  The dark matter halos that host galaxies and clusters form out of initial
high-density patches, providing a biased tracer of the linear matter density
field. In the simplest local bias approximation, the halo field is treated as a
perturbative series in the average overdensity of the Lagrangian patch. In more
realistic models, however, additional quantities will affect the clustering of
halo-patches, and this expansion becomes a function of several stochastic
variables. In this paper, we present a general multivariate expansion scheme
that can parametrize the clustering of any biased Lagrangian tracer, given only
the variables involved and their symmetry (in our case rotational invariance).
This approach is based on an expansion in the orthonormal polynomials
associated with the relevant variables, so that no renormalization of the
coefficients ever occurs. We provide explicit expression for the series
coefficients, or Lagrangian bias parameters, in the case of peaks of the linear
density field. As an application of our formalism, we present a simple
derivation of the original BBKS formula, and compute the non-Gaussian bias in
the presence of a primordial trispectrum of the local shape.
