Conditional entropy H(X|Y) equals unconditional entropy H(X) if and only if the random variables X and Y are statistically independent. In this case, knowing Y provides no information about X, so the uncertainty remains unchanged.
Conditions: X and Y are discrete random variables.
Conditional entropy H(X|Y) equals unconditional entropy H(X) if and only if the random variables X and Y are statistically independent. In this case, knowing Y provides no information about X, so the uncertainty remains unchanged.
Conditions: X and Y are discrete random variables.
Joint entropy H(X,Y) is not equal to H(X)+H(Y) because adding the marginal entropies double-counts the mutual information I(X;Y) shared between the variables. The correct relationship is H(X,Y)=H(X)+H(Y)−I(X;Y).
Conditions: X and Y are random variables that may share information (i.e., they are not necessarily independent).
Joint entropy H(X,Y) is not equal to H(X)+H(Y) because adding the marginal entropies double-counts the mutual information I(X;Y) shared between the variables. The correct relationship is H(X,Y)=H(X)+H(Y)−I(X;Y).
Conditions: X and Y are random variables that may share information (i.e., they are not necessarily independent).
When conditional entropy H(X|Y) is zero, it means there is no remaining uncertainty or surprise about X once Y is known. This implies that X is a deterministic function of Y, so all information about X is completely contained within Y.
Conditions: X and Y are discrete random variables.
When conditional entropy H(X|Y) is zero, it means there is no remaining uncertainty or surprise about X once Y is known. This implies that X is a deterministic function of Y, so all information about X is completely contained within Y.
Conditions: X and Y are discrete random variables.
Conditional entropy H(X|Y) is bounded by 0≤H(X|Y) ≤ H(X) because learning the value of Y provides information that can only reduce or leave unchanged the average uncertainty about X; it cannot increase it on average.
Conditions: X and Y are discrete random variables.
Conditional entropy H(X|Y) is bounded by 0≤H(X|Y) ≤ H(X) because learning the value of Y provides information that can only reduce or leave unchanged the average uncertainty about X; it cannot increase it on average.
Conditions: X and Y are discrete random variables.