\cdot)`
   respectively.

Outputs
-------

-  Estimates of :math:`\mu_{t+1}` and :math:`V_{t+1}`

Procedure
---------

We describe a Monte Carlo procedure using :math:`N` Monte Carlo iterations.
For discussion of the choice of :math:`N`, see the discussion page on Monte
Carlo estimation (:ref:`DiscMonteCarlo<DiscMonteCarlo>`).

#. For :math:`i=1,\ldots,N`, sample :math:`w_t^{(i)}` from :math:`N(\mu_t,V_t)`
#. Estimate :math:`\mu_{t+1}` by
   :math:`\hat{\mu}_{t+1}=\frac{1}{N}\sum_{i=1}^N
   m^*(w_t^{(i)},a_{t+1},\phi)`
#. Estimate :math:`V_{t+1}` by

   .. math::
      \hat{V}_{t+1}=\frac{1}{N}\sum_{i=1}^N
      v^*\{(w_t^{(i)},a_{t+1},\phi),(w_t^{(i)},a_{t+1},\phi)\}
      +\frac{1}{N-1}\sum_{i=1}^N\left(m^*(w_t^{(i)},a_{t+1},\phi)-\hat{\mu}_{t+1}\right)^2