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The Bhattacharyya distance is the negative natural logarithm of the Bhattacharyya coefficient, with $-\ln 0=\infty$. It is zero for identical probability distributions and infinite for disjoint probability supports. Despite the name, it need not satisfy the triangle inequality required of a metric.

For probability mass functions $P,Q$, their Chernoff affinity at $0<s<1$ is the sum $Z(s)={\sum }_{x}P(x{)}^{s}Q(x{)}^{1-s}$. It weights pointwise agreement between the two laws. The weighted arithmetic-geometric mean inequality gives $P(x{)}^{s}Q(x{)}^{1-s}\le sP(x)+(1-s)Q(x)$, hence $Z(s)\le 1$. 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 $1/2$.