I have a simple question regarding the transmission of binary data over a Binary Symmetric Channel (BSC) using LDPC codes.
Let's assume that the input message $x\in[0,1]$ is encoded using binary LDPC codes and transmitted over BSC. At the receiver side, the LDPC decoder based on belief propagation (sum-product algorithm) decodes the message and output the soft information $y$ in the form of log-likelihood ratios.
My question is that is it possible to determine $P(x=0|y)$ or $P(x=1|y)$ directly using the soft information $y$ i.e. the reliability about the decoded information is also reflected in the above mentioned probability calculation.
PS: The a priori probability $P(x)$ is also known at the decoder.
For example:
If we represent the variable nodes (output) of LDPC decoder by a random variable $L$ which can take on values from $+\infty$ to $-\infty$ (log likelihood ratios), for example, how to determine $Pr(x=0|l=+2.00)$ or $Pr(x=1|l=-3.00)$
$$\text{LLR}=\log\left(\frac{\Pr(x=0|y)}{\Pr(x=1|y)}\right)$$ with known LLR, and using the fact that $$\Pr(x=0|y)+\Pr(x=1|y)=1$$ You can find each of the two probabilities.