## 2 The \(\alpha\)-Divergences Induced by a Pair of Strictly Comparable Weighted Means
### The \((m,n)\)\(\alpha\)-Divergences
Thus, \((A,G)\) is a pair of strictly comparable weighted means._  
For a weighted mean \(M\), define \(M_{\lambda}^{\prime}(x,y):=M_{1-\lambda}(x,y)\). We are ready to state the definition of generalized \(\alpha\)-divergences:  
**Definition 2** (\((M,N)\)\(\alpha\)-divergences). : _The \((M,N)\)\(\alpha\)-divergences \(I_{\alpha}^{M,N}(p:q)\) between two positive densities \(p\) and \(q\) for \(\alpha\in(0,1)\) is defined for a pair of strictly comparable weighted means \(M_{\alpha}\) and \(N_{\alpha}\) with \(M_{\alpha}\geq N_{\alpha}\) by:_  
\[I_{\alpha}^{M,N}(p:q) := \frac{1}{\alpha(1-\alpha)}\int(M_{1-\alpha}(p,q)-N_{1-\alpha}(p, q))\mathrm{d}\mu,\quad\alpha\in(0,1) \tag{22}\] \[= \frac{1}{\alpha(1-\alpha)}\int\big{(}M_{\alpha}^{\prime}(p,q)-N_ {\alpha}^{\prime}(p,q)\big{)}\mathrm{d}\mu,\quad\alpha\in(0,1). \tag{23}\]  
Using \(\alpha=\frac{1-\alpha_{A}}{2}\), we can rewrite this \(\alpha\)-divergence as  
\[A_{\lambda}(x,y) = (1-\lambda)x+\lambda y,\]0 \[A_{\lambda}(x,y) = (1-\lambda)x+\lambda y,\]1  
It is important to check the conditions on the weighted means \(M_{\alpha}\) and \(N_{\alpha}\) which ensures the law of the indiscernibles of a divergence \(D(p:q)\), namely, \(D(p:q)=0\) iff \(p=q\) almost \(\mu\)-everywhere. This condition rewrites as \(\int M_{\alpha}(p,q)\mathrm{d}\mu=\int N_{\alpha}(p,q)\mathrm{d}\mu\) if and only if \(p(x)=q(x)\)\(\mu\)-almost everywhere. A sufficient condition is to ensure that \(M_{\alpha}(x,y)\
eq N_{\alpha}(x,y)\) for \(x\
eq y\). In particular, this condition holds if the weighted means \(M_{\alpha}\) and \(N_{\alpha}\) are strictly comparable weighted means. Instead of taking the difference \(M_{1-\alpha}(x:y)-N_{1-\alpha}(x:y)\) between two weighted means, we may also measure the gap logarithmically, and thus define the family of \(\log\frac{M}{N}\)\(\alpha\)-divergences as follows:  
**Definition 3** (\(\log\frac{M}{N}\)\(\alpha\)-divergence). : _The \(\log\frac{M}{N}\)\(\alpha\)-divergences \(L_{\alpha}^{M,N}(p:q)\) between two positive densities \(p\) and \(q\) for \(\alpha\in(0,1)\) is defined for a pair of strictly comparable weighted means \(M_{\alpha}\) and \(N_{\alpha}\) with \(M_{\alpha}\geq N_{\alpha}\) by:_  
\[A_{\lambda}(x,y) = (1-\lambda)x+\lambda y,\]2 \[A_{\lambda}(x,y) = (1-\lambda)x+\lambda y,\]3  
Note that this definition is different from the skewed Bhattacharyya type distance [39; 40], which rather measures  
\[A_{\lambda}(x,y) = (1-\lambda)x+\lambda y,\]4 \[A_{\lambda}(x,y) = (1-\lambda)x+\lambda y,\]5  
The ordinary \(\alpha\)-skewed Bhattacharyya distance [39] is recovered when \(N_{\alpha}=G_{\alpha}\) (weighted geometric mean) and \(M_{\alpha}=A_{\alpha}\) the arithmetic mean since \(\int A_{1-\alpha}(p,q)\mathrm{d}\mu=1\). The Bhattacharyya type divergences \(B_{\alpha}^{M,N}\) were introduced in [41] in order to upper bound the probability of error in Bayesian hypothesis testing. A weighted mean \(M_{\alpha}\) is said symmetric if and only if \(M_{\alpha}(x,y)=M_{1-\alpha}(y,x)\). When both the weighted means \(M\) and \(N\) are symmetric, we have the following reference duality [11]:  
\[A_{\lambda}(x,y) = (1-\lambda)x+\lambda y,\]6  
We consider symmetric weighted means in the remainder. In the limit cases of \(\alpha\to 0\) or \(\alpha\to 1\), we define the 0-divergence \(I_{0}^{M,N}(p:q)\) and the 1-divergence \(I_{1}^{M,N}(p:q)\), respectively, by  
\[A_{\lambda}(x,y) = (1-\lambda)x+\lambda y,\]7 \[A_{\lambda}(x,y) = (1-\lambda)x+\lambda y,\]8  
provided that those limits exist. Notice that the ordinary \(\alpha\)-divergences are defined for any \(\alpha\in\mathbb{R}\) but our generic quasi-arithmetic \(\alpha\)-divergences are defined in general on \((0,1)\). However, when the weighted means \(M_{\alpha}\) and \(N_{\alpha}\) admit weighted extrapolations (e.g., the arithmetic mean \(A_{\alpha}\) or the geometric mean \(G_{\alpha}\)) the quasi-arithmetic \(\alpha\)-divergences can be extended to \(\mathbb{R}\backslash\{0,1\}\).