Multi-response Approach to Improving Identifiability in Model Calibration

Abstract

In physics-based engineering modeling, two primary sources of model uncertainty that account for the differences between computer models and physical experiments are parameter uncertainty and model discrepancy. One of the main challenges in model updating results from the difficulty in distinguishing between the effects of calibration parameters versus model discrepancy. In this chapter, this identifiability problem is illustrated with several examples that explain the mechanisms behind it and that attempt to shed light on when a system may or may not be identifiable. For situations in which identifiability cannot be achieved using only a single response, an approach is developed to improve identifiability by using multiple responses that share a mutual dependence on the calibration parameters. Furthermore, prior to conducting physical experiments but after conducting computer simulations, in order to address the issue of how to select the most appropriate set of responses to measure experimentally to best enhance identifiability, a preposterior analysis approach is presented to predict the degree of identifiability that will result from using different sets of responses to measure experimentally. To handle the computational challenges of the preposterior analysis, we also present a surrogate preposterior analysis based on the Fisher information of the calibration parameters.

Appendices

Appendix A: Estimates of the Hyperparameters for the Computer Model MRGP

To obtain the MLEs of the hyperparameters for the computer model MRGP model, the multivariate normal likelihood function is first constructed as:

$$\displaystyle{ \begin{array}{ll} &p(vec(\mathbf{Y}^{m})\vert \mathbf{B}^{m},\boldsymbol{\Sigma }^{m},\boldsymbol{\upomega }^{m}) = (2\pi )^{-qN_{m}/2}\left \vert \Sigma ^{m}\right \vert ^{-N_{m}/2}\left \vert \mathbf{R}^{m}\right \vert ^{-q/2} \\ & \quad \quad \quad \quad \times \exp \left \{-\frac{1} {2}vec(\mathbf{Y}^{m} -\mathbf{H}^{m}\mathbf{B}^{m})^{T}\left (\Sigma ^{m} \otimes \mathbf{R}^{m}\right )^{-1}vec(\mathbf{Y}^{m} -\mathbf{H}^{m}\mathbf{B}^{m})\right \},\end{array} }$$

(4.18)

where vec(⋅ ) is the vectorization of the matrix (stacking of the columns), ⊗ denotes the Kronecker product, R ^m is a N _m × N _m correlation matrix whose ith-row, jth-column entry is $R^{m}((\mathbf{x}_{i}^{m},\boldsymbol{\uptheta }_{i}^{m}),(\mathbf{x}_{j}^{m},\boldsymbol{\uptheta }_{j}^{m}))$, and $\mathbf{H}^{m} = [\mathbf{h}^{m}(\mathbf{x}_{1}^{m},\boldsymbol{\uptheta }_{1}^{m})^{T},\,\,\ldots,\mathbf{h}^{m}(\mathbf{x}_{N_{m}}^{m},\boldsymbol{\uptheta }_{N_{m}}^{m})^{T}]^{T}$. Taking the log of Eq. (4.18) yields:

$$\displaystyle{ \begin{array}{l} \ln (p(vec(\mathbf{Y}^{m})\vert \mathbf{B}^{m},\boldsymbol{\Sigma }^{m},\boldsymbol{\upomega }^{m})) = -\frac{qN_{m}} {2} \ln (2\pi ) -\frac{N_{m}} {2} \ln \left (\left \vert \Sigma ^{m}\right \vert \right ) -\frac{q} {2}\ln \left (\left \vert \mathbf{R}^{m}\right \vert \right ) \\ \qquad \qquad \qquad \qquad -\frac{1} {2}vec(\mathbf{Y}^{m} -\mathbf{H}^{m}\mathbf{B}^{m})^{T}\left (\Sigma ^{m} \otimes \mathbf{R}^{m}\right )^{-1}vec(\mathbf{Y}^{m} -\mathbf{H}^{m}\mathbf{B}^{m}). \end{array} }$$

(4.19)

The MLE of B ^m is found by setting the derivative of Eq. (4.19) with respect to B ^m equal to zero, which gives:

$$\displaystyle{ \hat{\mathbf{B}} ^{m} = [(\mathbf{H}^{m})^{T}(\mathbf{R}^{m})^{-1}\mathbf{H}^{m}]^{-1}(\mathbf{H}^{m})^{T}(\mathbf{R}^{m})^{-1}\mathbf{Y}^{m}. }$$

