Statement

Lemma

Suppose we have two independent random variables $X$ and $Y$ over different domains $A$ and $B$. Then the Conditional entropy is excludes the dependent

$$H(Y \vert X) = H(Y).$$

Proof

This follows from the definitions

$$ \begin{align*} H(Y \vert X) = & - \sum_{a \in A} \sum_{b \in B} \mathbb{P}[X = a, Y = b] \log(\mathbb{P}[Y = b \vert X = a])\\ = &- \sum_{a \in A} \sum_{b \in B} \mathbb{P}[X = a] \mathbb{P}[Y = b] \log(\mathbb{P}[Y = b])\\ = &\sum_{a \in A} \mathbb{P}[X = a] \left ( - \sum_{b \in B} \mathbb{P}[Y = b] \log(\mathbb{P}[Y = b]) \right )\\ = &H(Y). \end{align*} $$