# The Bayesian Inference of Pareto Models Based on Information Geometry

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. α-Parallel Prior

**Definition**

**1**

**.**For a statistical manifold $M=\left\{p\right(x;\theta \left)\right|\int p(x;\theta )\mathrm{d}x=1,p(x;\theta )>0,$$\theta \in \Theta \subset {\mathbb{R}}^{d}\}$, define an affine connection ${\nabla}^{\left(\alpha \right)}$ on M with the following coefficients

**Definition**

**2**

**.**An affine connection ∇ is called locally equiaffine if around each point x of M, there is a parallel volume element, that is, a nonvanishing d-form w such that $\nabla w=0$.

**Definition**

**3**

**.**In a statistically equiaffine manifold, for a fixed $\alpha \in \mathbb{R}$, we call the above form of π an α-parallel prior.

**Proposition**

**1**

**.**For a statistical manifold M with the α connection ${\nabla}^{\left(\alpha \right)}$, if $\alpha \ne 0$, then there exists an α-parallel prior if and only if

#### 2.2. Bayesian Inference

- Predict the future observations of the same population $p(\tilde{x};\theta )$$$\begin{array}{c}m\left(\tilde{x}\right|\mathbf{x})={\int}_{\Theta}p(\tilde{x};\theta )\pi \left(\theta \right|\mathbf{x})\phantom{\rule{0.166667em}{0ex}}\mathrm{d}\theta .\end{array}$$
- Predict the observations of another population $g(z;\theta )$$$\begin{array}{c}m\left(z\right|\mathbf{x})={\int}_{\Theta}g(z;\theta )\pi \left(\theta \right|\mathbf{x})\phantom{\rule{0.166667em}{0ex}}\mathrm{d}\theta ,\end{array}$$

## 3. The Geometric Approaches for Bayesian Inference

#### 3.1. The Geometric Prior

#### 3.2. The Geometric Loss Functions

**Proposition**

**2.**

**Proof.**

**Definition**

**4.**

## 4. Bayesian Inference on Pareto Model

#### 4.1. The Geometric Structure of Pareto Two-Parameter Model

#### 4.2. The Existence of α-Parallel Prior on Pareto Two-Parameter Model

**Theorem**

**1.**

**Proof.**

#### 4.3. Bayesian Estimations of Pareto Model

#### 4.3.1. Mean Geodesic Estimation

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

#### 4.3.2. Bayesian Estimations under Al-Bayyati’s Loss Function

**Proposition**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

**Remark**

**1.**

**Theorem**

**5.**

**Proof.**

#### 4.4. Bayesian Posterior Prediction

- When neither $\alpha $ nor $\beta $ are unknown, then we have$$\begin{array}{cc}m\left(\tilde{x}\right|x)& ={\int}_{0}^{+\infty}{\int}_{0}^{+\infty}\pi (\tilde{x};\alpha ,\beta )\pi (\alpha ,\beta |x)\mathrm{d}\alpha \mathrm{d}\beta \hfill \\ \phantom{\rule{1.em}{0ex}}& =-{\int}_{0}^{+\infty}{\int}_{0}^{+\infty}\frac{n{\left(\right)}^{{q}_{2}}n}{}\tau \left(n\right){\beta}^{n}{\alpha}^{n\beta -1}exp\left(\right)open="("\; close=")">-{q}_{2}\left(x\right)\beta \hfill & {I}_{[0\le \alpha \le {q}_{1}\left(x\right)]}\xb7\frac{\beta {\alpha}^{\beta}}{{\tilde{x}}^{\beta +1}}{I}_{[\tilde{x}\ge \alpha ]}\mathrm{d}\alpha \mathrm{d}\beta \end{array}\phantom{\rule{1.em}{0ex}}& =-{\int}_{0}^{+\infty}\frac{n{\left(\right)}^{{q}_{2}}n}{}\tau \left(n\right)exp\left(\right)open="("\; close=")">-{q}_{2}\left(x\right)\beta \hfill & \frac{{\beta}^{n+1}}{{\tilde{x}}^{\beta +1}}\mathrm{d}\beta {\int}_{0}^{min\left(\right)open="("\; close=")">\tilde{x},{q}_{1}\left(x\right)}\\ {\alpha}^{(n+1)\beta -1}\mathrm{d}\alpha $$$$\begin{array}{c}N(x,\tilde{x})=\left(\right)open="\{"\; close>\begin{array}{c}{\left(\right)}^{{q}_{2}}-(n+1),\hfill \\ 0\tilde{x}{\widehat{\alpha}}_{MLE}\left(x\right)\hfill \end{array}{\left(\right)}^{(n+1)}-(n+1),\hfill \\ \tilde{x}\ge {\widehat{\alpha}}_{MLE}\left(x\right).\hfill \end{array}$$
- When α is known and $\beta $ is unknown, we have$$\begin{array}{cc}m\left(\tilde{x}\right|x,\alpha )& ={\int}_{0}^{+\infty}\pi (\tilde{x};\alpha ,\beta )\pi \left(\beta \right|x,\alpha )\mathrm{d}\beta =\frac{(n+1){\left(\right)}^{{q}_{2}}n+1}{}\tilde{x}{\left(\right)}^{-}n+2\hfill & {I}_{[\tilde{x}\alpha ]}.\end{array}$$
- When $\beta $ is known and $\alpha $ is unknown, we have$$\begin{array}{cc}m\left(\tilde{x}\right|x,\beta )& ={\int}_{0}^{+\infty}\pi (\tilde{x};\alpha ,\beta )\pi \left(\alpha \right|x,\beta )\mathrm{d}\alpha \hfill \\ \phantom{\rule{1.em}{0ex}}& =\frac{n}{n+1}\beta {\widehat{\alpha}}_{MLE}^{-n\beta}\left(x\right)\xb7{\tilde{x}}^{-\beta -1}{\left(\right)}^{min}(n+1)\beta \hfill & {I}_{[\tilde{x}0]}\end{array}\phantom{\rule{1.em}{0ex}}& =\frac{n}{n+1}\beta \xb7\left(\right)open="\{"\; close>\begin{array}{cc}{\widehat{\alpha}}_{MLE}^{-n\beta}\left(x\right)\xb7{\tilde{x}}^{n\beta -1},\hfill & 0\tilde{x}{\widehat{\alpha}}_{MLE}\left(x\right)\hfill \\ {\widehat{\alpha}}_{MLE}^{\beta}\left(x\right)\xb7{\tilde{x}}^{-\beta -1},\hfill & \tilde{x}\ge {\widehat{\alpha}}_{MLE}\left(x\right).\hfill \end{array}\hfill $$

