## Optimal lossy coding under known utility function

Given an alphabet $A$ and a distribution $P:A\rightarrow \mathbb{R}$, as well as a loss function $L:A \times A \rightarrow \mathbb{R}$, I want to find a lossy (possibly non-invertible) coding $C:A\rightarrow \{0,1\}^n$ minimizing

$\mathbb{E}[A, C^*(C(A))]$

where $C^*(b)$ is the maximum-likelihood decoding of $b$, i.e.

$\arg\max_{A~:~C(A)=b} P(A)$

Is it known how this coding relates to $P$ and $L$?

Is there an optimal algorithm known for creating such a coding? (Huffman or the like?)

How does this question relate to the source coding theorem and the channel coding theorem?

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 Don't you need a constraint on the rate of the code $C$? Otherwise, you can transmit $A$ uncoded and choose the reconstruction $\hat{A}$ such that $L(A,\hat{A})$ is minimized. If you constrain the rate of $C$ to $R$, the minimum distortion that you can achieve is the Shannon rate-distortion function $R(D)$. To approach this, you typically need to code over $A^n$ where the block length $n$ is very large. Except in some special cases, the optimal codebooks have little structure to them and aren't easy to build/describe. – Dinesh Dec 12 2011 at 20:39