I am wondering if there is way to derive the correlation between Jensen Shannon divergence and KL divergence for two distributions: P and Q, in order to show that if JSD(P,Q) decreases, KLD(P,Q) decreases as well.

JSD(P,Q) = 0.5*KLD(P,(P+Q)/2) + 0.5*KLD(Q,(P+Q)/2)


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