(4.20)

The MLE of $\boldsymbol{\Sigma }^{m}$ is found using result 4.10 of Ref. [52], which yields:

$$\displaystyle{ \boldsymbol{\hat{\Sigma } }^{m} = \frac{1} {N_{m}}(\mathbf{Y}^{m} -\mathbf{H}^{m}\hat{\mathbf{B}} ^{m})^{T}(\mathbf{R}^{m})^{-1}(\mathbf{Y}^{m} -\mathbf{H}^{m}\hat{\mathbf{B}} ^{m}). }$$

(4.21)

Finally, the MLE of $\boldsymbol{\upomega }^{m}$, denoted by $\hat{\boldsymbol{\upomega }}^{m}$, is found by numerically maximizing Eq. (4.19) after plugging in the MLEs of B ^m and $\boldsymbol{\Sigma }^{m}$.

Appendix B: Posterior Distributions of the Computer Responses

After observing Y ^m, the posterior of the computer response $y_{i}^{m}(\mathbf{x},\boldsymbol{\uptheta })$ given Y ^m (and given $\boldsymbol{\upomega }^{m}$ and $\boldsymbol{\Sigma }^{m}$ and assuming a non-informative prior for B ^m) is Gaussian with mean and covariance:

$$\displaystyle{ \mathbb{E}[\mathbf{y}^{m}(\mathbf{x},\boldsymbol{\uptheta })\vert \mathbf{Y}^{m},\boldsymbol{\upphi }^{m}] = \mathbf{h}^{m}(\mathbf{x},\boldsymbol{\uptheta })\hat{\mathbf{B}} ^{m} + \mathbf{r}^{m}(\mathbf{x},\boldsymbol{\uptheta })^{T}(\mathbf{R}^{m})^{-1}(\mathbf{Y}^{m} -\mathbf{H}^{m}\hat{\mathbf{B}} ^{m}) }$$

(4.22)

$$\displaystyle{ \begin{array}{ll} &\mbox{ Cov}[\mathbf{y}^{m}(\mathbf{x},\boldsymbol{\uptheta })),\mathbf{y}^{m}(\mathbf{x}^{{\prime}},\boldsymbol{\uptheta }')\vert \mathbf{Y}^{m},\boldsymbol{\upphi }^{m}] = \boldsymbol{\Sigma }^{m}\left \{R^{m}((\mathbf{x},\boldsymbol{\uptheta }),(\mathbf{x}^{{\prime}},\boldsymbol{\uptheta }')\right.) \\ &\quad -\mathbf{r}^{m}(\mathbf{x},\boldsymbol{\uptheta })^{T}(\mathbf{R}^{m})^{-1}\mathbf{r}^{m}(\mathbf{x}^{{\prime}},\boldsymbol{\uptheta }') + [\mathbf{h}^{m}(\mathbf{x},\boldsymbol{\uptheta })^{T} - (\mathbf{H}^{m})^{T}(\mathbf{R}^{m})^{-1}\mathbf{r}^{m}(\mathbf{x},\boldsymbol{\uptheta })]^{T} \\ &\quad \times \left.[(\mathbf{H}^{m})^{T}(\mathbf{R}^{m})^{-1}\mathbf{H}^{m}]^{-1}[\mathbf{h}^{m}(\mathbf{x},\boldsymbol{\uptheta })^{T} - (\mathbf{H}^{m})^{T}(\mathbf{R}^{m})^{-1}\mathbf{r}^{m}(\mathbf{x},\boldsymbol{\uptheta })]\right \}\end{array} }$$

(4.23)

where $\mathbf{r}^{m}(\mathbf{x},\boldsymbol{\uptheta })$ is a N _m × 1 vector whose ith element is $R^{m}((\mathbf{x}_{i}^{m},\boldsymbol{\uptheta }_{i}^{m}),(\mathbf{x},\boldsymbol{\uptheta }))$. Using an empirical Bayes approach, the MLEs of the hyperparameters from Appendix A are plugged into Eqs. (4.22) and (4.23) to calculate the posterior distribution of the computer responses. Notice that Eqs. (4.22) and (4.23) are analogous to the single-response GP model results.

