# Bayesian Updating for Random Tensile Force Identification of Ancient Tie Rods Using Modal Data
## 2 Mathematical Setting for Bayesian Updating Framework Using Dynamic Data  
Consider a structural system (e.g., beam, frame, building) whose dynamic modal behavior can be modeled using a set of partial differential equations. The computational model, \(\mathcal{M}\), is characterized by a set of known measurable quantities gathered in the vector \(\mathbf{m}\) and by a real valued random vector, \(\boldsymbol{\xi}=\{\boldsymbol{x},\boldsymbol{\Theta}\}\in\mathbb{R}^{P+N}\), consisting of a set of latent independent random variables, \(\mathbf{x}=\{x_{1},\ldots,x_{p}\}\in\mathbb{R}^{P}\), and an input random vector, \(\mathbf{\Theta}=\{\theta_{1},\ldots,\theta_{N}\}\in\mathbb{R}^{N}\). The probabilistic model can be written as  
\[\mathcal{M}(\mathbf{\mathbf{m}},\mathbf{\mathbf{\xi}})=\hat{\mathcal{M}}(\mathbf{\mathbf{m }},\mathbf{\mathbf{\xi}})+\mathbf{e}_{\mathbf{m}}(\mathbf{\mathbf{\xi}}), \tag{1}\]  
where \(\mathcal{M}(\mathbf{\mathbf{m}},\mathbf{\mathbf{\xi}})\) is the computational model output dynamic response or its transformation, and \(\hat{\mathcal{M}}(\mathbf{\mathbf{m}},\mathbf{\mathbf{\xi}})\) is the deterministic model output dynamic response or its transformation, \(\mathbf{e}_{\mathbf{m}}(\mathbf{\mathbf{\xi}})\) is the total model error term. Further details about this mathematical formulation can be found in [40; 41; 42]. Let us denote by \(\mathbf{\mathbf{D}}=\{D_{1},\ldots,D_{m}\}\in\mathbb{R}^{M}\) some observations related to the dynamic modal behavior of the system (i.e., natural frequencies). The observations typically differ from the real process for some measurement errors. Thus, the probabilistic description can be written as  
\[\mathbf{\mathbf{D}}=\mathcal{M}(\mathbf{\mathbf{m}},\mathbf{\mathbf{\xi}})+\mathbf{\overline{ e}}_{m}, \tag{2}\]  
where \(\mathbf{\overline{e}}_{m}\) is the total measurement error term, modeled using a normal random variable with zero mean and unit variance. Substituting Equation (1) into Equation (2), the following description of the mathematical framework for the uncertainty quantification problem is obtained:  
\[\mathbf{\mathbf{D}}=\hat{\mathcal{M}}(\mathbf{\mathbf{m}},\mathbf{\mathbf{\xi}})+\mathbf{e}_{ \mathbf{m}}(\mathbf{\mathbf{\xi}})+\mathbf{\overline{e}}_{m}. \tag{3}\]  
In the Bayesian context, the probabilities of the unknown parameters to be updated given knowledge of observations \(\mathbf{\mathbf{D}}\in\mathbb{R}^{M}\) are given by the joint PDF using [PERSON] theorem,  
\[p(\mathbf{\mathbf{\xi}}|\mathbf{\mathbf{D}})=\frac{p(\mathbf{D}|\mathbf{\mathbf{\xi}})p(\mathbf{ \mathbf{\xi}})}{\int p(\mathbf{D}|\mathbf{\mathbf{\xi}})p(\mathbf{\mathbf{\xi}})d\mathbf{\mathbf {\xi}}}, \tag{4}\]  
which are known as posterior distributions. The term \(p(\mathbf{\mathbf{\xi}})\) denotes the prior probability distribution of the unknown parameters, which quantifies the initial plausibility of the initial hypotheses on the vector \(\mathbf{\mathbf{\xi}}\). The term \(p(\mathbf{D}|\mathbf{\mathbf{\xi}})\) is the likelihood function and quantifies the probability of the data conditional to the uncertain parameters resulting from the probabilistic model assumed for the total error. Finally, the denominator makes the integration of the posterior joint probability distribution over the parameter space equal to 1 [40].