###### Abstract
When \(\alpha=\frac{1}{2}\), the \(\alpha\)-divergence is symmetric and called the squared Hellinger divergence [15]:  
\[I_{\frac{1}{2}}(p:q):=4\bigg{(}1-\int\sqrt{p}q\mathrm{d}\mu\bigg{)}=2\int( \sqrt{p}-\sqrt{q})^{2}\mathrm{d}\mu. \tag{7}\]  
The \(\alpha\)-divergences belong to the family of Ali-Silvey-Csizar's \(f\)-divergences [16; 17] which are defined for a convex function \(f(u)\) satisfying \(f(1)=0\) and strictly convex at \(1\):  
\[I_{f}(p:q):=\int p\,f\bigg{(}\frac{q}{p}\bigg{)}\mathrm{d}\mu. \tag{8}\]  
We have  
\[I_{\alpha}(p:q)=I_{I_{\alpha}}(p:q), \tag{9}\]  
with the following class of \(f\)-generators:  
\[f_{\alpha}(u):=\left\{ \begin{array}{ll}\frac{1}{\alpha(1-\alpha)}(\alpha+(1 -\alpha)u-u^{1-\alpha}),&\alpha\in\alpha\in\mathbb{R}\backslash\{0,1\}\\ u-1-\log u,&\alpha=1\\ 1-u+u\log u,&\alpha=0\end{array} \right. \tag{10}\]  
In information geometry, \(\alpha\)-divergences and more generally \(f\)-divergences are called invariant divergences [6], since they are provably the only statistical divergences which are invariant under invertible smooth transformations of the sample space. That is, let \(Y=m(X)\) be a smooth invertible transformation and let \(\mathcal{Y}=m(\mathcal{X})\) denote the transformed sample space. Denote by \(p_{Y}(y)\) and \(p_{Y^{\prime}}(y)\) the densities with respect to \(y\) corresponding to \(p_{X}(x)\) and \(p_{X^{\prime}}(x)\), respectively. Then, we have \(I_{f}(p_{X}:p_{X^{\prime}})=I_{f}(p_{Y}:p_{Y^{\prime}})\)[18]. The dualistic information-geometric structures induced by these invariant \(f\)-divergences between densities of a same parametric family \(\{p_{\theta}(x)\ :\ \theta\in\Theta\}\) of statistical models yield the Fisher information metric and the dual \(\pm\alpha\)-connections for \(\alpha=3+2\frac{f^{m}(1)}{f^{m}(1)}\), see [6] for details. It is customary to rewrite the \(\alpha\)-divergences in information geometry using rescaled parameter \(\alpha_{A}=1-2\alpha\) (i.e., \(\alpha=\frac{1-\alpha_{A}}{2}\)). Thus, the extended \(\alpha_{A}\)-divergence in information geometry is defined as follows:  
\[\hat{I}^{+}_{\alpha_{A}}(p:q)=\left\{ \begin{array}{ll}\frac{4}{1-\alpha_{A} ^{2}}\int\left(\frac{1-\alpha_{A}}{2}p+\frac{1+\alpha_{A}}{2}q-p^{\frac{1- \alpha_{A}}{2}}q^{\frac{1+\alpha_{A}}{2}}\right)\mathrm{d}\mu,&\alpha_{A}\in \mathbb{R}\backslash\{-1,1\}\\ I_{1}(p:q)=\mathrm{KL}^{+}(p:q),&\alpha_{A}=1\\ \hat{I}_{-1}(p:q)=\mathrm{KL}^{+}(q:p),&\alpha_{A}=-1\end{array} \right., \tag{11}\]  
and the reference duality is expressed by \(\hat{I}^{+}_{\alpha_{A}}(q:p)=\hat{I}^{+}_{-\alpha_{A}}(p:q)\). A statistical divergence \(D(\cdot:\cdot)\) when evaluated on densities belonging to a given parametric family \(\mathcal{P}=\{p_{\theta}\ :\ \theta\in\Theta\}\) of densities is equivalent to a corresponding contrast function \(D_{\mathcal{P}}\)[7]:  
\[D_{\mathcal{P}}(\theta_{1}:\theta_{2}):=D(p_{\theta_{1}}:p_{\theta_{2}}). \tag{12}\]  
**Remark 1**. : _Although quite confusing, those contrast functions [7] have also been called divergences in the literature [6]. Any smooth parameter divergence \(D(\theta_{1}:\theta_{2})\) (contrast function [7]) induces a dualistic structure in information geometry [6]. For example, the KLD on the family \(\Delta\) of probability mass functions defined on a finite alphabet \(\mathcal{X}\) is equivalent to a Bregman divergence, and thus induces a dually flat space [6]. More generally, the \(\alpha_{A}\)-divergences on the probability simplex \(\Delta\) induce the \(\alpha_{A}\)-geometry in information geometry [6]._  
We refer the reader to [3] for a richly annotated bibliography of many common statistical divergences investigated in signal processing and statistics. Building and studying novel statistical/parameter divergences from first principles is an active research area. For example, [PERSON] [19; 20] recently introduced some new divergence functionals based on the framework of transport information geometry [21], which considers information entropy functionals in Wasserstein spaces.