Mechanical equivalent of logical inference from correlated uncertain information

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. Particularly, we investigate the case of single parameter estimation, where the analogy is stated as follows: the expected value of the parameter corresponds to the position of a bar with one degree of freedom; an uncertain information on the parameter is modelled as a linear elastic spring of stiffness and pre-stretch equal to its inverse-variance and mean value, respectively; multiple sources of uncertainties on the same information are modelled as serial springs, each with stiffness equal to its variance; the resulting position of the bar corresponds to the posterior mean value of the parameter; the resulting flexibility of the bar corresponds to its posterior variance. We show how, through this analogy, we can easily reproduce a complex inference scheme, including multiple sources of information and correlation, in the form of a series/parallel spring model, and solve it using the classical methods of structural mechanics.