## 5. Simulation

#### 5.1. The Influence of Parameters on Sea Clutter

#### 5.2. Various Types of Bayesian Estimation on Sea Clutter Models

#### 5.2.1. Mean Geodesic Estimation and the Common Bayesian Estimations

**Case 1.**Both scale parameter $\alpha $ and shape parameter $\beta $ are unknown.

**Case 2.**Either shape parameter $\beta $ or scale parameter $\alpha $ is known.

#### 5.2.2. The Estimations under Al-Bayyati’s Loss Function

**Case 1.**Both scale parameter $\alpha $ and shape parameter $\beta $ are unknown.

**Case 2.**Either shape parameter β or scale parameter α is known.

#### 5.3. Simulation of Posterior Predictive Distribution

**Case 1.**Scale parameter α and shape parameter β are unknown

**Case 2.**α is known and β is unknown

**Case 3.**β is known and α is unknown

## 6. Conclusions and Future Work

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 5.**The variation of various Bayesian estimations of $\alpha $ with Al-Bayyati’s loss function parameter c.

**Figure 6.**The variation of various Bayesian estimations of $\beta $ with Al-Bayyati’s loss function parameter c.

**Figure 7.**The variation of various Bayesian estimations of $\alpha $ with loss function parameter c when $\beta ={\beta}_{0}$.

**Figure 8.**The variation of various Bayesian estimations of $\beta $ with loss function parameter c when $\alpha ={\alpha}_{0}$.

**Figure 9.**Posterior predictive distribution and underlying distribution ($\alpha $ and $\beta $ are unknown).

**Table 1.**Mean geodesic estimations and the common Bayesian estimations ($\alpha ,\beta $ are not known).

$\left(\right)open="("\; close=")">\mathit{\alpha},\mathit{\beta}$ | $\left(\right)open="("\; close=")">{\widehat{\mathit{\alpha}}}_{\mathit{MLE}},{\widehat{\mathit{\beta}}}_{\mathit{MLE}}$ | $\left(\right)open="("\; close=")">{\widehat{\mathit{\alpha}}}_{\mathit{Me}},{\widehat{\mathit{\beta}}}_{\mathit{Me}}$ | $\left(\right)open="("\; close=")">{\widehat{\mathit{\alpha}}}_{\mathit{E}},{\widehat{\mathit{\beta}}}_{\mathit{E}}$ | $\left(\right)open="("\; close=")">{\widehat{\mathit{\alpha}}}_{\mathit{MGE}},{\widehat{\mathit{\beta}}}_{\mathit{MGE}}$ |
---|---|---|---|---|

(0.5,0.5) | (0.5024,0.5171) | (0.5017,0.5169) | (0.5014,0.5171) | (0.5014,0.5171) |

(0.5,1.0) | (0.5001,1.0292) | (0.4998,1.0288) | (0.4997,1.0292) | (0.4997,1.0292) |

(0.5,1.5) | (0.5003,1.5566) | (0.5001,1.5561) | (0.5000,1.5566) | (0.5000,1.5566) |

(1.0,0.5) | (1.0053,0.4743) | (1.0038,0.4742) | (1.0032,0.4743) | (1.0032,0.4743) |

