This likelihood / bias model is implemented in libLSS/samplers/borg/borg_nb_likelihood.cpp

Likelihood model

The likelihood is given by the Poisson model: {$$ P(N | \lambda, r) = \frac{\Gamma(N+1/r)}{\Gamma(1/r) \Gamma(N+1)} \frac{(r\lambda)^N}{(1+r\lambda)^{1/r + N}} $$} where each 3d grid elements holding the {} galaxies are supposed to be statistically independent realizations of the galaxy field {$\lambda$}. This parameterization recovers a Poisson distribution for {$r \rightarrow 0$}.

Bias model

The galaxy density {$\rho^g$} is derived from the dark matter field {$\rho^m$} using the following relation: {$$ \rho^g = A ({\rho_m})^{\alpha} \exp{} \left(-B (\rho^m)^{-\epsilon} \right) $$} with {$A$}, {$B$} and {$\epsilon$} free parameters.