Identifying the relevant degrees of freedom is a key to developing an effective theory. RG provides a framework for this task, but its practical execution in unfamiliar systems is difficult. Machine learning (ML) approaches, on the other hand, though promising, lack formal interpretability: it is often unclear what relation, if any, the architecture- and training-dependent "relevant" features bear to standard objects of physical theory. I will present results addressing both issues. We develop a fast algorithm, the RSMI-NE, employing recent results in ML-based estimation of information-theoretic quantities to identify the most relevant field theory operators describing a statistical system. We show how comprehensive information about the phase diagram, correlations, and symmetries (also emergent!) can be obtained, and validate the approach on the example of the interacting dimer model. I will also discuss formal results underlying the algorithm: we establish an equivalence between the information-theoretic notion of relevance defined in the Information Bottleneck (IB) approach of compression theory, and the field-theoretic relevance of the RG. We show analytically that for statistical physical systems the "relevant" degrees of freedom found using IB (and RSMI-NE) indeed correspond to operators with the lowest scaling dimensions. This provides a formal dictionary connecting two distinct theoretical toolboxes, and conceptually paves the way towards automated theory-building.