Let (**x**) be the
field of mass-density fluctuations and *g ( x)*
the corresponding field of galaxy-density
fluctuations, at a given time and for a given type of object.
The fields are both smoothed with a fixed window
which defines the term ``local''.
The local biasing relation is considered to be
a

Define the *mean biasing function* *b*() by the conditional
mean,

This function is plotted in Figure 1.
It is a natural generalization of the deterministic linear biasing relation,
*g* = *b*_{1} .
The function *b*() allows
for any possible *non-linear* biasing.
We find it useful to characterize the function *b*()
by its moments and
defined by

It will become clear that is the
natural extension of *b*_{1}
and that /
is the relevant measure of
non-linearity, independent of stochasticity.

The local *statistical* character of the biasing can be expressed
by the conditional moments of higher order about the mean at a given
Define the random *biasing field* by
*g* - < *g* | >, with < | > = 0.
The local variance of at a
given defines the biasing
*scatter
function* _{b}() and by averaging over
one obtains the local biasing *scatter parameter*:

The scaling by ^{2} is
for convenience.
The function < ^{2}
| >^{1/2} is marked by
error bars in
Figure 1.
Here and below we make use of a straightforward lemma,
valid for any functions *p(g)* and *q*
():

From the three basic parameters defined above one can derive other
biasing parameters.
A common one is the ratio of *variances*,

The second equality is a result of Eq. (4).
It immediately shows that *b*_{var} is sensitive both to
non-linearity and
to stochasticity, with *b*_{var} .
This makes *b*_{var} biased compared to
,

Using Eq. (4), the mean parameter
is related to the *covariance*,

Thus, is the slope of the linear regression of *g* on
,
which makes it a natural generalization of *b*_{1}.
Unlike the variance _{g}^{2} in Eq. (5), the covariance in Eq. (7)
has no contribution from _{b}.
A complementary parameter to *b*_{var} is the linear
*correlation coefficient*,

The ``inverse" regression, of
on *g*, yields another biasing parameter:

Thus, *b*_{inv} is biased relative to
, even more than
*b*_{var}.
The parameter *b*_{inv} is close to what is measured in practice
by two-dimensional linear regression
[52],
because the errors in are
larger than in *g*.
Note that and
_{b} nicely separate the
non-linearity and stochasticity, while *b*_{var}, *r*
and *b*_{inv} mix them.

In the case of *linear*, stochastic biasing,
the above parameters reduce to

Thus, *b*_{1}
*b*_{var}
*b*_{inv}.
In the case of non-linear *deterministic* biasing:

In the fully degenerate case of linear and deterministic biasing,
all the *b* parameters are the same, and only then r = 1.

In actual applications,
the above local biasing parameters are involved when the parameter
``''
is measured from observational data.
For linear and deterministic biasing this parameter is defined unambiguously
as _{1}
*f*
() /
*b*_{1}, but any deviation from this
model causes us to measure different
's by the
different methods. For example,
it is _{var}
*f* () / *b*_{var} which is
determined from _{g}
and *f* ().
The former is typically determined from a redshift survey,
and the latter either from an analysis of peculiar velocity data,
from the abundance of rich clusters,
or by COBE normalization of a specific power-spectrum shape.
In the case of stochastic biasing *b*_{var}
is always an overestimate of ,
Eq. (5), and when the biasing is linear
*b*_{var} is an overestimate of
*b*_{1}. Therefore _{var} is underestimated
accordingly.

Another useful way of estimate
is via the *linear regression* of the fields in our cosmological
neighborhood, e.g., - **.**
**v**(**x**) on *g ( x)*
[17,
33,
52].
In the mildly-non-linear regime, -