  We present a comprehensive study of the information rate loss of the max-log
approximation for $M$-ary pulse-amplitude modulation (PAM) in a bit-interleaved
coded modulation (BICM) system. It is widely assumed that the calculation of
L-values using the max-log approximation leads to an information loss. We prove
that this assumption is correct for all $M$-PAM constellations and labelings
with the exception of a symmetric 4-PAM constellation labeled with a Gray code.
We also show that for max-log L-values, the BICM generalized mutual information
(GMI), which is an achievable rate for a standard BICM decoder, is too
pessimistic. In particular, it is proved that the so-called "harmonized" GMI,
which can be seen as the sum of bit-level GMIs, is achievable without any
modifications to the decoder. We then study how bit-level channel
symmetrization and mixing affect the mutual information (MI) and the GMI for
max-log L-values. Our results show that these operations, which are often used
when analyzing BICM systems, preserve the GMI. However, this is not necessarily
the case when the MI is considered. Necessary and sufficient conditions under
which these operations preserve the MI are provided.
