Next: Maximumentropy priors
Up: General principle based priors
Previous: General principle based priors
Transformation invariance
An important class of priors arises from the requirement
of transformation invariance.
We shall consider two specific cases, translation invariance
and scale invariance.
 Translation invariance

Let us assume we are indifferent over a transformation
of the kind
, where is our
variable of interest
and a constant.
Then
is an infinitesimal mass element of probability
for to be in the interval
. Translation invariance
requires that this mass element remains unchanged when expressed
in terms of , i.e.
since
.
It is easy to see that in order for Eq. (101) to hold for any ,
must be equal to a constant for all values of
from to . It is therefore an improper prior.
As discussed above,
this is just a convenient modelling. For practical purposes this prior
should always be regarded as the limit for
of
,
where is a large finite range around the values of interest.
 Scale invariance

In other cases, we could be indifferent about a scale transformation,
that is
, where is a constant. This invariance
implies, since
in this case,
i.e.
The solution of this functional equation is
as can be easily proved using Eq. (104) as test solution in
Eq. (103).
This is the famous Jeffreys' prior, since it was first proposed
by Jeffreys. Note that this prior also can be stated as
, as can be easily verified.
The requirement of scale invariance also
produces an improper prior, in the range
.
Again, the improper prior must be understood as the limit of a proper prior
extending several orders of magnitude around the values of interest.
[Note that we constrain to be positive because, traditionally,
variables which are believed to satisfy this invariance are associated with
positively defined quantities. Indeed, Eq. (104)
has a symmetric solution for negative quantities.]
According to the supporters of these invariance motivated priors
(see e.g. Jaynes 1968, 1973, 1998, Sivia 1997, and Fröhner 2000,
Dose 2002) variables associated to translation invariance are
location parameters, as the parameter in a Gaussian model;
variables associated to scale invariance are scale parameters,
like in a Gaussian model or in a Poisson model.
For criticism about the (mis)use of this kind
of prior see (D'Agostini 1999d).
Next: Maximumentropy priors
Up: General principle based priors
Previous: General principle based priors
Giulio D'Agostini
20030513