# Bayesian Updating for Random Tensile Force Identification of Ancient Tie Rods Using Modal Data
## 2 Mathematical Setting for Bayesian Updating Framework Using Dynamic Data
Gathering the model and the measurement error terms \(\mathbf{e}_{\mathbf{m}}(\mathbf{\mathbf{\xi}})\) and \(\mathbf{\overline{e}}_{m}\) in Equation (3) in the total error term \(\mathbf{\mathbf{e}}\) and considering that the object of inference is the real valued random vector \(\mathbf{\Theta}\in\mathbb{R}^{N}\), the likelihood function can be modeled assuming a normally distributed total error via marginalization as  
\[p(\mathbf{\mathbf{D}}|\mathbf{\Theta},\mathbf{x})=\int p(\mathbf{\mathbf{D}}|\mathbf{e},\mathbf{ \mathbf{\Theta}},\mathbf{x})p(\mathbf{e}|\mathbf{\mathbf{\Theta}},\mathbf{x})d\mathbf{e}, \tag{5}\]  
and the posterior distribution of the updating parameters \(\mathbf{\Theta}\) is given by  
\[\begin{split} p(\mathbf{\Theta}|\mathbf{\mathbf{D}})&=\int p (\mathbf{\Theta},\mathbf{x}|\mathbf{\mathbf{D}})d\mathbf{x}\\ &=\frac{\int p(\mathbf{\mathbf{\Theta}})p(\mathbf{x})p(\mathbf{\mathbf{D}}| \mathbf{\mathbf{\Theta}},\mathbf{x})d\mathbf{x}}{\int\int p(\mathbf{\mathbf{\Theta}})p(\mathbf{ \mathbf{\Theta}})p(\mathbf{\mathbf{D}}|\mathbf{\mathbf{\Theta}},\mathbf{x})d\mathbf{x}  
The determination of the likelihood function and the posterior distribution of the updating parameters, \(\mathbf{\Theta}\), is a crucial issue since a large number of numerical evaluations of the deterministic computational model solution \(\hat{\mathcal{M}}(\mathbf{\mathrm{m}},\mathbf{\mathrm{\xi}})\) are required, making the whole procedure computationally unfeasible when complicated computational models are used, when a large space of input parameter \(\mathbf{\mathrm{\xi}}\) is considered and when a large space of reference data, \(\mathbf{\mathrm{D}}\), is used as a target for the updating procedure. The Markov Chain Monte Carlo (MCMC) method [43; 44] is one of the most widely used method for posterior sampling. The term MCMC describes all procedures based on stationary chains of samples to approximate the posterior parameter distributions. Metropolis Hastings (MH) is the most popular MCMC method [43; 45]. It is based on generating samples from a target posterior probability distribution up to a constant factor in three main steps: (i) choose an initial state from prior distributions; (ii) generate a candidate state by making a random walk from the current state using a proposal distribution (e.g., a normal distribution centered at the current state); (iii) calculate an acceptance probability ratio for the candidate state and accept it or reject it as the current state of the chain. The algorithm is able to generate a Markov Chain and after a burn-in period the accepted samples converge to the target posterior distribution of the updating parameters, \(\mathbf{\Theta}\). When latent random variables are considered in the mathematical formulation of the probabilistic model in Equation (3), the MCMC-MH method allows one to obtain samples from a posterior distribution defined on an augmented space, \(p(\mathbf{\mathrm{\xi}}|\mathbf{\mathrm{D}})=p(\mathbf{\Theta},\mathbf{x})|\mathbf{D})\propto p( \mathbf{\Theta},\mathbf{x})p(\mathbf{\mathrm{D}}|\mathbf{\Theta},\mathbf{x})\). The MCMC-MH method is summarized in Algorithm 1. ```
Data: Initiate the algorithm with a value \(\mathbf{\Theta}^{0}\) for Each \(t=1\to N\)do  Draw \(\mathbf{\Theta}^{*}\) from the proposal distribution density function  Draw \(\mathbf{x}^{*}\) from the proposal distribution density function  Estimate the computational model solution for \(\hat{\mathcal{M}}(\mathbf{\mathrm{m}},\mathbf{\Theta}^{*},\mathbf{x}^{*})\)  Compute the acceptance rate \(\rho(\mathbf{\Theta}^{*},\mathbf{\Theta}^{t})\)  Draw \(u\) from a uniform distribution \(u\)\(\sim\)\(\mathcal{U}(0,1)\) if\(u<\rho(\mathbf{\Theta}^{*},\mathbf{\Theta}^{t})\)then  Accept the state \(\mathbf{\Theta}^{t+1}=\mathbf{\Theta}^{*}\) else  Reject the state \(\mathbf{\Theta}^{t+1}=\mathbf{\Theta}^{t}\)
```  
**Algorithm 1**Metropolis Hastings (MH) MCMC algorithm.