Probabilistic Calibration

Bayesian Calibration

We model the experimental data \(\boldsymbol{D} = \{E'(x_i), E''(x_i)\}_{i=1}^{N_d}\) using the statistical relation

\[ \boldsymbol{D} = \boldsymbol{\mathcal{M}}(\boldsymbol{x}, \boldsymbol{\theta}_m) + \boldsymbol{\varepsilon}, \]

where \(\boldsymbol{\mathcal{M}}\) denotes the deterministic model predictions and \(\boldsymbol{\varepsilon}\) represents measurement error. Both model parameters \(\boldsymbol{\theta}_m\) and errors are treated as random variables.

Error Model

Measurement errors are assumed independent and Gaussian:

\[ \varepsilon_{i,E'} \sim \mathcal{N}(0, \sigma_{E'}), \quad \varepsilon_{i,E''} \sim \mathcal{N}(0, \sigma_{E''}), \]

with unknown standard deviations \(\boldsymbol{\theta}_e = [\sigma_{E'}, \sigma_{E''}]\). The full parameter vector is

\[ \boldsymbol{\theta} = [\boldsymbol{\theta}_m, \boldsymbol{\theta}_e]. \]

Bayesian Inference

The posterior distribution is obtained via Bayes’ rule:

\[ \pi(\boldsymbol{\theta} \mid \boldsymbol{D}) = \frac{\pi(\boldsymbol{D} \mid \boldsymbol{\theta}) \, \pi_0(\boldsymbol{\theta})} {\pi(\boldsymbol{D})}. \]

Under Gaussian error assumptions, the likelihood becomes

\[ \pi(\boldsymbol{D} \mid \boldsymbol{\theta}) = \frac{1}{(2\pi)^{N_d} \sigma_{E'}^{N_d} \sigma_{E''}^{N_d}} \exp\left( -\sum_{i=1}^{N_d} \left[ \frac{r_{i,E'}^2}{2\sigma_{E'}^2} + \frac{r_{i,E''}^2}{2\sigma_{E''}^2} \right] \right), \]

with logarithmic residuals

\[ r_{i,E'} = \log\!\left(\frac{E'(x_i)}{\mathcal{M}_{E'}(x_i, \boldsymbol{\theta}_m)}\right), \quad r_{i,E''} = \log\!\left(\frac{E''(x_i)}{\mathcal{M}_{E''}(x_i, \boldsymbol{\theta}_m)}\right). \]

This formulation is consistent with the deterministic calibration objective while incorporating uncertainty.

Prior Distributions

Weakly-informative uniform priors are assigned:

\[ \theta_i \sim \mathcal{U}(a_i, b_i), \quad (a_i, b_i) = (\mu_i - \sqrt{3}\sigma_i,\; \mu_i + \sqrt{3}\sigma_i), \quad \sigma_i = 0.2\,\mu_i. \]

Error parameters are assigned broad priors:

\[ \sigma_{E'} \sim \mathcal{U}(0,1), \quad \sigma_{E''} \sim \mathcal{U}(0,1). \]

Dimensionality Reduction

Sensitivity analysis identifies \(\{\tau_{c_1}, \tau_{c_2}, \beta_1\}\) as non-influential; these are fixed at deterministic values. A physical constraint enforces

\[ E_{c_2} = E_{c_1} \left(\frac{\tau_{c_1}}{\tau_{c_2}}\right)^2, \]

reducing the inference problem to

\[ \boldsymbol{\theta} = [E_{c_1}, \alpha_1, \alpha_2, \sigma_{E'}, \sigma_{E''}]. \]

Please refer to our previous and current papers regarding the sensitivity analysis, influential parameter identification, and proposed constraint that links the phases.

Sampling

Posterior inference is performed using PyMC with the No-U-Turn Sampler (NUTS). Four chains are run with 4,000 tuning iterations and \(10^6\) posterior samples per chain, enabling efficient exploration of the parameter space.

Schematic of the Bayesian Calibration Framework

Figure 1: Graph representation of the Bayesian calibration framework for the FMM–FMG model