 Zero-rate distortion threshold

For the finite alphabets considered here, the zero-rate distortion threshold is the smallest expected distortion obtainable by always outputting one fixed reconstruction. Such a reconstruction is independent of the source, so its mutual information is zero. A randomized output that is independent of the source averages the costs of fixed reconstructions and cannot improve their minimum. Thus $R(D)=0$ exactly when $D\ge D_{\mathrm{zero}}$. For a fair bit under Hamming distortion, $D_{\mathrm{zero}}=1/2$; for a Bernoulli random variable of parameter $p$, it is $\min \{p,1-p\}$. The threshold depends on the source probability distribution, rather than only the alphabet.