The main defect of the appraising methods with the use of appraisal of the first kind is in the fact that they do not allow to use the available information completely. But the introduction of the expert appraisal of the second kind will allow to describe the values being considered in a more complete way. a loss of the information (which may be avoided) is considered, especially in the statistics to be a serious deficiency.
On the other hand, and this is also necessary to have in mind, in a number of the sociological inquests those opinions of the interrogated ones are of interest where these opinions are quite illegible or, let's say, not shaped owing to a lack of the necessity to think over the corresponding subject, and therefore such types of appraisals are used of which the degree of a reliability is a problematic one.
Let’s remind that a regulated pair of appraisal is called as the expert appraisal of the second kind if the first component of this pair is the expert appraisal of the first kind, but the second component – a degree (measure) of the experts confidence in his appraisal of the first kind expressed in the following types of the simplest appraisals of the first kind: 1, 7, 8, 9, 10 and 11 (these numbers are shown in the last but one lecture).
In addition, it is desirable to substantiate the expediency of the introduction of a new type of the expert appraisals and their combinations in both a theoretical and experimental way.
The expert appraisals which satisfy a definition of the appraisal of the second kind are discussed now in the literature on the expert appraisals.
Moreover, there are known the attempts of a generalized representation of the expert appraisal of the second kind. So, for example, the binary metered ratio introduced in the work [47] gives an opportunity to discuss a number of appraisals of the second type.
Let’s remind that a regulated pair = <P, W> is called as the binary metered ratio where P – is a binary ratio on A (PÌA2) and W is a real function characterizing an intensity of relations between elements of certain great number A ={A1,A2,…An}, W: P→ R [47].
Let's an appraisal of the first kind to be defined through a binary ratio, is, for example, a pair comparison or a certain grouping, but a degree of a confidence is expressed with a unique real number. Therefore, we are approaching a certain realization of a binary metered ratio.
In this lecture we are describing various approaches for a definition of a confidence degree of the expert in his appraisal.
As a rule, the experts are coming across unique situations and events for which there are no probability appraisals which are of a frequency character and nevertheless they have to appraise them at an intuitive level.
Certainly, this fact does not mean that there is no share of objectivity in them because an intuition is a non-formalized experience of experts drawn in for making appraisals.
First of all let's consider the most wide-spread approach related to the so-called "subjective" probability of results of some or other events. Moreover, often the measure of a confidence or conviction of a certain person or a group of people in the fact that this event will take place in reality is just called as a subjective probability [37]. Such a definition has a serious restriction related to the fact that in this case the degree of confidence is measured on a scale which is not weaker than the scale of relations.
Such prominent scientists as Laplas and Bernulli took up the positions of a subjective approach towards a probability.
Evidently, one of the first significant works on a subjective probability was the work of Jacob Bernulli "The art of the foresight" according to which the probability is "a degree of confidence" or, as it was written later, a level of the "confidence" of an individual in relation to an indefinite event: and this degree depends on a conversance of the individual and may be changed from one individual to another [123].
The subjective probability is described quite in detail in the English and French literature.
The subjective probability theory is described in the Russian literature in an insufficient way. In one of the most interesting articles [62] there are mentioned several original works devoted to a partially qualitative subjective probability.
An analysis of the main directions in the probability theory when studying the human appraisals of an authenticity of unique events is discussed in the work of A.A. Fedoulov, Yu.G. Fedoulov and V.N. Tsyhichko [95] and also in the work of E. Limer [45].
In a historical way the progress of the subjective probability concept was progressing in parallel to investigations of the subjective usefulness.
These two concepts became a basis for a development of the decision making models under the conditions of an uncertainty. The first one who indicated a connection between the subjective probability and usefulness was F.Ramsey [170] who used the usefulness for building a subjective probability.
The subjective probability may be also defined within the framework of a real or imaginary decision making problem under the conditions of an uncertainty. Frank Ramsey [170] was the first who suggested a decision making theory built upon the dual interconnected notions of an appraisal probability and usefulness.
According to Ramsey the probability is defined by operations as a degree of a readiness of a subject to accomplish one or another action in a decision making situation with possible unreliable gains.
Subsequently in the works of numerous authors the subjective probability appears not just a measure of a confidence on a great number of events, but it is co-ordinated with the preference system of the decision making person and in the end with a usefulness function [97].
The works in the sphere of subjective probability and usefulness were greatly influenced with the Newman - Morgenstern axiomatic approach which was used by them for constructing the usefulness functions for the decision making problems under the conditions of a risk under which the presence of the function of usefulness is deduced from the preference axioms in a multitude of lotteries with the known probabilities of results [63].
While developing this scheme and leaning upon the ideas of de-Finetti [145,146,147], L.J. Savage in 1954 constructed a system of axioms in which the probabilities are not supposed to be known, but are drawn together with the usefulness function from the preference axioms [175].
The "subjectivists" themselves are divided into two groups: personalists and rationalists.
The personalists, and among them de-Finetti [146], Ramsey [170] and Savage [175] affirm that if some individuals possess non-equal fund of knowledge then a quantitative measure of the knowledge should change from an individual to another one. It is recommended that the confidence levels of everyone of them for their own expert appraisals were subjected to the probability axioms, but in all other respects we are free to fix them as we like them (to be more correct, according to our knowledge).
The rationalists, as Jeffry [157] and Keines [159], on the other hand, assert that the subjective probability is a level of a confidence which it is "rational" to have in relation to a certain indefinite statement under a condition that there are also other statements (quoted from [95]).
