Intuitive explanation of the definition : According to the definition, H ( Y | X ) = E ( f ( X , Y ) ) {\displaystyle \displaystyle H(Y|X)=\mathbb {E} (\ f(X,Y)\ )} where f : ( x , y ) → − log ⁡ ( p ( y | x ) ) . {\displaystyle \displaystyle f:(x,y)\ \rightarrow -\log(\ p(y|x)\ ).} f {\displaystyle \displaystyle f} associates to ( x , y ) {\displaystyle \displaystyle (x,y)} the information content of ( Y = y ) {\displaystyle \displaystyle (Y=y)} given ( X = x ) {\displaystyle \displaystyle (X=x)} , which is the amount of information needed to describe the event ( Y = y ) {\displaystyle \displaystyle (Y=y)} given ( X = x ) {\displaystyle (X=x)} . According to the law of large numbers, H ( Y | X ) {\displaystyle \displaystyle H(Y|X)} is the arithmetic mean of a large number of independent realizations of f ( X , Y ) {\displaystyle \displaystyle f(X,Y)} .