Mechanical equivalent of logical inference: Application to monitoring data analysis

Structural health monitoring requires engineers to understand the state of a structure from its observed response. When this information is uncertain, Bayesian probability theory provides a consistent framework for making inferences. However, structural engineers are often unenthusiastic in regards to using Bayesian formal logic, finding its application complicated and burdensome, and prefer to make inference using heuristics. Here, we propose a quantitative method for logical inference based on a formal analogy between linear elastic mechanics and Bayesian inference with liner Gaussian variables. We investigate the case of two parameters estimation, where the analogy is stated as follows. The expected value of the parameters corresponds to the position and rotation of a bar (with two degrees of freedom). An uncertain information is modelled as a linear elastic spring of stiffness and pre-stretch equal to its inverse-variance and mean value, respectively. The resulting position of the bar corresponds to the posterior mean value of the first parameter and the resulting rotation to the posterior mean value of the second parameter. The resulting translational and rotational stiffness of the bar gives us the posterior standard deviation of each parameter. We show how, through this analogy, we can easily reproduce a complex inference scheme and solve it using the classical methods of structural mechanics.