Bayesian calibration, process modeling and uncertainty quantification in biotechnology
Fig 10
Independent variable PDFs in various observation scenarios.
Posterior densities inferred from various numbers of observations corresponding to different biomass concentrations are shown (A). The ends of the drawn lines in A indicate the 95% equal-tailed interval. Near biomass concentrations of 0, the posterior density is asymmetric (A, blue), indicating that very low concentrations cannot be distinguished. As the number of observations grows, the probability mass is concentrated and the ETIs shrink (A, oranges). The choice of a Student-t distribution model can lead to a multi-modality of the inferred posterior density when observations lie far apart (B). For asymmetric distributions, the median (dashed line) does not necessarily coincide with a mode and equal-tailed and highest-density intervals (ETI, HDI) can be different. Maximum likelihood estimates from individual observations, as obtained via predict_independent are shown as arrows. Note: and the model’s ν parameter were chosen at extreme values for illustrative purposes.