Left panels: the occurrence rate of hot Neptunes (top), warm Neptunes (middle), and cold Neptunes (bottom) as a function of stellar metallicity [Fe/H] from the synthetic NGPPS population by Bern planet formation and evolution model at 5 Gyr. The solid lines denote the best fits of Equation (1). Right panels: marginalized posterior probability density distributions of the parameters (C,β) for the occurrence rates of hot Neptunes (top), warm Neptunes (middle), and cold Neptunes (bottom) conditioned on the synthetic data generated by the Generation III Bern model.