## 5 Discussion  
The output of the inverse analysis is influenced by the ranges of the model spaces. The solution derived here from the inverse analysis is not unique [27]. The process of determining the likelihood function is essentially the process of optimizing the local model space. The prior information determines the range of the model space and influences the characteristics and numbers of the solution. As a result, the prior information concerning the parameters should be evaluated properly first.  
It is difficult to establish a proper relationship between monitoring data and structural conditions, especially for the relationship between concrete stress and monitored strain. In the pre-stressed concrete continuous beam, stress was influenced by many factors, such as the construction sequence, vehicle load, initial and secondary stresses caused by pre-stress, pre-stress loss due to creep and shrinkage of concrete, etc. These factors have effects on the stress \(\sigma(t)\) in the stress-dependent strain formula. This will generate instantaneous elastic strain and creep strain. Meanwhile, the generated strain will influence the stress through pre-stress loss or secondary stress. The relation between strain and stress in concrete beam is iteratively coupled. In this study, the concrete stress at different construction stages--instead of the process of stress development--is considered. The obtained stress can be compared with concrete strength to conduct structure evaluation. However, the constitution of the stress in concrete is ignored.  
Through inversion analysis, posterior stress and the constitution of strain can be obtained, as shown in Figures 6 and 8. On one hand, the case study verified the effectiveness of the proposed method. On the other hand, the result can be referred for other pre-stress concrete beam bridge to conduct rational structural design. The proposed method can decompose the monitoring data and determine the weight proportion of each component. This is important for monitoring the structural performance and modifying prediction models. It can be indicated from Figure 8 that creep strain dominates the total strain of concrete structures. This is helpful for long-term performance evaluation of bridges with other structural types, such as simply supported PC bridge decks or steel-concrete composite sections.  
Figure 8: Components of the total strain.  
Additionally, unlike conventional methods, the solution derived from the proposed method is not unique. For a definitive system, the relationship among the physical variables is unique. However, for the case stated in this paper, the relationship between the model space and data space is not unique, and the latter is a subspace of the former. The stress based on the monitored strain is not unique. From a mathematical point of view, a monitored strain may correspond to multiple stresses. In practice, when the actual information is not sufficient, the specific quantile value of stress can be used to identify the adverse structural condition (such as a crack occurring) in maintenance decision making.  
Accurate stress estimation is of great important for bridge design and maintenance. On the one hand, the obtained posterior information of concrete stress can be compared with the predicted stress in the design stage, which will provide guidance for designers updating stress prediction models. On the other hand, stress in concrete is an important index for the health evaluation of existing bridges. Through an inversion analysis based on the monitored strain, the initial stress of an as-built bridge and time-dependent stress curve could be obtained. This is helpful for making decisions in bridge management. Moreover, the proposed method can be used for evaluating other structural parameters based on monitored data, such as strength reduction in the concrete due to structural degradation.