 Binary rate-distortion function

For a stationary source of independent and identically distributed fair bits, with Hamming distortion and binary reconstruction alphabet,The binary entropy convention is $0{\log }_{2}0=0$. Once $D\ge 1/2$, always reconstructing zero meets the distortion constraint without sending any bits. The formula $1-h_2(D)$ must not be extended to $D>1/2$: a looser constraint cannot increase the required rate.^[ref]

The short argument identifies both sides of the optimum. For any feasible binary reconstruction, put $E=X\oplus \hat{X}$, using exclusive or. Knowing $\hat{X}$ makes $X$ and $E$ determine one another, so the conditional entropy obeyswhen $0\le D\le 1/2$. The last step uses the increase of binary entropy on that interval. Hence $I(X;\hat{X})=1-H(X\mid \hat{X})\ge 1-h_2(D)$. Conversely, choose $\hat{X}=X\oplus N$ with an independent Bernoulli random variable $N$ of parameter $D$. This binary symmetric channel has expected distortion $D$ and mutual information $1-h_2(D)$, attaining the bound.