# The Calibrate stage

Calibration of the computer model entails finding an optimal parameter $\theta^*$ that maximizes the posterior probability

\[\rho(\theta\vert y, \Gamma_y) = \dfrac{e^{-\mathcal{L}(\theta; y)}}{Z(y\vert\Gamma_y)}\rho_{\mathrm{prior}}(\theta), \qquad \mathcal{L}(\theta, y) = \langle \mathcal{G}(\theta) - y \, , \, \Gamma_y^{-1} \left ( \mathcal{G}(\theta) - y \right ) \rangle,\]

where $\mathcal{L}$ is the loss or negative log-likelihood, $Z(y\vert\Gamma)$ is a normalizing constant, $y$ represents the data, $\Gamma_y$ is the noise covariance matrix and $\rho_{\mathrm{prior}}(\theta)$ is the prior density. Calibration is performed using ensemble Kalman processes, which generate input-output pairs $\{\theta, \mathcal{G}(\theta)\}$ in high density from the prior initial guess to the found optimal parameter $\theta^*$. These input-output pairs are then used as the data to train an emulator of the forward model $\mathcal{G}$.

## Ensemble Kalman Processes

Calibration can be performed using different ensemble Kalman processes: ensemble Kalman inversion (Iglesias et al, 2013), ensemble Kalman sampler (Garbuno-Inigo et al, 2020), and unscented Kalman inversion (Huang et al, 2022). All algorithms are implemented in EnsembleKalmanProcesses.jl. Documentation of each algorithm is available in the EnsembleKalmanProcesses docs.

## Typical construction of the EnsembleKalmanProcess

Documentation on how to construct an EnsembleKalmanProcess from the computer model and the data can be found in the EnsembleKalmanProcesses docs.

## Julia-free forward model

One draw of our approach is that it does not require the forward map to be written in Julia. To aid construction of such a workflow, EnsembleKalmanProcesses.jl provides a documented example of a BASH workflow for the sinusoid problem, with source code here. The forward map interacts with the calibration tools (EKP) only though TOML file reading an writing, and thus can be written in any language; for example, to be used with slurm HPC scripts, with source code here.