(1.0,1.0) | (1.0009,1.0209) | (1.0002,1.0205) | (0.9999,1.0209) | (0.9999,1.0209) |

(1.0,1.5) | (1.0001,1.4639) | (0.9996,1.4634) | (0.9994,1.4639) | (0.9994,1.4639) |

(1.5,0.5) | (1.5033,0.4883) | (1.5012,0.4881) | (1.5002,0.4883) | (1.5002,0.4883) |

(1.5,1.0) | (1.5003,1.0226) | (1.4993,1.0223) | (1.4988,1.0226) | (1.4988,1.0226) |

(1.5,1.5) | (1.5010,1.4978) | (1.5003,1.4973) | (1.5003,1.4978) | (1.5000,1.4978) |

$\mathit{\beta}={\mathit{\beta}}_{\mathbf{0}}$ | ${\widehat{\mathit{\alpha}}}_{\mathit{Me}}\left(\mathit{x}\right|{\mathit{\beta}}_{\mathbf{0}})$ | ${\widehat{\mathit{\alpha}}}_{\mathit{E}}\left(\mathit{x}\right|{\mathit{\beta}}_{\mathbf{0}})$ | ${\widehat{\mathit{\alpha}}}_{\mathit{E}}\left(\mathit{x}\right|{\mathit{\beta}}_{\mathbf{0}})$ | ${\widehat{\mathit{\alpha}}}_{\mathit{MGE}}\left(\mathit{x}\right|{\mathit{\beta}}_{\mathbf{0}})$ |
---|---|---|---|---|

(0.5,0.5) | 0.5024 | 0.5017 | 0.5014 | 0.5014 |

(0.5,1.0) | 0.5001 | 0.4998 | 0.4996 | 0.4996 |

(0.5,1.5) | 0.5003 | 0.5000 | 0.4999 | 0.4999 |

(1.0,0.5) | 1.0053 | 1.0039 | 1.0033 | 1.0033 |

(1.0,0.5) | 1.0009 | 1.0002 | 0.9999 | 0.9999 |

(1.0,1.5) | 1.0001 | 0.9996 | 0.9994 | 0.9994 |

(1.5,0.5) | 1.5033 | 1.5012 | 1.5003 | 1.5003 |

(1.5,1.0) | 1.5003 | 1.4992 | 1.4988 | 1.4988 |

(1.5,1.5) | 1.5010 | 1.5003 | 1.5000 | 1.5000 |

$\mathit{\alpha}={\mathit{\alpha}}_{\mathbf{0}}$ | ${\widehat{\mathit{\beta}}}_{\mathit{MLE}}\left(\mathit{x}\right|{\mathit{\alpha}}_{\mathbf{0}})$ | ${\widehat{\mathit{\beta}}}_{\mathit{Me}}\left(\mathit{x}\right|{\mathit{\alpha}}_{\mathbf{0}})$ | ${\widehat{\mathit{\beta}}}_{\mathit{E}}\left(\mathit{x}\right|{\mathit{\alpha}}_{\mathbf{0}})$ | ${\widehat{\mathit{\beta}}}_{\mathit{MGE}}\left(\mathit{x}\right|{\mathit{\alpha}}_{\mathbf{0}})$ |
---|---|---|---|---|

(0.5,0.5) | 0.5158 | 0.5162 | 0.5163 | 0.5161 |

(0.5,1.0) | 1.0289 | 1.0295 | 1.0299 | 1.0294 |

(0.5,1.5) | 1.5553 | 1.5563 | 1.5568 | 1.5560 |

(1.0,0.5) | 0.4732 | 0.4735 | 0.4736 | 0.4734 |

(1.0,0.5) | 1.0199 | 1.0206 | 1.0210 | 1.0205 |

(1.0,1.5) | 1.4638 | 1.4647 | 1.4652 | 1.4645 |

(1.5,0.5) | 0.4877 | 0.4881 | 0.4882 | 0.4880 |

(1.5,1.0) | 1.0224 | 1.0231 | 1.0234 | 1.0229 |

(1.5,1.5) | 1.4921 | 1.4931 | 1.4936 | 1.4928 |

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Sun, F.; Cao, Y.; Zhang, S.; Sun, H.
The Bayesian Inference of Pareto Models Based on Information Geometry. *Entropy* **2021**, *23*, 45.
https://doi.org/10.3390/e23010045

**AMA Style**

Sun F, Cao Y, Zhang S, Sun H.
The Bayesian Inference of Pareto Models Based on Information Geometry. *Entropy*. 2021; 23(1):45.
https://doi.org/10.3390/e23010045

**Chicago/Turabian Style**

Sun, Fupeng, Yueqi Cao, Shiqiang Zhang, and Huafei Sun.
2021. "The Bayesian Inference of Pareto Models Based on Information Geometry" *Entropy* 23, no. 1: 45.
https://doi.org/10.3390/e23010045