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Properties of the Unified (r,s)-Divergence Measures

We shall present some properties of the measures given by (4.6), (4.7), (4.10) and (4.11). These properties are similar to one given in Chapter 2 for the unified $(r,s)-$relative information measure ${\ensuremath{\boldsymbol{\mathscr{D}}}}^s_r(P\vert\vert Q)$. Also the proof follows on similar lines, so their details are excluded. For simplicity of notations, we shall take $^1{\ensuremath{\boldsymbol{\mathscr{I}}}}^s_r(P\vert\vert Q)=\ ^1{\ensuremath{\boldsymbol{\mathscr{L}}}}^s_r(P\vert\vert Q)$, $^2{\ensuremath{\boldsymbol{\mathscr{I}}}}^s_r(P\vert\vert Q)=\ ^2{\ensuremath{\boldsymbol{\mathscr{L}}}}^s_r(P\vert\vert Q)$,$^1{\ensuremath{\boldsymbol{\mathscr{J}}}}^s_r(P\vert\vert Q)=\ ^3{\ensuremath{\boldsymbol{\mathscr{L}}}}^s_r(P\vert\vert Q)$, and $^2{\ensuremath{\boldsymbol{\mathscr{J}}}}^s_r(P\vert\vert Q)=\ ^4{\ensuremath{\boldsymbol{\mathscr{L}}}}^s_r(P\vert\vert Q)$.

Similar to expression (4.3) we can write 

$$^\alpha{\ensuremath{\boldsymbol{\mathscr{L}}}}^s_r(P\vert\vert ......\ensuremath{\boldsymbol{\mathscr{N}}}}^1_r(P\vert\vert Q)\big),\ \alpha=1,2,3\$$   and$$\ 4,$$

where ${\ensuremath{\boldsymbol{\mathscr{N}}}}_s$ is as given in (4.19).

Property 4.24. (Nonnegativity)$^\alpha{\ensuremath{\boldsymbol{\mathscr{L}}}}^s_r(P\vert\vert Q) \geq 0\ (\alpha =1,2,3\$   and$\ 4)$ with equality iff $P=Q$.

Property 4.25. (Continuity)$^\alpha{\ensuremath{\boldsymbol{\mathscr{L}}}}^s_r(P\vert\vert Q)\ (\alpha =1,2,3\$   and$\ 4)$ are continuous functions of the pair $(P,Q)$ and are also continuous with respect to the parameters $r$ and $s$.

Property 4.26. (Symmetry)$^\alpha{\ensuremath{\boldsymbol{\mathscr{L}}}}^s_r(P\vert\vert Q)\ (\alpha =1,2,3\$   and$\ 4)$ are symmetric functions of their arguments in pair, i.e., 

$$^\alpha {\ensuremath{\boldsymbol{\mathscr{L}}}}^s_r(p_1,...,p_n\v......r{L}}}}^s_r(p_{\tau(1)},...,p_{\tau(n)}\vert\vert q_{\tau(1)},...,q_{\tau(n)}),$$

$(\alpha =1,2,3$ and 4), where $\tau$ is any permutation from $1$ to $n$.

Property 4.27. (Nonnaditivity). We have

$^\alpha {\ensuremath{\boldsymbol{\mathscr{L}}}}^s_r(P_1*P_2\vert\vert Q_1*Q_2)......1)\,+ \, ^\alpha {\ensuremath{\boldsymbol{\mathscr{L}}}}^s_r(P_2\vert\vert Q_2)$

$$+\, (1-2^{1-s})\ ^\alpha {\ensuremath{\boldsymbol{\mathscr{L}}}}^......t Q_1)\ ^\alpha {\ensuremath{\boldsymbol{\mathscr{L}}}}^s_r(P_2\vert\vert Q_2),$$

for all $P_1,\ Q_{1}\in \Delta_n$, $P_2,\Q_{2} \in \Delta_m$ and $P_1*P_2, Q_1*Q_{2} \in \Delta_{nm}$, $\alpha =1,2,3$ and $4$.

