Supplementary Material for "A rate-distortion scenario for the emergence and evolution of noisy molecular codes"

In principle, to nd the optimal code one needs to optimize the tness functionH with respect to all its 2 m s degrees of freedom, the m s entries of the encoder e i and the m s entries of the decoder dj!, taking into account the m conservation relations for the encoder P i e i = 1 and the s conservation relations for the decoder P ! dj! = 1. However, although the degrees of freedom of the encoder and the decoder are independent, most encoder-decoder combinations would yield high distortion and therefore an improbable code. Instead, it proves convenient to sum only over the most probable codes by considering for each encoder e i only the most probable decoder dj! (This is somewhat similar to a saddle-point approximation). An optimal decoder must take into account the knowledge about the encoder, that is if the encoder e i tends to encode the meaning ! as the symbol j then it is likely that the decoder dj! will interpret j as !. The mathematical manifestation of this intuitive observation is through the Bayes theorem, P (j ! !) = P (j)P (jj!) = P (!)P (jj!) = P (! ! j). For the present channel, Bayes theorem can be expressed as