Appendix C: Estimates of the Hyperparameters for the Discrepancy Functions MRGP

To estimate the hyperparameters $\boldsymbol{\upphi }^{\delta } =\{ \mathbf{B}^{\delta }$, $\boldsymbol{\Sigma }^{\updelta }$, $\boldsymbol{\upomega }^{\delta }$, $\boldsymbol{\uplambda }\}$ of the MRGP model representing the discrepancy functions, the procedure outlined by Kennedy and O’Hagan [1] is used and modified to handle multiple responses. This procedure begins by obtaining a posterior of the experimental responses given the simulation data and the hyperparameters from Module 1, which has prior mean and covariance:

$$\displaystyle{ \mathbb{E}[\mathbf{y}^{e}(\mathbf{x})\vert \mathbf{Y}^{m},\hat{\boldsymbol{\upphi }}^{m},\boldsymbol{\uptheta } = \boldsymbol{\uptheta }^{{\ast}}] = \mathbb{E}[\mathbf{y}^{m}(\mathbf{x},\boldsymbol{\uptheta }^{{\ast}})\vert \mathbf{Y}^{m},\hat{\boldsymbol{\upphi }}^{m}] + \mathbf{h}^{\delta }(\mathbf{x})\mathbf{B}^{\delta }, }$$

(4.24)

$$\displaystyle{ \begin{array}{l} \mbox{ Cov}[\mathbf{y}^{e}(\mathbf{x}),\mathbf{y}^{e}(\mathbf{x'})\vert \mathbf{Y}^{m},\hat{\boldsymbol{\upphi }}^{m},\boldsymbol{\uptheta } = \boldsymbol{\uptheta }^{{\ast}}] = \boldsymbol{\Sigma }^{\delta }R^{\delta }(\mathbf{x},\mathbf{x}) + \boldsymbol{\uplambda } \\ \qquad \qquad \qquad \quad + \mbox{ Cov}[\mathbf{y}^{m}(\mathbf{x},\boldsymbol{\uptheta }^{{\ast}})),\mathbf{y}^{m}(\mathbf{x'},\boldsymbol{\uptheta }^{{\ast}}))\vert \mathbf{Y}^{m},\hat{\boldsymbol{\upphi }}^{m}],\end{array} }$$

(4.25)

where $\hat{\boldsymbol{\upphi }}^{m}$ are the MLEs of the hyperparameters for the computer model MRGP model. Since Eqs. (4.24) and (4.25) depend on the unknown true value of $\boldsymbol{\uptheta }^{{\ast}}$, these two equations are integrated with respect to the prior distribution of $\boldsymbol{\uptheta }(p(\boldsymbol{\uptheta }))$ via:

$$\displaystyle{ \begin{array}{ll} &\mathbb{E}[\mathbf{y}^{e}(\mathbf{x})\vert \mathbf{Y}^{m},\hat{\boldsymbol{\upphi }}^{m}] =\int \mathbb{E}[\mathbf{y}^{e}(\mathbf{x})\vert \mathbf{Y}^{m},\hat{\boldsymbol{\upphi }}^{m},\boldsymbol{\uptheta }]p(\boldsymbol{\uptheta })\mbox{ d}\boldsymbol{\uptheta }, \\ &\mbox{ Cov}[\mathbf{y}^{e}(\mathbf{x}),\mathbf{y}^{e}(\mathbf{x'})\vert \mathbf{Y}^{m},\hat{\boldsymbol{\upphi }}^{m}] =\int \mbox{ Cov}[\mathbf{y}^{e}(\mathbf{x}),\mathbf{y}^{e}(\mathbf{x'})\vert \mathbf{Y}^{m},\hat{\boldsymbol{\upphi }}^{m},\boldsymbol{\uptheta }]p(\boldsymbol{\uptheta })\mbox{ d}\boldsymbol{\uptheta }.\end{array} }$$

(4.26)

Kennedy and O’Hagan [53] provide closed form solutions for Eq. (4.26) under the conditions of Gaussian correlation functions, constant regression functions, and normal prior distributions for $\boldsymbol{\uptheta }$ (for details, refer to Section 3 of [53] and Section 4.5 of [1]). In this chapter, similar closed form solutions are used except that a uniform prior distribution is assumed for $\boldsymbol{\uptheta }$.

