  For a layered probabilistic shaping (PS) scheme with a general decoding
metric, an achievable rate is derived using Gallager's error exponent approach
and the concept of achievable code rates is introduced. Several instances for
specific decoding metrics are discussed, including bit-metric decoding,
interleaved coded modulation, and hard-decision decoding. It is shown that
important previously known achievable rates can also be achieved by layered PS.
A practical instance of layered PS is the recently proposed probabilistic
amplitude shaping (PAS).
