Definition of Term: Bayesian

The adjective ‘Bayesian’ refers to ideas and methods arising in the field of Bayesian Statistics.

Bayesian statistics is an approach to constructing statistical inferences that is fundamentally and philosophically different from the approach that is more commonly taught, known as frequentist statistics.

In Bayesian statistics, all inferences are probability statements about the true, but unknown, values of the parameters of interest. The formal interpretation of those probability statements is as the personal beliefs of the person providing them.

The fact that Bayesian inferences are essentially personal judgements has been the basis of heated debate between proponents of the Bayesian approach and those who espouse the frequentist approach. Frequentists claim that Bayesian methods are subjective and that subjectivity should have no role in science. Bayesians counter that frequentist inferences are not truly objective, and that the practical methods of Bayesian statistics are designed to minimise the undesirable aspects of subjectivity (such as prejudice).

As in any scientific debate, the arguments and counter-arguments are many and complex, and it is certainly not the intention of this brief definition to go into any of that detail. In the context of the MUCM toolkit, the essence of an emulator is that it makes probability statements about the outputs of a simulator, and the frequentist approach does not formally allow such statements. Therefore emulators are necessarily Bayesian.

In the MUCM field, there is a recognised alternative to the fully Bayesian approach of characterising uncertainty about unknown parameters by complete probability distributions. This is the Bayes linear approach, in which uncertainty is represented only through means, variances and covariances.