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Identifiability of discrete statistical models with latent variables is knownto be challenging to study, yet crucial to a model's interpretability andreliability. This work presents a general algebraic technique to investigateidentifiability of complicated discrete models with latent and graphicalcomponents. Specifically, motivated by diagnostic tests collecting multivariatecategorical data, we focus on discrete models with multiple binary latentvariables. In the considered model, the latent variables can have arbitrarydependencies among themselves while the latent-to-observed measurement graphtakes a "star-forest" shape. We establish necessary and sufficient graphicalcriteria for identifiability, and reveal an interesting and perhaps surprisingphenomenon of blessing-of-dependence geometry: under the minimal conditions forgeneric identifiability, the parameters are identifiable if and only if thelatent variables are not statistically independent. Thanks to this theory, wecan perform formal hypothesis tests of identifiability in the boundary case bytesting certain marginal independence of the observed variables. Our resultsgive new understanding of statistical properties of graphical models withlatent variables. They also entail useful implications for designing diagnostictests or surveys that measure binary latent traits.

Blessing of Dependence: Identifiability and Geometry of Discrete Models with Multiple Binary Latent Variables