Multicriteria estimation of probabilities on basis of expert non-numeric, non-exact and non-complete knowledge
Introduction
Theory and practice of decision aiding by use of decision support systems (DSS) pose a common problem of alternatives’ probabilities estimation, the information on these probabilities being taken from sources of different “weight” (relevance, significance, value, etc.) – e.g., see contemporary surveys (Figueira et al., 2005, Turban et al., 2005). A decision-maker (DM) aggregates an alternatives’ probability estimations into a resulting estimation taking into account weights of all sources of information. In some cases, statistical methods can be applied to estimate probabilities of the alternatives. But often a DM has insufficient amount of historical data or believes that future alternatives’ probabilities will change sufficiently, what makes all historical data out of date. In such case of statistical information deficiency, a DM may appeal to experts as to needed sources of information.
Experts rarely can provide precise numeric estimations and usually give information about alternatives’ probabilities in the form of comparative statements like “alternative Ai is more probable than alternative Aj” or “alternative Ai has the same probability as alternative Aj”. In a similar manner a DM, treated as a “super-expert”, formulates his/her knowledge about weights of information-sources (i.e., experts): “the probabilities estimations, given by kth expert, are more weighty then analogous estimations, given by lth expert”, “the weights of probabilities estimations, given by kth and lth experts, are equal”. Such non-numeric (ordinal) character of human estimations of probabilities and weights was explicitly expressed at least as early as in times of Keynes, 1952, Knight, 1921, who have predecessors (Fioretti, 2001).
Numerous psychological experiments and practical observations of last decades showed that the ordinal form of information representation is rather natural for experts – e.g., see (Barron and Barrett, 1996, Coletti, 1994, Kirkwood and Corner, 1993, Laine et al., 1986, Larichev, 1992, Moshkovich et al., 2002, Schoemaker and Waid, 1982, Weymark, 1997). Furthermore, an attempt to force experts and/or a DM to express their knowledge about probabilities and weights in precise numerical form may result in misleading assessment and biases (Erev and Cohen, 1990, Hogarth, 1975, Kirkwood and Corner, 1993, Larichev and Moshkovich, 1997, Tversky and Kahneman, 1974).
Another widespread model of imprecise subjective probabilities and weights representation supposes that experts and a super-expert (DM) have information on intervals where values of the probabilities and/or weights may vary (Engemann and Yager, 2001, Yager and Kreinovich, 1999). Such interval (non-exact) information (knowledge) may be joined with the above-mentioned ordinal (non-numeric) expert information (knowledge). But even this joined information may be incomplete, i.e. an amount of this information is not enough for unambiguous (unique) determination of probabilities and weights under consideration. So, it will be realistic to suppose that a DM has at his/her disposal only non-numeric (ordinal), non-exact (interval) and non-complete (incomplete) expert knowledge (NNN-knowledge, NNN-information) about alternatives’ probabilities and weights of different information sources (Hovanov et al., 1994).
Such NNN-information on probabilities and weights often is treated in a DSS by appropriate non-numeric methods directly – see, e.g., rather detailed survey (Moshkovich et al., 2005) of similar methods, which are used in VDA (Verbal Decision Analysis). But our wide practical experience assured us that a more transparent (for a DM) results may be achieved by an indirect method ofNNN-information treating. Namely, we propose, following basic idea of well known Th. Bayes’ work (Bayes, 1958), to model uncertain choice of admissible (from the point of view of appropriate NNN-information) probabilities and weights by a random choice from corresponding sets of probabilities and weights – see, e.g., (Hovanov et al., 1999). By such Bayesian randomization of uncertainty we obtain random alternatives’ probabilities and random experts’ weights, mathematical expectations of these numerical random variables being interpreted as the needed numerical image of corresponding NNN-information. Additional statistical characteristics of the random probabilities and weights (variance, standard deviation, probability of stochastic dominance, etc.) may aid to a DM to evaluate quality of resulting numerical estimations of alternatives’ probabilities and experts’ weights – cf., e.g., (Lahdelma et al., 1998, Lahdelma et al., 2003, Miettinen and Salminen, 1999, Rietveld and Ouwersloot, 1992).
In Section 1, we present a general scheme of alternatives’ probabilities estimation on a base of NNN-information, taken from a single expert (from a super-expert, i.e. DM). Further, we propose a simple technique for aggregation of these random probabilities’ estimations taking into account randomized weights of corresponding experts (Section 2). An algorithm of all admissible probability-vectors (weight-vectors) generation and corresponding calculating formulas are presented in Section 3. In the end of the paper (Section 4), a “real life” example illustrates how probabilities of alternatives of oil shares price movements may be estimated by NNN-information obtained from different investment firms. In Section 6, main results of the paper are outlined.
Section snippets
General scheme of the method
Consider at a present time-point t1 a complex system, which can proceed to one of the finite number of alternatives A1, … , Ar at a future time-point t2. Suppose that a DM has m different sources of information (experts) about probabilities pi = P(Ai), i = 1, … , r, pi ⩾ 0, p1 + ⋯ + pr = 1 of the system transition into alternative states A1, … , Ar at the time-point t2. Each source of information can provide two types of expert knowledge. First, an expert can express his/her opinion with comparative statements like
Aggregation of random probabilities with random weights
Consider a matrix , of randomized estimations of alternatives’ probabilities. Rows of this matrix are random vectors and their components can be considered as randomized estimations of alternatives’ probabilities according to NNN-information Ij, obtained from corresponding jth source of information. Transposed column of the matrix is a random vector with components which are randomized estimations , of
The method’s algorithm
On the base of formulas (2), (3) we can split calculation of probabilities’ estimations into two stages. On the first stage, NNN-knowledge I1, … , Im from m different sources of information is taken into account and matrix , of mean probabilities’ estimations is built. Transposed column of matrix is multi-criteria estimation of probability pi of alternative Ai. On the second stage, obtained estimations
Example: Oil share price forecasting
Let us consider an example of the method application to oil share price change forecasting. We will examine share of LUKoil company (Russia). For the share price movement from January 1, 2005 (“present” time-point t1) to March 1, 2005 (“future” time-point t2) five (r = 5) alternatives (intervals) A1, … , A5 are fixed:
A1 = [20, 35] – price increases significantly (from 20% to 35%),
A2 = [5, 20] – price increases (from 5% to 20%),
A3 = [−5, 5] – price does not change significantly (increases or decreases for
Conclusions
In the paper an important problem of alternatives’ probabilities estimation on the base of information from various sources (expert) is considered. It is assumed that expert information about alternatives’ probabilities can be expressed with verbal statements like “alternative Ai is more probable than alternative Aj” or “alternative Ai has the same probability as alternative Aj” and formalized in form of system of equalities and inequalities for components of probability-vector. Obtained type
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