| abstract
(eng) |
A quantitative observation assigns numerical values to a phenomen on 𝑝∈𝒑 e.g. a system s property To ensure a proper observation process, any hidden feedback must be avoided. It means that the u ncertainty 𝑢∈𝒖 affect ing the assignment must not depend on the phenomen on itself. Since quantification implicitly involves compar isons e.g. 𝑎 is smaller than 𝑏””, 𝑐 is more desired tha n 𝑑 etc.etc.)), it assume s the existence of a transitive and complete ordering ≼ on 𝒑 It can be shown, that i ts completeness is always attainable under uncertainty. The result [1] implies existence of a continuous, ordering preserving, quantitative observation iff the topology of open intervals in (≺,𝒑) does not require more complexity than the natural order ing of real numbers . Hence , it is possible to distinguish a countable number of realizations of the quantitatively described phenomenon and a countable number of uncertainties that can be associated. Therefore , the observation mapping 𝒪:(𝒑,𝒖)↦𝒐 has a matrix structure 𝒪=[𝑂(𝑝,𝑢)], 𝑝∈𝒑,𝑢∈𝒖 To mitigate the influence of indices corresponding to phenomenon and uncertainty , the s ingular value decomposition (SVD) is applied 𝑂=𝑆𝑉𝑁∗ w it h 𝑁∗ denoting transposition and conjugat ion of 𝑁, [ Structurally, this implies that the uncertainty modelling unitary matrix 𝑁 spans complex Hilbert s space. Subspaces of this space are projected onto quantitative observations in 𝒐. These subspaces represent the relevant, distinguishable random events . Thus, the quantitative observation is to be handled as an observable [ 3]. Th e proposed work elaborates on and discusses this idea The twin work [4] addresses this viewpoint within the context of decision making. It demonstrates that a probabilistic model applied to subspaces model ling uncertainties is appropriate. The present study suggests that the findings of [4] are applicable to any quantitative observation (measurement).\n[1]G. Debreu. Representation of a pr eference ordering by a numerical function. In R.M. Thrall,\nC.H. Coombs, and R.L. Davis, editors, Decision Processes 159 65, Wiley, 1954.\n[2 ] G.H. Golub and C.F. Van Loan. Matrix Computations . Johns Hopkins , Univ. Press, 2012.\n[3] A. Dvurečenskij . Gleasons Theorem and Its Applications Mathematics and Its\nApplications , vol 60 Kluwer Academic Publishers, Dordrecht/Boston/London, 1993.\n[4] A. Gaj and M. Kárný. Quantum like modelling of uncertainty in dynamic decision making. In\nQuantum Information and Probability: from Foundations to Engineering (QIP24), 2024\n |