Understanding the structure of neural circuits requires distinguishing between correlations that arise from direct interactions between neurons and those mediated by the broader network. Pairwise correlations are straightforward to measure experimentally, yet they conflate direct and indirect dependencies. Partial correlations, by contrast, quantify pairwise dependencies conditioned on the activity of all other neurons and thus provide a more principled route to inferring network connectivity. We present a theoretical framework that bridges these two measures by showing that the total correlation between any two neurons can be decomposed into a sum over all network paths connecting them. Each path contributes additively, with its weight defined as the product of the corresponding partial correlations along the route. This decomposition establishes a direct link between experimentally accessible correlations and the underlying conditional dependencies. Building on this insight, we introduce a Bayesian method for estimating partial correlations in high-dimensional neural data. We validate our approach in neural simulations, demonstrating its ability to recover network structure and disentangle direct from indirect interactions. Our results provide a principled strategy for moving from correlation-based observations to connectivity-based interpretations in population neuroscience.