 Distortion measure (information theory)

A distortion measure assigns a nonnegative cost to replacing a source symbol $x$ by a reconstruction $\hat{x}$. It need not be a metric: the two alphabets may differ, and neither symmetry nor the triangle inequality is required. Choosing the cost is part of the model. For numerical symbols, $d(x,\hat{x})=(x-\hat{x}{)}^2$ gives mean squared error after averaging; for exact symbol agreement, use Hamming distortion.