# Investigating Bayesian Parameter Identification Using Non-Standard Laboratory Specimens
## 1 Introduction
Finally, the existence of various uncertainties, such as measurement and numerical model errors cannot be incorporated into the direct laboratory identification process which leads to unreliable parameter identification. The limitations of relating parameters to direct laboratory measurements can be overcome by solving an inverse problem, where numerical model predictions are compared to experimental measurements to identify the optimal set of model parameters. There are two distinct frameworks for solving the inverse problem: deterministic and probabilistic [16,17]. Deterministic approaches, also known as optimization techniques, rely on defining a loss function that quantifies the discrepancy between the numerical output of the model and the experimental measurements. The parameter set that minimizes the loss function is considered the \"best\" estimate and is assumed to approximate the true values [8,9,18]. This approach is effective when experimental data are highly informative and the inverse problem is well-posed. However, in more complex settings, especially those involving nonlinearity, deterministic methods have important limitations. They do not account for uncertainties in the identification process, such as measurement inaccuracies and modeling errors. Furthermore, due to the ill-posed nature of many inverse problems, multiple parameter sets may yield similarly low loss values, leading to non-unique solutions. To address this issue, additional constraints or regularization techniques are typically employed to ensure a stable and well-posed solution [9,19]. An example of such a parameter identification procedure with nonstandard metallic specimens by instrumented indentation and small punch can be found in [20; 21; 22]. Probabilistic approaches, on the other hand, represent parameters through probability distributions. This approach ensures that the inverse problem is well defined. A widely used probabilistic method is the Bayesian inverse method (Bayesian inference) [9], which is based on [PERSON]'s theorem of conditional probability. Here, the unknown parameters are treated as random variables, described by a prior distribution based on existing knowledge or experience. The prior distributions are then updated using experimental measurements from a specimen or structure. Notably, the required parameters do not need to be directly measured. Instead, the combinations of simpler measurements can be used to indirectly provide valuable information for parameter estimation. This enables the incorporation of a broader range of experimental data into the identification process. The end result of the identification is the posterior distributions that reflect the updated parameter estimates. The key strength of this approach lies in the ability to account for uncertainties by treating the discrepancy between numerical model predictions and experimental measurements as a variable within the error model. The error model is defined by accounting measurement and modeling errors. By directly incorporating these uncertainties into the identification process, the Bayesian approach provides a more comprehensive and realistic estimation of model parameters [9; 23; 24; 25]. Solving the inverse problem of parameter identification across the full range of material behavior, from the elastic and plastic phases to complete failure, requires advanced nonlinear numerical models and methods. Several approaches have been developed for this purpose, including the Extended Finite Element Method (X-FEM) [26; 27], the Embedded Strong Discontinuity Finite Element Method (ED-FEM) [11; 28; 29], the Phase-Field Method [30; 31], the Lattice Element Method [32; 33; 34; 35] and cohesive zone models [36]. Various applications of Bayesian parameter identification can be found in the literature. Namely, the combined isotropic/kinematic hardening model was calibrated based on the cyclic response of cantilever steel specimens, employing a Gaussian process model [37].