After observing the experimental data, Y ^e, one can construct a multivariate normal likelihood function with mean and variance from Eq. (4.26). The MLEs of $\boldsymbol{\upphi }^{\delta }$ maximize this likelihood function. The MLE of B ^δ can found by setting the analytical derivative of this likelihood function with respect to B ^δ equal to zero (see Section 2 of Ref. [53]). However, there are no analytical derivatives with respect to the hyperparameters $\boldsymbol{\Sigma }^{\delta }$, $\boldsymbol{\upomega }^{\delta }$, and $\boldsymbol{\lambda }$. Therefore, numerical optimization techniques are needed to find these MLEs.

Appendix D: Posterior Distribution of the Calibration Parameters

The posterior for the calibration parameters in Eq. (4.12) involves the likelihood function $p(\mathbf{d}\vert \boldsymbol{\uptheta },\hat{\boldsymbol{\upphi }})$ and the marginal posterior distribution for the data $p(\mathbf{d}\vert \hat{\boldsymbol{\upphi }})$. The likelihood function is multivariate normal with mean vector and covariance matrix defined as:

$$\displaystyle\begin{array}{rcl} \mathbf{m}(\boldsymbol{\uptheta })& =& \mathbf{H}(\boldsymbol{\uptheta })\hat{\mathbf{B}} \\ \quad \quad \;& =& \left [\begin{array}{*{20}c} \mathbf{I}_{q} \otimes \mathbf{H}^{m} & \mathbf{0} \\ \mathbf{I}_{q} \otimes \mathbf{H}^{m}(\mathbf{X}^{e},\boldsymbol{\uptheta })&\mathbf{I}_{q} \otimes \mathbf{H}^{\delta }\\ \end{array} \right ]\left [\begin{array}{*{20}c} vec(\hat{\mathbf{B}} ^{m}) \\ vec(\hat{\mathbf{B}} ^{\delta } )\\ \end{array} \right ],{}\end{array}$$

(4.27)

$$\displaystyle{ \mathbf{V}(\boldsymbol{\uptheta }) = \left [\begin{array}{*{20}c} \boldsymbol{\hat{\Sigma } } ^{m} \otimes \mathbf{R}^{m} & \boldsymbol{\hat{\Sigma } } ^{m} \otimes \mathbf{C}^{m} \\ \boldsymbol{\hat{\Sigma } } ^{m} \otimes \mathbf{C}^{m\;T}&\boldsymbol{\hat{\Sigma } } ^{m} \otimes \mathbf{R}^{m}(\mathbf{X}^{e},\boldsymbol{\uptheta }) + \boldsymbol{\hat{\Sigma } } ^{\delta }\otimes \mathbf{R}^{\delta } + \boldsymbol{\hat{\lambda }} \otimes \mathbf{I}_{N_{e}}\\ \end{array} \right ], }$$

(4.28)

where $\hat{\mathbf{B}}= \left (\mathbf{H}(\boldsymbol{\uptheta })^{T}\mathbf{V}(\boldsymbol{\uptheta })^{-1}\mathbf{H}(\boldsymbol{\uptheta })\right )^{-1}\mathbf{H}(\boldsymbol{\uptheta })^{T}\mathbf{V}(\boldsymbol{\uptheta })^{-1}\mathbf{d}$, which is calculated based on the entire data set (instead of using the estimates from Modules 1 and 2 for B ^m and B ^δ) as detailed in Section 4 of [53]. $\mathbf{H}^{m}(\mathbf{X}^{e},\boldsymbol{\uptheta }) = [\mathbf{h}^{m}(\mathbf{x}_{1}^{e},\boldsymbol{\uptheta })^{T},\,\,\ldots,\mathbf{h}^{m}(\mathbf{x}_{N_{e}}^{m},\boldsymbol{\uptheta })^{T}]_{}^{T}$ and $\mathbf{H}^{\delta } = [\mathbf{h}^{\delta }(\mathbf{x}_{1}^{e})^{T},\,\,\ldots,\mathbf{h}^{\delta }(\mathbf{x}_{N_{e}}^{\delta })^{T}]_{}^{T}$ denote the specified regression functions for the computer model and the discrepancy functions at the input settings X ^e. C ^m denotes the N _m × N _e matrix with ith-row, jth-column entries $R^{m}((\mathbf{x}_{i}^{m},\boldsymbol{\uptheta }_{i}^{m}),(\mathbf{x}_{j}^{e},\boldsymbol{\uptheta }))$. $\mathbf{R}^{m}(\mathbf{X}^{e},\boldsymbol{\uptheta })$ denotes the N _e × N _e matrix with ith-row, jth-column entries $R^{m}((\mathbf{x}_{i}^{e},\boldsymbol{\uptheta }),(\mathbf{x}_{j}^{e},\boldsymbol{\uptheta }))$. R ^δ denotes the N _e × N _e matrix with ith-row, jth-column entries R ^δ(x ^e _i , x ^e _j ). Finally, I _q and $\mathbf{I}_{N_{e}}$ denote the q × q and N _e × N _e identity matrices.

