Distribution of Lagrangian over-densities, δ, and the Laplacian, L (in units of h² Mpc⁻²) p(δ, L|g) (left), and the multi-variate bias function (right). The top row shows the distribution for matter particles for k_d = 0.2 h Mpc⁻¹ and the second row of haloes selected in the mass-range M_200b ∼ 2 × 10¹⁴ h M_⊙. The remaining rows show approximations to the halo distribution through a 3 parameter second-order expansion, a 5 parameter second-order expansion and a Gaussian bias model. The contours show the 1-,2- and 3-sigma regions of the matter distribution. The measurement for haloes has only reasonable statistics roughly inside of the 3-sigma region of the matter distribution (outermost grey contour). The multi-variate Gaussian bias seems to be a good approximation to the actual distribution and adequately respects the positivity constraint.