New Zealand Statistical Association 2024 Conference

Robin Willink

University of Otago, Wellington

A logical contradiction when considering an unknown constant to have a probability density function

The idea of attributing a probability density function (pdf) to an unknown constant is popular among some statisticians. But does this idea have a basis in logic? In this talk, we build on a result published in the measurement literature to show that it leads to a logical contradiction. The premise that an objective set of information about a constant can be accurately represented by a probability distribution seems accepted by some in the measurement community. In that context, this premise has recently been shown to be false: an arbitrary non-linear transformation leads to a logical contradiction when two pdfs that are said to represent disjoint sets of information about the same constant are combined [“On revision of the Guide to the Expression of Uncertainty in Measurement: Proofs of fundamental errors in Bayesian approaches”, R. Willink, Measurement: Sensors 24 (2022) 100416; “Paradox? What paradox?”, R. Willink, Accreditation and Quality Assurance, 29 (2024) 189-192]. This talk will extend this result to apply to the concept of subjective probability also – with far-reaching implications. The conclusion relates to the general idea of attributing a probability distribution to a constant that exists on a continuous scale, i.e., to the attribution of a probability density function. There remains the possibility that a constant on a discrete scale can logically be attributed a probability mass function.