  We present a framework that can exploit the tradeoff between the undetected
error rate (UER) and block error rate (BLER) of polar-like codes. It is
compatible with all successive cancellation (SC)-based decoding methods and
relies on a novel approximation that we call codebook probability. This
approximation is based on an auxiliary distribution that mimics the dynamics of
decoding algorithms following an SC decoding schedule. Simulation results
demonstrates that, in the case of SC list (SCL) decoding, the proposed
framework outperforms the state-of-art approximations from Forney's generalized
decoding rule for polar-like codes with dynamic frozen bits. In addition,
dynamic Reed-Muller (RM) codes using the proposed generalized decoding
significantly outperform CRC-concatenated polar codes decoded using SCL in both
BLER and UER. Finally, we briefly discuss three potential applications of the
approximated codebook probability: coded pilot-free channel estimation; bitwise
soft-output decoding; and improved turbo product decoding.