The marginal posterior distribution for the data $p(\mathbf{d}\vert \hat{\boldsymbol{\upphi }})$ is:

$$\displaystyle{ p(\mathbf{d}\vert \boldsymbol{\upphi }) =\int p(\mathbf{d}\vert \boldsymbol{\uptheta },\boldsymbol{\upphi })p(\boldsymbol{\uptheta })\mbox{ d}\boldsymbol{\uptheta }, }$$

(4.29)

which can be calculated using any numerical integration technique. In this chapter, Legendre-Gauss quadrature is used for the low-dimensional examples. Alternatively, Markov chain Monte Carlo (MCMC) could be used to sample complex posterior distributions such as those in Eq. (4.12).

Appendix E: Posterior Distribution of the Experimental Responses

Since a MRGP model represents the experimental responses, the conditional (given $\boldsymbol{\uptheta }$) posterior distribution at any point x is Gaussian with mean and covariance defined as (assuming a non-informative prior on B ^m and B ^δ and using the empirical Bayes approach that treats $\boldsymbol{\upphi }=\hat{\boldsymbol{\upphi }}$ as fixed):

$$\displaystyle{ \mathbb{E}[\mathbf{y}^{e}(\mathbf{x})^{T}\vert \boldsymbol{\uptheta },\mathbf{d},\hat{\boldsymbol{\upphi }}] = \mathbf{h}(\mathbf{x},\boldsymbol{\uptheta })\boldsymbol{\hat{\mathrm{B}} }+ \mathbf{t}(\mathbf{x},\boldsymbol{\uptheta })^{T}\mathbf{V}(\boldsymbol{\uptheta })^{-1}(\mathbf{d} -\mathbf{H}(\boldsymbol{\uptheta })\boldsymbol{\hat{\mathrm{B}} } ), }$$

(4.30)

$$\displaystyle{ \begin{array}{ll} &\mbox{ Cov}[\mathbf{y}^{e}(\mathbf{x})^{T},\mathbf{y}^{e}(\mathbf{x'})^{T}\vert \boldsymbol{\uptheta },\mathbf{d},\hat{\boldsymbol{\upphi }}] = \boldsymbol{\hat{\Sigma } } ^{m}R^{m}((\mathbf{x},\boldsymbol{\uptheta }),(\mathbf{x'},\boldsymbol{\uptheta })) + \boldsymbol{\hat{\Sigma } } ^{\delta } R^{\delta }(\mathbf{x},\mathbf{x'}) \\ &\quad + \boldsymbol{\hat{\lambda }} -\mathbf{t}(\mathbf{x},\boldsymbol{\uptheta })^{T}\mathbf{V}(\boldsymbol{\uptheta })^{-1}\mathbf{t}(\mathbf{x'},\boldsymbol{\uptheta }) + \left (\mathbf{h}(\mathbf{x},\boldsymbol{\uptheta })^{T} -\mathbf{H}(\boldsymbol{\uptheta })^{T}\mathbf{V}(\boldsymbol{\uptheta })^{-1}\mathbf{t}(\mathbf{x},\boldsymbol{\uptheta })\right )^{T} \\ &\quad \times \left (\mathbf{H}(\boldsymbol{\uptheta })^{T}\mathbf{V}(\boldsymbol{\uptheta })^{-1}\mathbf{H}(\boldsymbol{\uptheta })\right )^{-1}\left (\mathbf{h}(\mathbf{x'},\boldsymbol{\uptheta })^{T} -\mathbf{H}(\boldsymbol{\uptheta })^{T}\mathbf{V}(\boldsymbol{\uptheta })^{-1}\mathbf{t}(\mathbf{x'},\boldsymbol{\uptheta })\right ),\\ \end{array} }$$

