  We consider a periodic review inventory control problem having an underlying
modulation process that affects demand and that is partially observed by the
uncensored demand process and a novel additional observation data (AOD)
process. We present an attainability condition, AC, that guarantees the
existence of an optimal myopic base stock policy if the reorder cost $K=0$ and
the existence of an optimal $(s, S)$ policy if $K>0$, where both policies
depend on the belief function of the modulation process. Assuming AC holds, we
show that (i) when $K=0$, the value of the optimal base stock level is constant
within regions of the belief space and that each region can be described by two
linear inequalities and (ii) when $K>0$, the values of $s$ and $S$ and upper
and lower bounds on these values are constant within regions of the belief
space and that these regions can be described by a finite set of linear
inequalities. A heuristic and bounds for the $K=0$ case are presented when AC
does not hold. Special cases of this inventory control problem include problems
considered in the Markov-modulated demand and Bayesian updating literatures.
