Now let’s pass on to a discussion of the simplest types of the expert appraisals of the second kind described in the literature.
Appraisals of the type (1.1)
The example of the appraisals of the type (1.1) is a weather forecast in the Russia. For example, "In a number of regions of the West Siberia a transitory warming is possible in the end of April". In this case a degree of a certitude is defined by an identificator "possible", and the expert appraisal of the first kind (a final judgement) was expressed by the use of the words "transitory warming".
Appraisals of the type (1.7)
In a number of the works of the sociologists the expert appraisals of the type (1.7) are discussed. They are the works with the tests of a legibility [137,169] and in the experiments for an identification of signals [160]. In these experiments it is required from the expert, although in this context it is better to speak of the examinee, that in addition to the basic answer expressed in the form of a final opinion he should every time indicate a degree of his certitude in it. The numbers were the form of the expert appraisals in which the examinee estimated his certitude.
In the experimental work of Watson and other authors the examinee fixed a movable sighting device fixed on the ruler according to his own sensation of the confidence degree, and therefore the simplest mechanical device was used in the appraisal experiment [189].
Appraisals of the type (1.9)
The weather forecast is an example of the appraisals of the type (1.9) if confidence is given as a probability of one or another event expressed in a percentage. In spite of complicated mathematical models and dozens of thousands of initial data to be input into computers a short-term weather forecast satisfies us not always. This is partially related to the fact that, unfortunately, for the time being the numerical methods do not ensure a rather exact calculation of the development of the processes in the lowest one of the six layers of the atmosphere. The laws of the development of this layer are by far more complicated than those of the upper layers.
And just here an interference of the expert-prognosist is required. The experts submit their appraisals either of the type (1.1) as in Russia or (1.9) as in the USA and Canada. If, for example, the expert is asked: "What will the weather be exactly in a week?" - then the expert may answer in such a way: "Evidently, exactly in a week there would be a warm day", that is an appraisal of the type (1.1), or "With the 80 per cent confidence I think that it will be a warm day", that is an appraisal of the type (1.9). Generally speaking, the verbal answer is more reliable while taking into account, firstly, that the warm day is a non-exact notion and, secondly, for the time being we understand the dynamics of the weather changing not quite well. On the other hand, the second answer is more informative.
The idea of the weighted weather forecasts was considered by Cooc even in 1906 and was gradually developed into a probability prognostication concept which was seldom used in the beginning, but has already became a generally used in such countries as the USA, Canada [138].
According to the opinion of Sheridan and Ferrell definition of the accuracy of forecasts expressed in such a form [105] is one of the most important but puzzlingproblems. So, for example, if every day to forecast a clear weather in Tucson (Arizona) then such a prognosis will be exact in 86 per cent of cases. The forecast of the clear weather in 86 per cent of cases corresponds to the average data and in this sense is absolutely correct. But certainly it does not indicate a high exactness of the expert prognosis of the specialist.
An attempt to avoid this difficulty was undertaken in the work of Merphi and Winkler [164] when both the ability of a meteorologist to discern real conditions and a correspondence between a probability mentioned in the prognosis and an observed frequency of precipitation were taken into account when defining the quality of the weather forecasts.
Another example of the appraisals of the type (1.9) is using of logical (conditional) links with the indication of a probability of their realization contained in the data base of the MICIN expert system [141].
Appraisals of the type (2.9)
The appraisals of the type (2.9) are discussed, for example, in the work [92]. "Let’s imagine, Yu.N. Tyurin writes, - that a probability is distributed in a probable great number of classifications of a given final multitude of objects. Every expert, while compiling "his" classification independently of ether experts, derives a classification M’ from the multitude {M} with a probability P(M’) ". In case when the probability P(M’) is given by the expert as a degree of an authenticity of his classification M’ we are getting the appraisal of the type (2.9).
Appraisals of the type (2.10)
As an example of the appraisals of the type (2.10) it is possible to consider the appraisals in which the expert defines an entire group of events (A1, A2, ..., An) and the vector of probabilities p = (p1, p2, ……pn) while indicating forevery Ai a probability pi = p(Ai) in such a way that the sum ∑1n pi = 1. Such appraisals may be considered as the p – mixtures [63].
Appraisals of the type (3.7)
The appraisals of this type are discussed, for example, by B.G. Mirkin. In this case the expert is supposed to appraise an intensity of his preference in figures for every pair of objects [58].
