**2.2.1. Mathematical Details**

We now describe the method in detail. We
assume that the relevant distance indicator is the TF relation; with minor
changes, the formalism could be adapted to comparable distance indicators
such as *D*_{n} -
.
We use the terminology of
Willick (1994)
and Willick et
al. (1995): briefly, we denote by *m* and
log
*v*_{rot} - 2.5 a galaxy's corrected apparent magnitude and
velocity width parameter, respectively; by *cz*, its Local Group frame
radial velocity ("redshift")
in units of km s^{-1}; and by *r*, its true distance in units
of km s^{-1}. We define the distance modulus as
*µ* 5
log *r*, and absolute magnitudes as *M* = *m* -
*µ*. We write the forward and inverse TF relations as linear
expressions,
*M*()
= *A* -
*b*
and ^{0}(*M*)
= -*e*(*M* - *D*), and denote their rms
scatters _{TF}
and _{}, respectively.

We seek an exact expression for the
probability that a galaxy at redshift *cz* possesses TF
observables (*m*,
), given
a model of the peculiar velocity and
density fields. ^{(5)} We first
consider the joint probability distribution of the TF observables,
redshift, and an *unobservable* quantity, the true distance
*r*. Later, we will integrate over *r* to obtain the probability
distribution of the observables. We may write

(5) |

The splitting into conditional probabilities reflects the fact that the
TF observables and the redshift couple with one another only via
their individual dependences on the true distance *r*.

The first of the three terms on the right-hand side of equation (5) depends on the luminosity function, the sample selection function, and the TF relation. We can express it in one of two ways, depending on whether we are using the forward or inverse form of the TF relation:

1. Forward relation:

(6) |

2. Inverse relation:

(7) |

where () and
(*M*)
are the (closely related) velocity width distribution function and
luminosity function, respectively, and *S*(*m*,
,
*r*) is the sample selection function; we have assumed Gaussian
scatter of the TF relation (cf. the discussion in
Section 4.1).

Detailed derivations of these expressions
are given by
Willick (1994).
^{(6)}
In equations (6) and (7), we have written only proportionalities, since the
normalization is straightforward and will occur at a later point in any
case.

The third term on the right-hand side
of equation (5) is simply the a priori probability of observing an
object at distance *r*,

(8) |

where *n*(*r*)
1
+ _{g}(**r**)
is the number density of the species of galaxies that makes up the sample.
The second term on the right-hand side of equation (5),
*P*(*cz*|*r*),
is the one that couples the TF observables to the velocity field model. We
assume that, for the correct *IRAS* velocity field reconstruction
(i.e., for the correct value of
_{I}
and other velocity field parameters, to be described below), the redshift
is normally distributed about the value predicted from the velocity model:

(9) |

where *u*(*r*)
^{.}
[**v**(**r**) - **v**(**0**)]
is the radial component of the predicted peculiar velocity field in the
Local Group frame (cf. eq. [A1]). We treat the velocity noise
_{v}
as a free parameter in our analysis; we discuss its origin in detail
in Section 3.2.
Although _{v}
must be position or density dependent at some level, we treat it as
spatially constant in this paper, except in the Virgo Cluster
(Section 4.3). The present data do not
enable us to model the possible position or density dependence
of _{v}
in terms of free parameters.

Substituting equations (6) (or eq. [7]), (8), and (9) into equation (5)
yields the joint probability distribution *P*(*m*,
,
*cz*, *r*). To obtain the joint probability distribution of
the observable quantities, one integrates over the
(unobserved) line-of-sight distance, i.e.,

(10) |

In practice, it is not optimal to base a likelihood analysis on the
joint distribution *P*(*m*,
,
*cz*) because of its sensitivity to terms, such as the luminosity
function, the sample selection function, and the density field, that are
not critical for our purposes. Instead, the desired probability
distributions are the *conditional* ones:

1. Forward TF relation:

(11) |

2. Inverse TF relation:

(12) |

where *P*(*cz*|*r*) is given by equation (9).

Although neither of these expressions is
independent of the density field *n*(*r*) or the selection
function *S*, their appearance in both the numerator and the
denominator much reduces their sensitivity to them. A similar statement
holds for the luminosity function
in
equation (12). The velocity width distribution function
,
however, has dropped out entirely from the forward relation probability. We
discuss these points further in
Section 2.2.2.

Equations (11) and (12) are the conditional
probabilities whose products we wish to maximize over all the galaxies in
the sample. In practice, we do so by *minimizing* the quantities

(13) |

or

(14) |

where the index *i* runs over all the objects in the TF sample.
We have assumed that the probabilities for each galaxy are independent;
we validate this assumption a posteriori (cf.
Section 5.2).

^{5} The
dependence of all quantities on the line-of-sight direction will remain
implicit. Back.

^{6}
Willick (1994)
assumed that the selection function depended on the TF
observables only. Here we acknowledge the possibility of an explicit
distance dependence; the origin of such a dependence was discussed by
SW, Section 6.5.3.
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