High fidelity mathematical model building with experimental data: A Bayesian approach

Abstract

Mathematical models of physicochemical systems are usually built in an iterative fashion during the course of an experimental investigation. In this paper, a novel Bayesian approach to model building is presented. This approach is now feasible because of breakthroughs in Monte Carlo sampling procedures and high performance computing, that make it possible to deal directly with the nonlinear mathematical models themselves instead of their linear approximations. By including an error model for experimental data, it is further possible to use nonlinear statistical concepts to test a given model for adequacy against experimental data and prior knowledge, and to place realistic confidence limits on the resulting model parameters. In this paper a model building work flow that takes advantage of these recent advances to enable high fidelity mathematical modeling is proposed. A set of models and their parameters are needed to initiate the process. Probability distributions for the models and their parameters based on available quantitative and subjective information must also be supplied. Finally, an error model describing the heteroscedasticity in the data along with probability distributions for the error model parameters must be generated from exploratory data. Then experiments are designed and data collected. Using Bayes’ theorem, Monte Carlo (MC) or Markov Chain Monte Carlo (MCMC) methods are used to generate a sequence of samples of parameter values for each postulated model. These sets of samples are then used to discriminate among the models using the criteria introduced in this paper. Once discrimination is achieved, a global lack of fit test is introduced to determine model adequacy. After a single adequate model is selected, highest probability density (HPD) intervals are determined for the individual parameters and HPD density regions are constructed for all model parameter pairs. Experiments are then designed to reduce the uncertainty in the joint posterior probability HPD regions. Finally, a sampling procedure is described to properly represent uncertainties in predictions made from the model. The proposed approach is demonstrated by an illustrative problem where three simple models are discriminated and the parameters in the most suitable ones are estimated rigorously.