  Predictive state representations (PSRs) are models of controlled non-Markov
observation sequences which exhibit the same generative process governing POMDP
observations without relying on an underlying latent state. In that respect, a
PSR is indistinguishable from the corresponding POMDP. However, PSRs
notoriously ignore the notion of rewards, which undermines the general utility
of PSR models for control, planning, or reinforcement learning. Therefore, we
describe a sufficient and necessary accuracy condition which determines whether
a PSR is able to accurately model POMDP rewards, we show that rewards can be
approximated even when the accuracy condition is not satisfied, and we find
that a non-trivial number of POMDPs taken from a well-known third-party
repository do not satisfy the accuracy condition. We propose reward-predictive
state representations (R-PSRs), a generalization of PSRs which accurately
models both observations and rewards, and develop value iteration for R-PSRs.
We show that there is a mismatch between optimal POMDP policies and the optimal
PSR policies derived from approximate rewards. On the other hand, optimal R-PSR
policies perfectly match optimal POMDP policies, reconfirming R-PSRs as
accurate state-less generative models of observations and rewards.
