  We connect two recent advances in the stochastic analysis of nonequilibrium
systems: the (loose) uncertainty principle for the currents, which states that
statistical errors are bounded by thermodynamic dissipation; and the analysis
of thermodynamic consistency of the currents in the light of symmetries.
Employing the large deviation techniques presented in [Gingrich et al., Phys.
Rev. Lett. 2016] and [Pietzonka et al., Phys. Rev. E 2016], we provide a short
proof of the loose uncertainty principle, and prove a tighter uncertainty
relation for a class of thermodynamically consistent currents $J$. Our bound
involves a measure of partial entropy production, that we interpret as the
least amount of entropy that a system sustaining current $J$ can possibly
produce, at a given steady state. We provide a complete mathematical discussion
of quadratic bounds which allows to determine which are optimal, and finally we
argue that the relationship for the Fano factor of the entropy production rate
$\mathrm{var}\, \sigma / \mathrm{mean}\, \sigma \geq 2$ is the most significant
realization of the loose bound. We base our analysis both on the formalism of
diffusions, and of Markov jump processes in the light of Schnakenberg's cycle
analysis.
