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  "body": "\n## Definition\n\nLet $P$ and $Q$ be [[probability measures|059TJN0NZ1]] on a measurable space $(\\mathcal{X}, \\mathcal{F})$.\nThe *total variation distance* between $P$ and $Q$ is\n$$\\delta(P, Q) = \\sup_{A \\in \\mathcal{F}} \\abs{P(A) - Q(A)}.$$\n\n## Remarks\n\nIt lies in $[0, 1]$, and it is a [[metric|0VKD1TP7F3]] on the probability measures on $(\\mathcal{X}, \\mathcal{F})$.\nIf $\\lambda$ is a measure with densities $p = dP/d\\lambda$ and $q = dQ/d\\lambda$, then $\\delta(P, Q) = \\frac{1}{2} \\int \\abs{p - q} \\, d\\lambda$, and the supremum is attained at $A = \\set{p \u003e q}$.\nEquivalently, $\\delta(P, Q) = \\frac{1}{2} \\norm{P - Q}$, where $\\norm{\\cdot}$ is the [[total variation|EPBND51CR5]] norm of the signed measure $P - Q$.\nSome authors call $\\norm{P - Q}$ itself the total variation distance, which differs by the factor $2$.\n\nExample: let $P$ and $Q$ be Bernoulli distributions with parameters $p$ and $q$.\nThe four events are $\\emptyset$, $\\set{0}$, $\\set{1}$ and $\\set{0, 1}$, with differences $0$, $\\abs{q - p}$, $\\abs{p - q}$ and $0$.\nSo $\\delta(P, Q) = \\abs{p - q}$.\n\nNon-example: the largest difference over single points, $\\sup_x \\abs{P(\\set{x}) - Q(\\set{x})}$, is not the total variation distance.\nFor the [[normal distributions|05AQE0QR0D]] with means $0$ and $1$ and variance $1$, every point has probability $0$, so this quantity is $0$.\nYet $\\delta(P, Q) \u003e 0$, since the two measures are different.\n",
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