In the exact sciences they avoided, as a rule, to give strict definitions of the subjective probabilities.
However, in the works of Savage [174] and Churchman [136] it was said that it was impossible to avoid a discussion of the subjective probabilities. More over, Churchman showed that it is necessary to use the subjective appraisals of probabilities for getting objective appraisals. In connection with this fact the methods of getting the subjective appraisals attracted a certain attention and became an object of wide research works.
As for a reliability and accuracy of the subjective appraisals of a probability there are exist directly contradictory points of view. So, in the experimental works of Slovik, Tverskoy and Kaneman and ether authors it was shown that during their own appraisal of the subjective probabilities the examinees are making serious mistakes including systematic ones [83,181,182,187].
On the other hand, there are works according to which the examinees gave extremely exact appraisals of the subjective probabilities. So, according to the opinion of T.B. Sheridan andU.R. Ferrell a lot of experiments lead us inevitably to a conclusion that the people, at least at an average, define a relative frequency of the events being observed efficiently. Moreover, they are even able to find quantitative values of probabilities [105]. In the case when the examinees are watching a succession of binary events and are informing orally on a percent correlation of results then their appraisals for a number or series of trials of one subject proved to be extremely exact in at an average.
In this connection the results of the experiments made by Erlik are typicaly, in them the examinees were given the letters "A" and "C" successively with the speed of four letters per second, that is too rapidly in order to count them [142]. The examinees had to define the frequency of getting one of the letters approximating it with the accuracy up to 5 per cent. The maximum mean error proved to be a small one, approximately 5 per cent. Analogous experiments were made by Atnev [121] who studied the appraisals of relative frequencies of the use of the letters of the English language given by the examinees from memory. The interesting experiments were made by Simpson and Voos [180] and by Pitz too [168]. In a general case a high accuracy is observed when is appraised the frequency of the events of a given type in the series of the events.
A direct appraisal of the percentage of the less frequent elements in the static spatial bases was studied by Shafford [179], and those appraisals proved to be also extremely correct. Robinson [171] suggested and verified the selection model for the problem of a non-interrupted appraisal of a probability being changed in time. His examinees had to move a pointer on the scale in order to indicate a seeming probability with which one of the two lamps was flashing. The probabilities of the lamp flashings underwent a succession of accidental changes, but remained to be constant during at least 30 flashes. The examinees did not know beforehand to what value these probabilities were changing in time. Even in such a complicated situation the results of the experiments showed an acceptable accuracy at an average.
A good review of the research works in which the people were considered to be "intuitive statisticians" was given by Peterson and Beach [166].
The availability of the directly contradictory results concerning an exactness and reliability of appraisals of the subjective probabilities rather confirms the concept of a relativity of the best type of an expert appraisal and a necessity to select the best type of the expert appraisal every time.
As a measure of the confidence of a man in a possible occurring some events the subjective probability may be in a formal way submitted in various ways: a distribution of probabilities on a great number of events, a binary ratio on a great number of events, a non-completely given distribution of probabilities or a partial binary relation and other methods [62].
The quantitative and qualitative subjective probability is singled out depending upon the form of a presentation.
The quantitative subjective probability is a probability measure for a great number of events while meeting the requirements of the same system of axioms as an objective probability [38]. Therefore, from a formal point of view, the quantitative subjective probability differs in no way from the objective probability. The difference is the sense which is put into these notions. Practically, a formation of the quantitative subjective probability requires an indication of the numerical values of the probability for a number of events by the expert.
However, it is known that such a quantitative information is very complicated for a man and unreliable in a number of cases [108].
The information consisting of answers to the questions on a comparable probability (possibility) of two events is considerably more simple and therefore a more trustworthy. In connection with this a non-numerical formalization of the subjective probability based upon the use of binary ratios of the superiority and equality of events according to a probability are of a greater practical interest. So, the formalized subjective probability has acquired the name of the qualitative one (comparative also [144,188]).
The problem on a possibility of forming a quantitative probability which is coordinated with the qualitative one in some sense is traditionally considered to be the basic question related to the concept of the qualitative probability. This fact became reflection of the fact just the quantitative probability [35]/ has been used for solving practical problems up to the recent times, but the qualitative probability engendered just a theoretical interest. However, recently in the decision making theory there appeared special procedures meant to an analysis of the qualitative information in connection with which the notion of the qualitative probability has acquired an independent practical meaning.
The second approachtoward the definition of the confidence degree is based upon normalized and non-normalized eroded numbers introduced by P.B. Shoshin [111]. The practical use of eroded numbers in expert appraisals as a certain presentation of subjective appraisals is expedient even now according to P.B. Shoshin's opinion.
There are also exist other approaches - through the use of illegible multitudes and finite casual multitudes. In the expert appraisals illegible multitudes were discussed in the works of M. Pinas [167], but casual multitudes - by A.I. Orlov [65]. All these approaches are near and related to the illegible multitudes theory. By the present time the method of the illegible multitudes theory is rather well developed and described in Russian in the works of L.A. Zade [32], L.A. Gousev and I.M. Smirnov [25], A.I. Orlov [65]and others. Unfortunately, the number of theoretical works is uncomparably greater than the number of quite prominent and convincing applied works on this subject.
= <P, W> is called as the binary metered ratio where P – is a binary ratio on A (PÌA2) and W is a real function characterizing an intensity of relations between elements of certain great number A ={A1,A2,…An}, W: P→ R [47].