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I'm reading the book by MacKay "Information theory, inference and learning algorithms" and I'm confused by how he introduces the decoding problem for LDPC codes (page 557).

given a channel output $\mathbf{r}$, we wish to find the codeword $\mathbf{t}$ that whose likelihood $P(\mathbf{r}|\mathbf{t})$ is biggest.

I can get on board with that, even if it seems a bit backwards since $\mathbf{r}$ is the evidence we have, so I would be more comfortable if we were trying to find $\mathbf{t}$ that maximizes $P(\mathbf{t}|\mathbf{r})$, and as a matter of fact, he goes on to say

there are two approaches to the decoding problem both of which lead to the generic problem 'find $\mathbf{x}$ that maximizes $$ P^*(\mathbf{x})=P(\mathbf{x})\mathbb{1}[\mathbf{Hx}=\mathbf{z}]$$

he exposes the two points of view in the following page, the first one especially confuses me

the codeword decoding point of view

First, we note the prior distribution over codewords $$ P(\mathbf{t})=\mathbb{1}[\mathbf{Ht}=\mathbf{0}]$$[...]. The posterior distribution over codewords is given by multiplying the prior by the likelihood [...] $$P(\mathbf{t}|\mathbf{r})\propto P(\mathbf{t})P(\mathbf{r}|\mathbf{t}) $$

from the generic decoding problem he gave, it looks like we're actually trying to maximize $P(\mathbf{x})=P(\mathbf{t}|\mathbf{r})$, instead of $P(\mathbf{r}|\mathbf{t})$.

Is it obvious that the two maxima are the same and with the same maximizer? It is not obvious to me since the maximizer $\mathbf{t_0}$ that maximizes $P(\mathbf{r}|\mathbf{t})$ might actually minimize $P(\mathbf{t})$, so the maximizer of $P(\mathbf{t})P(\mathbf{r}|\mathbf{t})$ might be different! I understand that in this case $P(\mathbf{t})$ is uniform, so this shouldn't be a problem, but it seems weird to me that this is simply not stated and I feel like I'm missing something.

Why should we start with the problem of maximizing $P(\mathbf{r}|\mathbf{t})$ and not $P(\mathbf{t}|\mathbf{r})$? Why does he seem to switch after a few sentences? Am I correct in thinking the algorithm presented actually maximizes $P(\mathbf{t}|\mathbf{r})$?

Thank you

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Maximizing $P(r|t)$ is known as the Maximum Likelihood (ML) principle, while maximizing $P(t|r)$ is known as the Maximum A Posteriori probability (MAP) principle.

Here are some facts that answer your question:

  • Optimal decoding (minimizing the probability of error) is achieved by the MAP principle. See p. 152 of the book.
  • However, using MAP would require a different decoding algorithm for every distribution on $t$. I.e. you would have to adapt decoding to the statistical properties of the source. To avoid this, it is typical to assume a uniform distribution on $t$. In this case, MAP and ML are equivalent because $P(t|r)$ becomes proportional to $P(r|t)$ (see p. 152 again).
  • I agree with you he should have started with the MAP problem of maximizing $P(t|r)$, since that's the correct goal anyway. But I presume that he is assuming a uniform distribution on $t$, in which case the difference doesn't matter anyway.
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