= Chernoff affinity
{c}
{title2=$Z(s)=\sum_xP(x)^sQ(x)^{1-s}$}
For <probability mass functions> $P,Q$, their <Chernoff affinity> at $0<s<1$ is the <sum> $Z(s)=\sum_xP(x)^sQ(x)^{1-s}$. It weights pointwise agreement between the two laws. The <weighted arithmetic-geometric mean inequality> gives $P(x)^sQ(x)^{1-s}\leq sP(x)+(1-s)Q(x)$, hence $Z(s)\leq1$. With positive common support, $Z(0)=Z(1)=1$. Smaller affinity means a stronger separation at that weighting.
At $s=1/2$, $Z(1/2)=\sum_x\sqrt{P(x)Q(x)}$ is the <Bhattacharyya coefficient>. The quantity $-\ln Z(1/2)$ is the <Bhattacharyya distance>. Allowing all weights $s$ gives <Chernoff information> at least this large. Symmetry under exchanging $P,Q$ sends $s$ to $1-s$, so <Chernoff information> is symmetric even though an optimizing weight need not be $1/2$.
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