Intuitionistic fuzzy (IF) evidence theory, as an extension of Dempster-Shafer theory of evidence to the intuitionistic fuzzy environment, is exploited to process imprecise and vague information. Since its inception, much interest has been concentrated on IF evidence theory. Many works on the belief functions in IF information systems have appeared. Although belief functions on the IF sets can deal with uncertainty and vagueness well, it is not convenient for decision making. This paper addresses the issue of probability estimation in the framework of IF evidence theory with the hope of making rational decision. Background knowledge about evidence theory, fuzzy set, and IF set is firstly reviewed, followed by introduction of IF evidence theory. Axiomatic properties of probability distribution are then proposed to assist our interpretation. Finally, probability estimations based on fuzzy and IF belief functions together with their proofs are presented. It is verified that the probability estimation method based on IF belief functions is also potentially applicable to classical evidence theory and fuzzy evidence theory. Moreover, IF belief functions can be combined in a convenient way once they are transformed to interval-valued possibilities.

The Dempster-Shafer theory of evidence, also called belief function theory, is an important method to deal with uncertainty in information systems. Since it was firstly presented by Dempster [

The theory of fuzzy set, proposed by Zadeh [

Relationship between fuzzy set theory and belief function theory has been focused on for a long time. Zadeh was the first to generalize the Dempster-Shafer theory to fuzzy sets, based on his work on the concept of information granularity and the theory of possibility [

All the above works on IF belief function theory focused on the determination of the basic probabilities assigned to IF events based on the known probabilistic distribution in the universe of discourse. However, its converse problem, that is, probability estimate through belief functions on IF events, is rarely addressed. Although belief functions on IF events can handle uncertainty better, it is not a good tool for decision making. Therefore, a transformation that can transform belief functions on IF events to probability distribution in the universe is desirable for the sake of sounder decision.

In this paper, we investigate the probability estimation of belief functions on IF event to cope with the issue of decision making in the framework of IF evidence theory. A method to estimate the probability distribution in the universe of discourse is proposed in this paper. Since the probability of each basic event tends to be defined as an interval value on the unit interval

The remainder of this paper is organized as follows. Section

The background material presented in this section deals with the following three main points: (1) the interpretation of Dempster-Shafer theory of evidence, which will be used in this paper to ease the exposition, (2) a brief review of definitions on fuzzy set and IF set, and (3) introduction of intuitionistic fuzzy evidence theory.

Dempster-Shafer theory of evidence was modeled based on a finite set of mutually exclusive elements, called the frame of discernment denoted by

Let

A subset

For a belief function

The pignistic transformation maps a belief function

Particularly, when

Given two belief functions

In this section, we briefly recall the basic concepts related to fuzzy sets and intuitionistic fuzzy sets.

Let

An IF set

The hesitancy degree

For

It is worth noting that, besides Definition

It has been well known that Dempster-Shafer evidence theory can express and deal with uncertainty in crisp sets. However, D-S theory itself can not represent and manage vague information such as “the price is high” or “his age is about 40.” To overcome this problem, fuzzy evidence theory was proposed to process imprecise and vague information [

Let

Let the probability of each element in

Let

Let the probability of each element in

It is easy to verify that

Suppose there is a discussion on the amount of money which should be assigned for advertisement of a new product. Let us assume that the IF event

The probability distribution of the amount of money is

From (

It means that the basic probability assigned to the event of “assigning about fifty thousands for advertisement” lies in the interval [0.48, 0.6].

Let

The normalization of classical belief functions has been investigated in many works, for example, [

It is apparent that the proposed BPA in (

The advantage of belief function theory in IF set lies not only in its ability to express degree of uncertainty (nonspecificity, discord, and fuzziness), but also in its capacity for facilitating the fusion of uncertain information. However, since decision making based on classical belief function is still under dispute, how to make decision based on IF belief function is also worth studying.

In many practical information systems, the final outputs may be expressed by belief functions on IF events such as “about fifty thousands,” which are given by experts without the probability distribution of basic events, or from the fusion result of several sensors with limit knowledge. Due to the fuzziness of IF events, few precise decisions can be made based on the probabilities assigned to them. In contrast, probability distribution on the specific events in the universe is more helpful for decision making. Therefore, the problem of probability estimation based on IF evidence arises. This issue can be illustrated by the following example.

For the situation discussed in Example

Experts’ assessment result can be written as

This assessment result is too approximate to leading to a precise decision for the decision maker. It can merely provide qualitative rather than quantitative guidance. Since the money assigned for the advertisement is a specific number in implement, such a result must be transformed to a probability distribution in the universe of discourse, which can also be regarded as a course of defuzzification in the IF belief function.

Let

Let

Let

Then, the probability of

The probability estimation

Given

Therefore,

The multielement focal elements for classical belief function can be also regarded as fuzzy sets, where the membership degrees of elements take values 0 or 1. For example, the focal element

Since a probability distribution generates interval-valued belief functions for IF events, the probability estimation through IF belief functions should also be interval values. Therefore, for each

Let

Then, the probability of

The interval probability estimation

The relationships between

Form the above relationships we can get

For,

We can also obtain

Finally we can get

Hence

Therefore,

The probability estimation

Since

Then

So

Considering Proposition

Now let us reinvestigate Example

The probability distribution can be applied for decision making directly by comparing these intervals. However, the comparison of intervals is challengeable, especially for intervals with overlaps. Many methods for comparison of intervals have been proposed [

Since the crisp focal elements containing more than one element can be represented by fuzzy focal elements, they can also be expressed by IF focal elements. The focal element

It is worth noticing that the combination of IF belief functions is still an open topic. If we transform IF belief functions to probability distribution on the universe, it can be considered as interval-valued probability distribution or interval-valued Bayesian belief functions [

Reconsider the situation discussed in Example

From Definition

Taking a closer examination on the result, we can find that it satisfies the condition of

Reconsider the situation discussed in Example

The events

By Definition

These interval-valued probabilities can be regarded as the normalized interval-valued Bayesian belief functions as defined in our earlier paper [

It can be verified that the final result satisfies the conditions in Definition

These examples illustrate that the probability estimation method is also suitable for both fuzzy belief functions and classical belief functions. By estimating the probability distribution in the universe, we can make decision more conveniently. Moreover, the probability estimation can provide an alternative way for the combination of IF belief functions.

The evidence theory has been extended to intuitionistic fuzzy environment to deal with imprecise and vague information. Intuitionistic fuzzy belief functions have received considerable interest for its capability of managing uncertainty in information systems. Uncertainty on the events can be expressed by belief functions on IF sets, while a sounder decision cannot be got based on IF belief functions. So, it is necessary to transform IF belief functions to probability distribution in the universe of discourse. In this paper we mainly investigated the probability estimation based on IF belief functions. The probability estimations based on fuzzy and IF belief functions, together with their proofs, are presented. It has been proved that the probabilities of basic events estimated form the fuzzy belief functions are precise value, while those from the IF belief functions are intervals. Decision making can be implemented based on the comparison between intervals. Moreover, when all IF belief functions degenerate into classical belief functions, the probability estimation is identical to the pignistic transformation. In such sense, the probability estimation based on IF belief functions can be regarded as the extension of pignistic transformation to the intuitionistic fuzzy environment. In addition, the proposed probability estimation method provides an alternative combination rule for IF belief functions.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work is supported by the National Natural Science Foundation of China under Grants 60975026 and 61273275. The authors would also like to send their sincere gratitude to the editors and anonymous reviewers.