Using Some “prior” When There Is No Prior

Abstract If a prior probability does not exist or is controversial, can we formulate probabilistic inference based only on the properties of the measurement? In the case of plain ‘location’ measurements the answer to this is known. Here we extend that answer to address the general problem of estimating a scalar parameter from a small data sample. That measurement is treated as if resulting from perturbations of a location measurement. A proof is provided, that always a “posterior” without a prior exists and is equal to the posterior based on “Jeffreys’ prior”. The latter density is utilised in this way only, without assuming that it is the real prior, because Jeffreys’ prior may be updated when we combine measurements, or when we revise the stopping rule. Prior-free inference is meaningful only if no prior exists or if it is provisionally suppressed. We discuss in what kind of problems this method is applicable.