# Log likelihhood ratio with soft information

The log likelihood ratio of two binary random variables $x$ and $y$ (only taking values $0$ or $1$) can be defined as

$LLR_x=\frac{P(x=0|y)}{P(x=1|y)}$

The above definition is OK if $y=0$ or $1$.

Now if $y$ is in the form of log likelihood ratio (soft information output from another system), how can it be incorporated in the above equation i.e.

$LLR_y=\frac{P(y=0|z)}{P(y=1|z)}$

so that,

$LLR_xnew=\frac{P(x=0|LLR_{y})}{P(x=1|LLR_{y})}$

Obviously, I don't want to take the hard decisions on the value of $y$ (it is straight forward for the hard decisions).

I further elaborate the question below.

1. The a priori probabilities about $x$ and $y$ are known i.e. it is possible to define both $LLR_{apriori}=\frac{P(x=0|y=0)}{P(x=1|y=0)}$ and $LLR_{apriori}=\frac{P(x=0|y=1)}{P(x=1|y=1)}$.
2. Now let's assume that $y$ is transmitted over a noisy channel and at the receiver side, it is recovered as soft information in the form of log likelihood ratios.
3. Next, on the basis of the received (recovered) value of $y$ referred to as $y^{'}$ and according to the defined decoder, I have to choose one of the above defined LLRs so that the decoder can further process or recover the value of $x$ (The information about the decoder is irrelevant here).
4. Now, if we take the hard decisions of $y^{'}$, it is very easy to choose one of the LLRs depending upon if $y^{'}=0$ or $y^{'}=1$ (Although the error will propagate in case of wrong decision)
5. However, I want to incorporate the soft value to choose the a priori LLRS e.g. if the recovered $LLR$ of $y$ is $+10$, the reliability of the decision must be reflected in the selected a priori LLRs. But the a priori probabilities are based on the distribution of the original data (not on the received data which may have errors)

I'll give a very simple example.

• Assume the a priori LLRs are $LLR_{apriori}=\frac{P(x=0|y=0)}{P(x=1|y=0)}=+0.84$ and $LLR_{apriori}=\frac{P(x=0|y=1)}{P(x=1|y=1)}=-0.84$
• After transmission, the $LLR$ of the recovered $y^{'}=+10$
• Being the hard decision of $y^{'}=0$, the decoder choose the a priori $LLR=+0.84$ and feed it to the next stage for the recovery of $x$.
• So again the question, is there anyway to incorporate the soft information ?and if it is, how to prove it mathematically?

$LLR_{x,new} =\log \frac{P(y|x=0)}{P(y|x=1)}+\log \frac{P(x=0)}{P(x=1)}$
$LLR_{x,new}=LLR(y|x)+LLR(x)$