Inference from Inadequate and Inaccurate Data, I

This paper describes how one can proceed when E is adequately described by one member m(E) of a Hilbert space [unk] of possible models of E, when he believes that the Hilbert norm of m( E) is very likely rather smaller than some known number M, and (except for section 6) when all the observed and sought-after properties of E are continuous linear functionals on [unk].

Abstract

Having measured D numerical properties of a physical object E which requires many more than D parameters for its complete description, we want to estimate P other numerical properties of E. Continuing the discussion in papers I(1) and II,(2) the present paper gives estimates when we believe it likely that we can guess an upper bound M on the Hilbert norm not of h(E), the model representing E in some Hilbert space, but of the orthogonal projection of h(E) onto a sufficiently large subspace. In addition, the present paper simplifies the notation of I, and makes explicit the application of Bayesian subjective probability when there are errors in the data and we want to find the joint probability distribution of more than one prediction.