Appraisals of the type (3.9)
In the case when an expert makes a pair comparison A1 > A2 and a probability p(X1 > X2) of the fact that the object A1 is more preferable than the object A2 where X1, X2 are the values of the objects A1 and A2 in the psychological continuum of the expert we are obtaining the appraisals of the type (3.9).
The statistical models of the pair comparisons are discussed in numerous works beginning from the work of Terstone [184]. The work of Yu.N. Tyurin, A.P. Vasiljevich, P.P. Androukovich [94] is an example of experimental research works in this direction.
Appraisals of the type (4.9)
By analogy with the appraisal of the type (3.9) the experts suggest an appraisal obtained with the multiple comparisons method, and in addition they give an estimation of a probability as a confidence degree of their appraisal. The appraisals of this type are described in the review [108].
Appraisals of the type (5.9)
As well as the appraisals of the type (3.9) and (4.9) the – appraisals of the type (5.9) are obtained in that case when an expert gives a ranging and an estimation of a probability as a degree of a certitude in his appraisal. A review of statistical models of the ranging is discussed in the works of D.S. Shmerling and others [108].
Appraisals of the type (10.10)
The appraisals of this type are, in particular, discussed in the work [36] of Yu.V. Kisselev. He gave the following grounds of a transition from the three-point appraisals to the appraisals of the type (10.10). "If an expert can appraise three characteristic points of a distribution then why it is impossible to go further and to build a pseudo-statistical function of a distribution while considering it as a result of a mental experiment". As Yu.V. Kisselev believes this fact would allow to give up any artificial assumptions concerning the form of a distribution law and its parameters.
yu.V. Kisselev made experiments on a formation pseudo-statistic distribution functions in conformity with an unknown time which is necessary for realization of a certain technical operation. The method of forming a pseudo-statistical function for a distribution was the following. The expert selected six or eight points on the time axis when the left point corresponded to a minimum probable value, but the right one - to maximum probable value of time necessary for a realization of an operation. Then for every chosen point the expert assigned a number in the interval 0 - 100 which served as an answer to the question how many chances out of 100 according to the opinion of the expert that the operation would be realized during a period which is less than that one which is defined by the given point.
In other words, the pseudo-statistic integral distribution function was defined with the number of points from six to eight. Ten engineers engaged in the planning of the balancing and commissioning in putting complexes of the new techniques into operation were used as experts. In conformity with the operations of such a kind every expert was forming a pseudo-statistical distribution function for the realization period of 10 operations. The obtained distribution functions were processed with the use of ordinary statistical methods.
The experiments showed that such an approach is acceptable only for the experts able to use the fundamentals of the theory of probability. The experts who are not able to use mathematical methods demonstrate an obvious tendency towards the use of linear interpolation when appraising the ordinates (chances) of the intermediate points. When taking the fact into account that the methods of drawing the pseudo-statistical functions of distribution by the experts, firstly, does not require a practical knowledge of the theory of chances and, secondly, leads to a sufficient complication of calculationsin its mass applications then its use in a general case looks like to be inexpedient according to the opinion of yu.V. Kisselev. Such an approach should be recommended only under the condition of an evident inapplicability of the beta-distribution model. In practice, it is usually taking place in the situations when it is intuitively clear that the function of the distribution density of an unknown characteristic cannot be a uni-modal one. The example of such situation is a necessity of estimating a duration of a seasonal work which may be finished either in the first or in the second season, but not in an interval between them.
It is not excluded that the difficulties during a discussion of the appraisal of the type (10.10) are conditioned with the use of just a pseudo-statistical distribution function for an indication of a confidence degree of the expert in his multi-point appraisal. Yu.I. Alimov in his work recommends to use in measurements not the distribution functions, but empirical distribution densities [2] while indicating the fact that the difference the distribution function from the densities is similar to the advantage of a non-sensitive device which gives a small dispersion of results of measurements in comparison with a sensitive device for which a dispersion of the results under the same conditions is considerably greater just because its sensitivity.
Subsequently, a theoretical and experimental investigation of the expert appraisal of the type (10.10) in which the histograms—empirical distribution densities will be used would be evidently necessary.
[1] The transitivity is one of the most important properties of the binary ratios. A ratio R for the multitude A is called transitive provided for any "a", "b" and "c" of A it follows from aRb and bRc that aRc. The relations of an equivalence and order are examples of the transitive binary relations [56].
