A relation between conditional and unconditional soft bit densities of binary input memoryless symmetric channels

We show that a general very simple relation holds between the variance of the unconditional and the mean of the conditional expected bit: both are identical for channels that are symmetric and in addition possess the property of exponential symmetry (also called consistency). Special cases are the binary erasure channel, the binary symmetric channel or the binary AWGN channel. This equality allows to obtain the conditional soft bit mean from observing the unconditional channel or decoder output which is of interest for estimators using expected bit or symbol values. We prove our result using results from Sharon et al., where properties of conditional soft bit densities had been used to study EXIT functions. We exploit the relation by applying it to the problem of soft turbo channel estimation.