On independence, exchangeability, and logical correlation of random variables in metrology

Abstract

Information about the set of input quantities to a measurement model comprises statements about correlation of the random variables associated with the quantities. In a Bayesian framework, underlying internationally agreed evaluation procedures applied to measurement data, the correlation coefficients, or equivalently the covariances, of a joint probability density function, PDF, are fixed and calculable parameters. It will be shown that correlation often is due to logical inference and not necessarily expresses physical cause and effect. A Bayesian understanding of measurements under repeatability conditions is presented, finally leading to the replacement of (complete) independence within the sequence of random variables generating the observations with a conditional independence, which means a hidden correlation of the random variables in the sequence. The concept of an exchangeable joint PDF is introduced to briefly discuss the relation of measurements under repeatability conditions to de Finetti’s purely mathematical General Representation Theorem that, moreover, calls for a Bayesian approach.