# Negation of Belief Function Based on the Total Uncertainty Measure

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Dempster-Shafer Theory

#### 2.2. Uncertainty Measurements of Basic Probability Assignment (BPA)

#### 2.3. Negation of Probability Distribution

- Repeated process of negation of probability distribution converges to a certain probability distribution.
- The maximum value of uncertainty of the system is calculated exactly for the convergent probability distribution.
- The entropy increases constantly till the maximum value of the total uncertainty attains.

## 3. Negation of BPA

#### 3.1. Definition of Negation

#### 3.2. Steps of Constructing the Negation

**Step 1:**Obtain the element in the $\overline{{A}_{j}}$ by

**Step 2:**Calculate the cardinality of intersection of ${A}_{i}$ and $\overline{{A}_{j}}$, which is the reallocation weight of negation process for ${H}_{j}$ denoted as ${c}_{j}$ and the sum $\sigma $ of the cardinality from $j=2$ to $j={2}^{N}-1$ (except for empty set and the whole set). Namely:

**Step 3:**Normalize these weights of negation process of ${H}_{j}$ to guarantee their sum is one.

- (1)
- ${\sum}_{i=1}^{i=2}{\overline{m}}_{i}=1$
- (2)
- ${\overline{m}}_{i}\in [0,1]$

#### 3.3. Numerical Examples of the Negation Process

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

#### 3.4. Discussion

**Proof.**

- The existing work tried to present the negation of a mass function the same as the negation of a probability distribution proposed by Yager [27], which means the mass is equally reallocated to other focal elements and the elements in the power set is ignored. However, we believe that the uncertainty of non-singleton elements should be taken into account and the negation of BPA should be extended to the power set. Thus, the proposed negation of a mass function reallocates the corresponding BPA in a weighted manner among the power set.
- The existing work of negation of a mass function is not based on the maximal uncertainty (entropy). Our work tried to refine this point and reflect the negation of a mass function by total uncertainty measure and proposed a negation method of a mass function based on the maximum total uncertainty mathematically, which is consistent with the negation of a probability distribution based on the maximum entropy proposed by Yager [27].

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Evolution of basic probability assignment (BPA) as iteration of negation process increases.

**Table 1.**BPA value for each element and the total uncertainty corresponding to each negation process.

Frequency of Iterations | $\mathit{m}(\mathit{a})$ | $\mathit{m}(\mathit{b})$ | $\mathit{m}(\mathit{c})$ | $\mathit{m}(\mathit{ab})$ | $\mathit{m}(\mathit{ac})$ | $\mathit{m}(\mathit{bc})$ | $\mathit{m}(\mathit{abc})$ | Total Uncertainty |
---|---|---|---|---|---|---|---|---|

0 | 0.1000 | 0.1500 | 0.0000 | 0.0000 | 0.3000 | 0.2500 | 0.2000 | 3.0952 |

1 | 0.1083 | 0.1167 | 0.0417 | 0.2250 | 0.1500 | 0.1583 | 0.2000 | 3.5305 |

2 | 0.0792 | 0.0750 | 0.1125 | 0.1542 | 0.1917 | 0.1875 | 0.2000 | 3.5647 |

3 | 0.0937 | 0.0958 | 0.0771 | 0.1896 | 0.1708 | 0.1729 | 0.2000 | 3.5723 |

4 | 0.0865 | 0.0854 | 0.0948 | 0.1719 | 0.1812 | 0.1802 | 0.2000 | 3.5742 |

5 | 0.0901 | 0.0906 | 0.0859 | 0.1807 | 0.1760 | 0.1766 | 0.2000 | 3.5747 |

6 | 0.0883 | 0.0880 | 0.0904 | 0.1763 | 0.1786 | 0.1784 | 0.2000 | 3.5748 |

7 | 0.0892 | 0.0893 | 0.0882 | 0.1785 | 0.1773 | 0.1775 | 0.2000 | 3.5749 |

8 | 0.0887 | 0.0887 | 0.0893 | 0.1774 | 0.1780 | 0.1779 | 0.2000 | 3.5749 |

9 | 0.0890 | 0.0890 | 0.0887 | 0.1780 | 0.1777 | 0.1777 | 0.2000 | 3.5749 |

10 | 0.0889 | 0.0888 | 0.0890 | 0.1777 | 0.1778 | 0.1778 | 0.2000 | 3.5749 |

11 | 0.0889 | 0.0889 | 0.0888 | 0.1778 | 0.1778 | 0.1778 | 0.2000 | 3.5749 |

12 | 0.0889 | 0.0889 | 0.0889 | 0.1778 | 0.1778 | 0.1778 | 0.2000 | 3.5749 |

13 | 0.0889 | 0.0889 | 0.0889 | 0.1778 | 0.1778 | 0.1778 | 0.2000 | 3.5749 |

14 | 0.0889 | 0.0889 | 0.0889 | 0.1778 | 0.1778 | 0.1778 | 0.2000 | 3.5749 |

15 | 0.0889 | 0.0889 | 0.0889 | 0.1778 | 0.1778 | 0.1778 | 0.2000 | 3.5749 |

Uncertainty Measures | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
---|---|---|---|---|---|---|---|---|

${H}_{b}(m)$ | 3.5084 | 3.5726 | 3.5743 | 3.5747 | 3.5748 | 3.5749 | 3.5749 | 3.5749 |

${H}_{rp}(m)$ | 2.5511 | 2.4349 | 2.4352 | 2.4352 | 2.4353 | 2.4353 | 2.4353 | 2.4353 |

${H}_{D}(m)$ | 4.1331 | 4.1291 | 4.1308 | 4.1312 | 4.1312 | 4.1313 | 4.1313 | 4.1313 |

$\overline{m}(a)$ | $\frac{1}{6}(b+c+2bc)$ |

$\overline{m}(b)$ | $\frac{1}{6}(a+c+2ac)$ |

$\overline{m}(c)$ | $\frac{1}{6}(a+b+2ab)$ |

$\overline{m}(ab)$ | $\frac{1}{6}[a+b+2(c+ac+bc)]$ |

$\overline{m}(ac)$ | $\frac{1}{6}[a+c+2(b+ab+bc)]$ |

$\overline{m}(bc)$ | $\frac{1}{6}[b+c+2(a+ab+ac)]$ |

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Xie, K.; Xiao, F.
Negation of Belief Function Based on the Total Uncertainty Measure. *Entropy* **2019**, *21*, 73.
https://doi.org/10.3390/e21010073

**AMA Style**

Xie K, Xiao F.
Negation of Belief Function Based on the Total Uncertainty Measure. *Entropy*. 2019; 21(1):73.
https://doi.org/10.3390/e21010073

**Chicago/Turabian Style**

Xie, Kangyang, and Fuyuan Xiao.
2019. "Negation of Belief Function Based on the Total Uncertainty Measure" *Entropy* 21, no. 1: 73.
https://doi.org/10.3390/e21010073