Detailed representation of the BHM used in this study for the inference of cosmological parameters ω from galaxy surveys. The hidden-box model corresponds to the second dashed orange rectangle and defines the true (unknown) likelihood L(θ) when Φ = Φ_O. The variables are ω (the target cosmological parameters), θ (the initial matter power spectrum), ψ (the nuisance parameters), o (the cosmological parameters, formally treated as nuisance parameters within the hidden-box), d (the full data vector containing the full galaxy count fields), Φ (the summary statistics), and $\stackrel{\sim }{\mathit{\omega }}$ (the compressed summaries). We provide further details about the roles and physical meanings of these variables in Table 1.