f X ( x ) {\displaystyle f_{X}(x)} is the prior density of X {\displaystyle X} , L X ∣ Y = y ( x ) = f Y ∣ X = x ( y ) {\displaystyle {\mathcal {L}}_{X\mid Y=y}(x)=f_{Y\mid X=x}(y)} is the likelihood function as a function of x {\displaystyle x} , ∫ − ∞ ∞ f X ( u ) L X ∣ Y = y ( u ) d u {\displaystyle \int _{-\infty }^{\infty }f_{X}(u){\mathcal {L}}_{X\mid Y=y}(u)\,du} is the normalizing constant, and f X ∣ Y = y ( x ) {\displaystyle f_{X\mid Y=y}(x)} is the posterior density of X {\displaystyle X} given the data Y = y {\displaystyle Y=y} .