Posterior Probability

$f\_\{X\backslash mid\; Y=y\}(x)=\{f\_X(x)\; L\_\{X\backslash mid\; Y=y\}(x)\; \backslash over\; \{\backslash int\_\{-\backslash infty\}^\backslash infty\; f\_X(x)\; L\_\{X\backslash mid\; Y=y\}(x)\backslash ,dx\}\}$

• $f\_X(x)$ is the prior density of X,
• $L\_\{X\backslash mid\; Y=y\}(x)\; =\; f\_\{Y\backslash mid\; X=x\}(y)$ is the likelihood function as a function of x,
• $\backslash int\_\{-\backslash infty\}^\backslash infty\; f\_X(x)\; L\_\{X\backslash mid\; Y=y\}(x)\backslash ,dx$ is the normalizing constant, and
• $f\_\{X\backslash mid\; Y=y\}(x)$ is the posterior density of X.

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