Mathematical Methods of Subjective Modeling in Scientific Research: I. The Mathematical and Empirical Basis

Abstract

mathematical formalism for subjective modeling, based on modelling of uncertainty, reflecting unreliability of subjective information and fuzziness that is common for its content. The model of subjective judgments on values of an unknown parameter x ∈ X of the model M(x) of a research object is defined by the researcher–modeler as a space¹ (X, p(X), $P{I^{\bar x}}$, $Be{l^{\bar x}}$) with plausibility$P{I^{\bar x}}$ and believability $Be{l^{\bar x}}$ measures, where x is an uncertain element taking values in X that models researcher—modeler’s uncertain propositions about an unknown x ∈ X, measures $P{I^{\bar x}}$, $Be{l^{\bar x}}$ model modalities of a researcher–modeler’s subjective judgments on the validity of each x ∈ X: the value of $P{I^{\bar x}}(\tilde x = x)$ determines how relatively plausible, in his opinion, the equality $(\tilde x = x)$ is, while the value of $Be{l^{\bar x}}(\tilde x = x)$ determines how the inequality $(\tilde x = x)$ should be relatively believed in. Versions of plausibility Pl and believability Bel measures and pl- and bel-integrals that inherit some traits of probabilities, psychophysics and take into account interests of researcher–modeler groups are considered. It is shown that the mathematical formalism of subjective modeling, unlike “standard” mathematical modeling, •enables a researcher–modeler to model both precise formalized knowledge and non-formalized unreliable knowledge, from complete ignorance to precise knowledge of the model of a research object, to calculate relative plausibilities and believabilities of any features of a research object that are specified by its subjective model $M(\tilde x)$, and if the data on observations of a research object is available, then it: •enables him to estimate the adequacy of subjective model to the research objective, to correct it by combining subjective ideas and the observation data after testing their consistency, and, finally, to empirically recover the model of a research object.