  Accurate performance benchmarking after forward error correction (FEC)
decoding is essential for system design in optical fiber communications.
Generalized mutual information (GMI) has been shown to be successful at
benchmarking the bit-error rate (BER) after FEC decoding (post-FEC BER) for
systems with soft-decision (SD) FEC without probabilistic shaping (PS).
However, GMI is not relevant to benchmark post-FEC BER for systems with SD-FEC
and PS. For such systems, normalized GMI (NGMI), asymmetric information (ASI),
and achievable FEC rate have been proposed instead. They are good at
benchmarking post-FEC BER or to give an FEC limit in bit-interleaved coded
modulation (BICM) with PS, but their relation has not been clearly explained so
far. In this paper, we define generalized L-values under mismatched decoding,
which are connected to the GMI and ASI. We then show that NGMI, ASI, and
achievable FEC rate are theoretically equal under matched decoding but not
under mismatched decoding. We also examine BER before FEC decoding (pre-FEC
BER) and ASI over Gaussian and nonlinear fiber-optic channels with
approximately matched decoding. ASI always shows better correlation with
post-FEC BER than pre-FEC BER for BICM with PS. On the other hand, post-FEC BER
can differ at a given ASI when we change the bit mapping, which describes how
each bit in a codeword is assigned to a bit tributary.
