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 emulating a dynamic simulator, as set out in the variant thread on dynamic emulation (ThreadVariantDynamic).
The approximation procedure for iterating the single step emulator (ProcApproximateIterateSingleStepEmulator) recursively defines
where the expectations and variances are taken with respect to \(w_{t}\), with \(w_{t} \sim N_r(\mu_{t},V_{t})\). If the single step emulator has a linear mean and a separable Gaussian covariance function, then \(\mu_{t+1}\) and \(V_{t+1}\) can be computed explicitly, as described in the procedure page for recursively updating the dynamic emulator mean and variance (ProcUpdateDynamicMeanAndVariance). Otherwise, simulation can be used, which we describe here.
Inputs¶
- \(\mu_{t}\) and \(V_{t}\)
- The single step emulator, conditioned on training inputs \(D \), and hyperparameters \(\theta\), with posterior mean and covariance functions \(m^*(\cdot)\) and \(v^*(\cdot,\cdot)\) respectively.
Outputs¶
- Estimates of \(\mu_{t+1}\) and \(V_{t+1}\)
Procedure¶
We describe a Monte Carlo procedure using \(N\) Monte Carlo iterations. For discussion of the choice of \(N\), see the discussion page on Monte Carlo estimation (DiscMonteCarlo).
For \(i=1,\ldots,N\), sample \(w_t^{(i)}\) from \(N(\mu_t,V_t)\)
Estimate \(\mu_{t+1}\) by \(\hat{\mu}_{t+1}=\frac{1}{N}\sum_{i=1}^N m^*(w_t^{(i)},a_{t+1},\phi)\)
Estimate \(V_{t+1}\) by
\[\hat{V}_{t+1}=\frac{1}{N}\sum_{i=1}^N v^*\{(w_t^{(i)},a_{t+1},\phi),(w_t^{(i)},a_{t+1},\phi)\} +\frac{1}{N-1}\sum_{i=1}^N\left(m^*(w_t^{(i)},a_{t+1},\phi)-\hat{\mu}_{t+1}\right)^2\]