3D Gaussians, as an explicit scene representation, typically involve thousands to millions of elements per scene. This makes it challenging to control the scene in ways that reflect the underlying semantics, where the number of independent entities is typically much smaller. Especially, if one wants to animate or edit objects in the scene, as this requires coordination among the many Gaussians involved in representing each object. To address this issue, we develop a mutual information shaping technique that enforces resonance and coordination between correlated Gaussians via a Gaussian attribute decoding network. Such correlations can be learned from putative 2D object masks in different views. By approximating the mutual information with the gradients concerning the network parameters, our method ensures consistency between scene elements and enables efficient scene editing by operating on network parameters rather than massive Gaussians. In particular, we develop an effective learning pipeline named InfoGS with lightweight optimization to shape the attribute decoding network ,while ensuring that the shaping (consistency) is maintained during continuous edits, avoiding re-shaping after parameter changes. Notably, our training only touches a small fraction of all Gaussians in the scene yet attains the desired correlated behavior according to the underlying scene structure. The proposed technique is evaluated on challenging scenes and demonstrates significant performance improvements in 3D object segmentation and promoting scene interactions, while inducing low computation and memory requirements. Our code is available at: https://github.com/StylesZhang/InfoGS.