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  "body": "\n## Definition\n\nLet $P$ and $Q$ be [[probability measures|059TJN0NZ1]] on a measurable space $(\\mathcal{X}, \\mathcal{F})$.\nLet $\\lambda$ be a [[σ-finite measure|WEBBT18D4X]] such that $P$ and $Q$ are absolutely continuous with respect to $\\lambda$, for example $\\lambda = P + Q$.\nLet $p = dP/d\\lambda$ and $q = dQ/d\\lambda$ be the [[Radon–Nikodym derivatives|1NTKG64F7N]].\nThe *Hellinger distance* between $P$ and $Q$ is the number $H(P, Q) \\ge 0$ given by\n$$H^2(P, Q) = \\frac{1}{2} \\int_{\\mathcal{X}} \\big( \\sqrt{p} - \\sqrt{q} \\big)^2 \\, d\\lambda.$$\n\n## Remarks\n\nThe value does not depend on the choice of $\\lambda$.\nExpanding the square gives $H^2(P, Q) = 1 - \\int \\sqrt{p q} \\, d\\lambda$, so $0 \\le H \\le 1$.\nThe map $H$ is a [[metric|0VKD1TP7F3]] on the set of probability measures on $(\\mathcal{X}, \\mathcal{F})$.\nSome authors omit the factor $\\frac{1}{2}$, which multiplies $H$ by $\\sqrt{2}$.\nWith the convention above, the [[total variation distance|3SZESKX3N5]] $\\delta$ satisfies $H^2(P, Q) \\le \\delta(P, Q) \\le \\sqrt{2} \\, H(P, Q)$.\n\nExample: on $\\set{0, 1}$ with counting measure, let $P$ give mass $\\frac{1}{2}$ to each point and let $Q$ be the point mass at $1$.\nThen $\\int \\sqrt{pq} = \\sqrt{0 \\cdot \\tfrac{1}{2}} + \\sqrt{1 \\cdot \\tfrac{1}{2}} = 1/\\sqrt{2}$, so $H^2(P, Q) = 1 - 1/\\sqrt{2}$.\n\nNon-example: the quantity $\\frac{1}{2} \\int (p - q)^2 \\, d\\lambda$, without square roots, is not the Hellinger distance.\nIt changes with $\\lambda$: replacing $\\lambda$ by $2\\lambda$ halves $p$ and $q$ and divides the quantity by $2$.\n",
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