Gibbs Sampling as a Natural Statistical Analog of Constraints Techniques: Prediction in Science under General Probabilistic Uncertainty

One of the main objectives of science is to predict future events, i.e., more precisely, the results of future measurements and observations. If we take into account the probabilistic uncertainty related to the inaccuracy of the measurement results, to the inaccuracy of the model, and to the inaccuracy of the prior information, then the most adequate approach is to generate a posterior distribution by using Bayes’ theorem. For the simplest posterior distributions, we can deduce explicit analytical formulas for the resulting statistical characteristics (mean, standard deviation, etc.) of the predict future measurement result. However, in general, such formulas are not possible, so we have to use a Monte-Carlo simulation of the corresponding joint distribution of the future measurement results and model parameters.