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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.