# Bayesian Update with Information Quality under the Framework of Evidence Theory

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Evidence Theory

**Definition**

**1.**

**Definition**

**2.**

**Example**

**1.**

#### 2.2. Pignistic Probability Transformation

**Definition**

**3.**

**Example**

**2.**

- ${m}_{1}\left(\{{\omega}_{1},{\omega}_{2}\}\right)=0.8000$ ${m}_{1}\left(\left\{{\omega}_{3}\right\}\right)=0.1000$ ${m}_{1}\left(\left\{{\omega}_{4}\right\}\right)=0.1000$
- Let B = $\left\{{\omega}_{1}\right\}$ then $Bet{P}_{1}\left(B\right)=0.4000$
- Let A = $\left\{{\omega}_{4}\right\}$ then $Bet{P}_{1}\left(A\right)=0.1000$

#### 2.3. Information Quality

**Definition**

**4.**

**Definition**

**5.**

**Example**

**3.**

## 3. Proposed Method

#### 3.1. Determine Weight

**Definition**

**6.**

**Example**

**4.**

#### 3.2. Generate Basic Probability Assignent

Algorithm 1: The algorithm to generate a basic probability assignment |

// To get all BPA, execute this algorithm n (total number of probability distributions) times as the algorithm is used to convert a probability distribution to a BPA.Input: The weight of the probability distribution, ${\omega}_{1}$$m\left(A\right)={\omega}_{1}\ast p\left(A\right)$ $m\left(B\right)={\omega}_{1}\ast p\left(B\right)$ ⋯ $m\left(N\right)={\omega}_{1}\ast p\left(N\right)$ $m(AB\cdots N)=1-{\sum}_{i=1}^{n}{\omega}_{1}\ast p\left(I\right)\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}I=A,B,C,\cdots ,N$ Output: ${m}_{1}=(\left\{m\left(A\right)\right\},\left\{m\left(B\right)\right\},\{\cdots \},\left\{m(AB\cdots N)\right\})$ |

#### 3.3. Fusion Method

Algorithm 2: The algorithm of fusion process |

## 4. Application

#### 4.1. Numerical Example

**Example**

**5.**

- ${m}_{1}$ and ${m}_{2}$ fusion provides ${m}^{\prime}$ = ({0.3000}, {0.1300}, {0.0900}, {0.4800})
- ${m}^{\prime}$ and ${m}_{3}$ fusion provides m = ({0.4000}, {0.1700}, {0.0900}, {0.3400})

**Example**

**6.**

#### 4.2. Target Recognition

- ${m}_{1}$ and ${m}_{2}$ fusion gives ${m}^{\prime}$,$${m}^{\prime}=(\left\{0.4700\right\},\{0.0600,\}\left\{0.0300\right\},\left\{0.4400\right\})$$
- ${m}^{\prime}$ and ${m}_{3}$ fusion gives m,$$m=\left(\right\{0.5300\},\{0.0700\},\{0.0500\},\{0.3500\left\}\right)$$

#### 4.3. Multi-Sensor Target Recognition

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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${\mathit{s}}_{1}$, ${\mathit{s}}_{2}$ | ${\mathit{s}}_{1}$, ${\mathit{s}}_{2}$, ${\mathit{s}}_{3}$ | ${\mathit{s}}_{1}$, ${\mathit{s}}_{2}$, ${\mathit{s}}_{3}$, ${\mathit{s}}_{4}$ | ${\mathit{s}}_{1}$, ${\mathit{s}}_{2}$, ${\mathit{s}}_{3}$, ${\mathit{s}}_{4}$, ${\mathit{s}}_{5}$ | |
---|---|---|---|---|

simple average | $p\left(A\right)=$ 0.6000 | $p\left(A\right)=$ 0.5800 | $p\left(A\right)=$ 0.5750 | $p\left(A\right)=$ 0.5800 |

$p\left(B\right)=$ 0.1500 | $p\left(B\right)=$ 0.1400 | $p\left(B\right)=$ 0.1250 | $p\left(B\right)=$ 0.1200 | |

$p\left(C\right)=$ 0.2500 | $p\left(C\right)=$ 0.2800 | $p\left(C\right)=$ 0.3000 | $p\left(C\right)=$ 0.3000 | |

proposed method | $p\left(A\right)=$ 0.5532 | $p\left(A\right)=$ 0.5924 | $p\left(A\right)=$ 0.6267 | $p\left(A\right)=$ 0.6428 |

$p\left(B\right)=$ 0.1899 | $p\left(B\right)=$ 0.1490 | $p\left(B\right)=$ 0.1185 | $p\left(B\right)=$ 0.1100 | |

$p\left(C\right)=$ 0.2569 | $p\left(C\right)=$ 0.2586 | $p\left(C\right)=$ 0.2548 | $p\left(C\right)=$ 0.2472 |

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**MDPI and ACS Style**

Li, Y.; Xiao, F.
Bayesian Update with Information Quality under the Framework of Evidence Theory. *Entropy* **2019**, *21*, 5.
https://doi.org/10.3390/e21010005

**AMA Style**

Li Y, Xiao F.
Bayesian Update with Information Quality under the Framework of Evidence Theory. *Entropy*. 2019; 21(1):5.
https://doi.org/10.3390/e21010005

**Chicago/Turabian Style**

Li, Yuting, and Fuyuan Xiao.
2019. "Bayesian Update with Information Quality under the Framework of Evidence Theory" *Entropy* 21, no. 1: 5.
https://doi.org/10.3390/e21010005