  Due to the unobservability of confoundings, there has been widespread concern
about how to compute causality quantitatively. To address this challenge,
proxy-based negative control approaches have been commonly adopted, where
auxiliary outcome variables $\bm{W}$ are introduced as the proxy of
confoundings $\bm{U}$. However, these approaches rely on strong assumptions
such as reversibility, completeness, or bridge functions. These assumptions
lack intuitive empirical interpretation and solid verification techniques,
hence their applications in the real world are limited. For instance, these
approaches are inapplicable when the transition matrix $P(\bm{W} \mid \bm{U})$
is irreversible. In this paper, we focus on a weaker assumption called the
partial observability of $P(\bm{W} \mid \bm{U})$. We develop a more general
single-proxy negative control method called Partial Identification via
Sum-of-ratios Fractional Programming (PI-SFP). It is a global optimization
algorithm based on the branch-and-bound strategy, aiming to provide the valid
bound of the causal effect. In the simulation, PI-SFP provides promising
numerical results and fills in the blank spots that can not be handled in the
previous literature, such as we have partial information of $P(\bm{W} \mid
\bm{U})$.
