Hierarchical representation of the Bayesian framework employed for diagnosing model misspecification and inferring the target parameters ω. The rounded green boxes denote probability distributions, whilst the purple squares represent deterministic functions. 𝒫(ω) is the prior on ω. 𝒯 is a deterministic function linking ω to θ. 𝒫(Φ|θ) denotes the probabilistic data model that maps the space of latent vectors θ to the survey space. The final layer, $\stackrel{\sim }{\mathcal{C}}$, is a deterministic compression step required for the ILI of the target parameters ω. In this study, ω is the vector of cosmological parameters, 𝒯 is derived from the Boltzmann solver CLASS, θ is the initial matter power spectrum (after recombination), and Φ corresponds to the power spectra of multiple galaxy number count fields.