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  "body": "\n## Definition\n\nLet $M_0$ and $M_1$ be two models for data $x$.\nFor $i = 0, 1$, model $M_i$ has a parameter $\\theta_i \\in \\Theta_i$, a density or mass function $f_i(x \\mid \\theta_i)$ and a proper [[prior|00NBBSXRQS]] $\\pi_i$ on $\\Theta_i$.\nThe *marginal likelihood* of $M_i$ is\n$$m_i(x) = \\int_{\\Theta_i} f_i(x \\mid \\theta_i) \\, \\pi_i(d\\theta_i).$$\nIf $m_0(x) \u003e 0$, the *Bayes factor* of $M_1$ against $M_0$ is\n$$B_{10}(x) = \\frac{m_1(x)}{m_0(x)}.$$\n\n## Remarks\n\nBy [[Bayes' theorem|0NM6E9VW04]], with prior model probabilities $P(M_0), P(M_1) \u003e 0$, the posterior odds are\n$$\\frac{P(M_1 \\mid x)}{P(M_0 \\mid x)} = B_{10}(x) \\cdot \\frac{P(M_1)}{P(M_0)}.$$\nIf both models are simple, that is each $\\Theta_i$ is a single point, then $B_{10}$ is the [[likelihood|05FSTXM92W]] ratio.\nExample: let $M_0$ say that a coin is fair and $M_1$ that its chance $p$ of heads is uniform on $[0, 1]$, and observe two heads in two tosses.\nThen $m_0 = 1/4$ and $m_1 = \\int_0^1 p^2 \\, dp = 1/3$, so $B_{10} = 4/3$.\nNon-example: with an improper prior $\\pi_1$, the integral $m_1(x)$ is defined only up to an arbitrary constant factor, so the ratio is not a well-defined Bayes factor.\n",
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