## 2 Materials and Methods
### Bayesian Calibration Procedure  
In this section, we introduce the key steps of the entire procedure used for the calibration of the stochastic systems examined below (for the numerical benchmark and for the case study). Specifically, a three-phase approach was used as follows:  
* Phase 1: definition of the seismic input excitation and of the computational model. First, a ground earthquake acceleration record must be selected as seismic input excitation of the system. Then, one specifies general options for the physical model and its governing laws. In this work the system of Ordinary Differential Equations (ODEs) governing the Bouc-Wen hysteretic oscillator (Section 3) are implemented and solved numerically with the explicit Runge-Kutta method.
* Phase 2: definition of the probabilistic prior information on the model parameters. Once the computational model \(\mathcal{M}(\mathbf{x})\) is defined, one has to select carefully which model parameters \(\mathbf{x}_{\mathbf{M}}\) to include in the calibration in order to get a reliable set of physical values from the resulting posteriors estimates after the Bayesian updating. Prior information on possible values of the hyperparameters are set by setting their prior probability distributions \(\pi(\mathbf{x}_{\mathbf{M}})\), defining for each hyperparameter the type of univariate distribution (i.e., uniform, Gaussian, lognormal distributions, etc.) and its statistical moments. The prior information is obtained making some considerations about the amount of the dissipated energy during the hysteresis.
* Phase 3: Bayesian model updating. At this stage, the Bayesian model updating can be carried out using the experimental data \(\mathbf{Y}\) inferring the posterior distributions of the hyperparameters. However, in many practical applications, a closed form of Equation (7) does not exist. For this reason, Markov Chain Monte Carlo (MCMC) simulations have been conducted herein, allowing for an approximate expectation in Equation (8).  
Specifically, the posterior distribution can also be seen as an intermediate quantity used to compute the conditional expectation of a certain quantity of interest (QoI) \(h(\mathbf{x}):\;D_{\mathbf{x}}\mapsto\mathbb{R}\)[27], so that:  
\[\mathbb{E}\Big{[}h(\mathbf{X})\big{|}\mathbf{Y}\Big{]}=\int_{D_{\mathbf{x}}}h(\mathbf{x})\;\pi (\mathbf{x}|\mathbf{Y})\;d\mathbf{x}\approx\frac{1}{T}\sum\
olimits_{t=1}^{T}h\Big{(}\mathbf{ x}^{(t)}\Big{)}, \tag{8}\]  
where \(\mathbf{x}^{(t)}\) is the step of the chain at iteration \(t\), whereas \(T\) is the total number of the generated MCMC sample point. In this way, the posterior is explored by realizing appropriate Markov chains over the prior support. Specifically, the affine invariant ensemble sampler (AIES) solver algorithm [28] has been used in this work, and an additive Gaussian discrepancy with zero mean and residual diagonal covariance matrix \(\mathbf{\Sigma}=\sigma^{2}\mathbf{I}\) with unknown residual variances \(\sigma^{2}\) has been introduced to connect the model response to the experimental observations. To infer \(\sigma^{2}\), a uniform prior distribution was assumed:  
\[\pi\Big{(}\sigma^{2}\Big{)}\sim U\Big{(}0,max|\mathbf{\gamma}|^{2}\Big{)}, \tag{9}\]  
whose standard deviation was set equal to the maximum of the absolute value over the time of the experimental observation \(\mathbf{Y}\) at the beginning of the calibration. The Bayesian updating is then conducted by minimizing this value (i.e., maximizing the likelihood function of Equation (6)) until the convergence of the Markov chains is reached. The obtained sample can be used to estimate output statistics by drawing samples from the posterior. Finally, having obtained model parameters posterior distributions, a deterministic model response prediction can be estimated choosing the maximum a posteriori (MAP) or the mean of the distributions as a point estimate. The aforementioned approach is summarized by the flowchart in Figure 1.