# Information Geometric Duality of ϕ-Deformed Exponential Families

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## Abstract

**:**

## 1. Introduction

## 2. Deformed Exponential Family

## 3. Deformed Divergences, Entropies, and Metrics

#### 3.1. Linear Constraints: Divergence a là Naudts

#### 3.2. Escort Constraints: Divergence a là Amari

#### 3.3. Cramér–Rao Bound of Naudts Type

## 4. The Information Geometric “Amari–Naudts” Duality

#### 4.1. Cramér–Rao Bound of the Amari Type

#### 4.2. Example: Duality of $(c,d)$-Entropy

## 5. Connection to the Deformed-Log Duality

## 6. Discussion

- (SK1) Entropy is a continuous function of the probabilities ${p}_{i}$ only and should not explicitly depend on any other parameters.
- (SK2) Entropy is maximal for the equi-distribution ${p}_{i}=1/W$.
- (SK3) Adding a state $W+1$ to a system with ${p}_{W+1}=0$ does not change the entropy of the system.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Fisher metric for $p=(p,1-p)$ corresponding to various $(c,d)$-deformations ($(c,d)=(1,1)$, $(1,1/2)$, $(1/2,0)$) for (

**a**) the Naudts type, (

**b**) the Amari type, and (

**c**) the Cramér–Rao bound corresponding to the metric.

**Figure 2.**Fisher metric for $(c,d)$-deformations as a function of c and d of Naudts type (

**a**) and Amari type (

**b**). The metric is evaluated at a point $p=(1/3,2/3)$.

**Table 1.**$\varphi $-deformation of divergence, entropy, and Fisher information corresponding to the $\varphi $-exponential family under linear and escort constraints. For the ordinary logarithm, $\varphi \left(x\right)=x$, the two entropies become the Shannon entropy, and the divergence is Kullback–Leibler.

$\mathit{\varphi}$-Deformation | Linear Constraints | Escort Constraints | |
---|---|---|---|

divergence | ${D}_{\varphi}(\mathit{p}\parallel \mathit{q})$ | ${\sum}_{j}{\int}_{{q}_{j}}^{{p}_{j}}\mathrm{d}x\left(\right)open="("\; close=")">{log}_{\varphi}\left(x\right)-{log}_{\varphi}\left({q}_{j}\right)$ | $\frac{{\sum}_{j}\varphi \left({p}_{j}\right)({log}_{\varphi}\left({p}_{j}\right)-{log}_{\varphi}\left({q}_{j}\right))}{{\sum}_{k}\varphi \left({p}_{k}\right)}$ |

entropy | ${S}_{\varphi}\left(\mathit{p}\right)$ | $-{\sum}_{i}{\int}_{0}^{{p}_{i}}{log}_{\varphi}\left(x\right)\mathrm{d}x$ | $-{\sum}_{i}\varphi \left({p}_{i}\right){log}_{\varphi}\left({p}_{i}\right)/{\sum}_{k}\varphi \left({p}_{k}\right)$ |

metric | ${g}_{ij}^{\varphi}\left(\mathit{p}\right)$ | $\frac{1}{\varphi \left({p}_{i}\right)}{\delta}_{ij}+\frac{1}{\varphi \left({p}_{0}\right)}$ | $\frac{1}{{\sum}_{k}\varphi \left({p}_{k}\right)}\left(\right)open="("\; close=")">\frac{{\varphi}^{\prime}\left({p}_{i}\right)}{\varphi \left({p}_{i}\right)}{\delta}_{ij}+\frac{{\varphi}^{\prime}\left({p}_{0}\right)}{\varphi \left({p}_{0}\right)}$ |

**Table 2.**Two important special cases of $(c,d)$-deformations and related quantities: Power laws (Tsallis) and stretched exponentials.

Tsallis q-Exponential [4] | Stretched $\mathit{\eta}$-Exponential [33] | |
---|---|---|

$\varphi \left(x\right)$ | ${x}^{q}$ | $x\eta log{\left(x\right)}^{1-1/\eta}$ |

${log}_{\varphi}\left(x\right)$ | $\frac{{x}^{1-q}-1}{1-q}$ | $log{\left(x\right)}^{1/\eta}$ |

${exp}_{\varphi}\left(x\right)$ | ${\left(\right)}^{1}$ | $exp\left({x}^{\eta}\right)$ |

${\chi}_{\varphi}\left(x\right)$ | $\frac{x}{q}$ | $\frac{x\eta log\left(x\right)}{(\eta -1)+\eta log\left(x\right)}$ |

${S}_{\varphi}^{N}\left(\mathit{p}\right)$ | $\frac{1}{q-1}\left(\right)open="("\; close=")">{\sum}_{i}\frac{{p}_{i}^{2-q}}{2-q}-1$ | ${\sum}_{i}\Gamma \left(\right)open="("\; close=")">\frac{\eta +1}{\eta},-log{p}_{i}$ |

${S}_{\varphi}^{A}\left(\mathit{p}\right)$ | $\frac{1}{q-1}\left(\right)open="("\; close=")">\frac{1}{{\sum}_{i}{p}_{i}^{q}}-1$ | $\frac{{\sum}_{i}{p}_{i}log{p}_{i}}{{\sum}_{i}{p}_{i}{(log{p}_{i})}^{1-1/\eta}}$ |

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Korbel, J.; Hanel, R.; Thurner, S.
Information Geometric Duality of *ϕ*-Deformed Exponential Families. *Entropy* **2019**, *21*, 112.
https://doi.org/10.3390/e21020112

**AMA Style**

Korbel J, Hanel R, Thurner S.
Information Geometric Duality of *ϕ*-Deformed Exponential Families. *Entropy*. 2019; 21(2):112.
https://doi.org/10.3390/e21020112

**Chicago/Turabian Style**

Korbel, Jan, Rudolf Hanel, and Stefan Thurner.
2019. "Information Geometric Duality of *ϕ*-Deformed Exponential Families" *Entropy* 21, no. 2: 112.
https://doi.org/10.3390/e21020112