Property 4.28. (Monotonicity).$^\alpha{\ensuremath{\boldsymbol{\mathscr{L}}}}^s_r(P\vert\vert Q)$ ($\alpha =1,2,3$ and $4$) are increasing functions of $r$ ($s$ fixed) and of $s$ ($r$ fixed). In particular, when $r=s$, the result still holds.

Property 4.29. (Convexity).$^\alpha{\ensuremath{\boldsymbol{\mathscr{L}}}}^s_r(P\vert\vert Q)$ ($\alpha =1,2,3$ and $4$) are convex functions of the pair of probability distributions $(P,Q) \in\Delta_n\times\Delta_n$ for $s\geq r> 0$.

Property 4.30. (Schur-convexity).$^\alpha{\ensuremath{\boldsymbol{\mathscr{L}}}}^s_r(P\vert\vert Q)$ ($\alpha =1,2,3$ and $4$) are Schur-convex functions in the pair $(P,Q) \in\Delta_n\times\Delta_n$.

Property 4.31. (Generalized data processing inequalities). We have 

$$^\alpha {\ensuremath{\boldsymbol{\mathscr{L}}}}^s_r\big(P(B)\vert......\ensuremath{\boldsymbol{\mathscr{L}}}}^s_r(P\vert\vert Q),\ \(\alpha =1,2,3\$$   and$$\ 4)$$ where$P(B)$ and $Q(B)$ are the probability distributions given by $$P(B)=\Big(\sum_{i=1}^n{p_ib_{i1}},...,\sum_{i=1}^n{p_ib_{im}}\Big)\\in\ \Delta_m,$$ and $$Q(B)=\Big(\sum_{i=1}^n{q_ib_{i1}},...,\sum_{i=1}^n{q_ib_{im}}\Big)\\in\ \Delta_m,$$ where $B=\{b_{ij}\}$, $b_{ij}\geq 0$, $\forall \ i=1,2,...,n$; $j=1,2,...,m$ is a stochastic matrix such that$\sum_{j=1}^n{b_{ij}}=1$, $\forall \ i=1,2,...,n$.

Property 4.32. (Strong generalized data processing inequalities). If the stochastic matrix B given in the property 4.31 is such that there exists an $i_o$ for which$b_{i_{o}j}\geq c>0$, $\forall\ j=1,...,m$, then we have 

$$^\alpha{\ensuremath{\boldsymbol{\mathscr{L}}}}^s_r\big(P(B)\ver......{\ensuremath{\boldsymbol{\mathscr{L}}}}^s_r(P\vert\vert Q),\ \ (\alpha =1,2,3\$$   and$$\ 4), \,\,\, s\geqr>0$$

Property 4.33. (Inequalities among the measures). We have

(i) $^\alpha {\ensuremath{\boldsymbol{\mathscr{L}}}}^s_r(P\vert\vert Q)\left\{\begi......mbol{\mathscr{L}}}}^1_r(P\vert\vert Q), & 1\leq s< \infty\end{array}\right.$ for $\alpha =1,2,3$ and 4.

(ii) $^\alpha {\ensuremath{\boldsymbol{\mathscr{L}}}}^s_r(P\vert\vert Q)\left\{\begi......mbol{\mathscr{L}}}}^s_1(P\vert\vert Q), & 1\leq r< \infty\end{array}\right.$ for $\alpha =1,2,3$ and 4.

(iii) $^1 {\ensuremath{\boldsymbol{\mathscr{L}}}}^s_r(P\vert\vert Q)\left\{\begin{arr......{\boldsymbol{\mathscr{L}}}}^s_r(P\vert\vert Q), & s\geq r\end{array}\right.$

(iv) $^3 {\ensuremath{\boldsymbol{\mathscr{L}}}}^s_r(P\vert\vert Q)\left\{\begin{arr......{\boldsymbol{\mathscr{L}}}}^s_r(P\vert\vert Q), & s\geq r\end{array}\right.$

(v) $^\alpha {\ensuremath{\boldsymbol{\mathscr{J}}}}^s_r(P\vert\vert Q)\geq 4\ ^\alpha{\ensuremath{\boldsymbol{\mathscr{I}}}}^s_r(P\vert\vert Q),$ $\alpha =1$ and $2,\ s\geq r > 0$.