(4.31)

where:

$$\displaystyle{ \mathbf{t}(\mathbf{x},\boldsymbol{\uptheta }) = \left [\begin{array}{*{20}c} \boldsymbol{\hat{\Sigma } } ^{m} \otimes \mathbf{R}^{m}((\mathbf{X}^{m},\boldsymbol{\uptheta }^{m}),(\mathbf{x},\boldsymbol{\uptheta })) \\ \boldsymbol{\hat{\Sigma } } ^{m} \otimes \mathbf{R}^{m}((\mathbf{X}^{e},\boldsymbol{\uptheta }),(\mathbf{x},\boldsymbol{\uptheta })) + \boldsymbol{\hat{\Sigma } } ^{\delta }\otimes \mathbf{R}^{\delta }(\mathbf{X}^{e},\mathbf{x})\\ \end{array} \right ], }$$

(4.32)

$$\displaystyle{ \mathbf{h}(\mathbf{x},\boldsymbol{\uptheta }) = \left [\begin{array}{*{20}c} \mathbf{I}_{q} \otimes \mathbf{h}^{m}(\mathbf{x},\boldsymbol{\uptheta })&\mathbf{I}_{ q} \otimes \mathbf{h}^{\delta }(\mathbf{x})\\ \end{array} \right ]. }$$

(4.33)

$\mathbf{R}^{m}((\mathbf{X}^{m},\boldsymbol{\Theta }^{m}),(\mathbf{x},\boldsymbol{\uptheta }))$ is a N _m ×1 vector whose ith entry is $R^{m}((\mathbf{x}_{i}^{m},\boldsymbol{\uptheta }_{i}^{m}),(\mathbf{x},\boldsymbol{\uptheta }))$, $\mathbf{R}^{m}((\mathbf{X}^{e},\boldsymbol{\uptheta }),(\mathbf{x},\boldsymbol{\uptheta }))$ is a N _e ×1 vector whose ith entry is $R^{m}((\mathbf{x}_{i}^{e},\boldsymbol{\uptheta }),(\mathbf{x},\boldsymbol{\uptheta }))$, and R ^δ(X ^e, x) is a N _e × 1 vector whose ith entry is R ^δ(x ^e _i , x).

To calculate the unconditional posterior distributions (marginalized with respect to $\boldsymbol{\uptheta }$) of the experimental responses, the conditional posterior distributions are marginalized with respect to the posterior distribution of the calibration parameters from Module 3. The mean and covariance of the unconditional posterior distributions can be written as:

$$\displaystyle\begin{array}{rcl} \mathbb{E}[\mathbf{y}^{e}(\mathbf{x})^{T}\vert \mathbf{d},\hat{\boldsymbol{\upphi }}] = \mathbb{E}[\mathbb{E}[\mathbf{y}^{e}(\mathbf{x})^{T}\vert \boldsymbol{\uptheta },\mathbf{d},\hat{\boldsymbol{\upphi }}]],& &{}\end{array}$$

(4.34)

$$\displaystyle\begin{array}{rcl} \begin{array}{l} \mbox{ Cov}[\mathbf{y}^{e}(\mathbf{x})^{T},\mathbf{y}^{e}(\mathbf{x'})^{T}\vert \mathbf{d},\hat{\boldsymbol{\upphi }}] = \mathbb{E}[\mbox{ Cov}[\mathbf{y}^{e}(\mathbf{x})^{T},\mathbf{y}^{e}(\mathbf{x'})^{T}\vert \boldsymbol{\uptheta },\mathbf{d},\hat{\boldsymbol{\upphi }}]] \\ \quad + \mbox{ Cov}[\mathbb{E}[\mathbf{y}^{e}(\mathbf{x})^{T}\vert \boldsymbol{\uptheta },\mathbf{d},\hat{\boldsymbol{\upphi }}], \mathbb{E}[\mathbf{y}^{e}(\mathbf{x'})^{T}\vert \boldsymbol{\uptheta },\mathbf{d},\hat{\boldsymbol{\upphi }}]],\\ \end{array} & &{}\end{array}$$

(4.35)

where the outer expectation and covariance are with respect to the posterior distribution of the calibration parameters. Equations (4.34) and (4.35) are derived using the law of total expectation and the law of total covariance [54]. Due to the complexity of the posterior distribution of the calibration parameters, the marginalization requires numerical integration methods. For the examples in this chapter, Legendre-Gauss quadrature is used.