Next: Invariant determinants
Up: Adapting prior covariances
Previous: Automatic relevance detection
  Contents
Local masses and gauge theories
The Bayesian analog of a mass term in quantum field theory
is a term
proportional to the identity matrix
in the inverse prior covariance .
Consider, for example,
|
(464) |
with real (so that )
representing a mass parameter.
For large masses
tends to copy the template locally,
and longer range effects of data points following from
smoothness requirements become less important.
A constant mass can be replaced by a mass function .
This allows to adapt locally that interplay between ``template copying''
and smoothness related influence of training data.
As hyperprior, one may
use a smoothness constraint on the mass function ,
e.g.,
|
(465) |
where denotes the diagonal mass operator
with diagonal elements .
Functional hyperparameters
like represent,
in the language of physicists,
additional fields entering the problem
(see also Sect. 5.6).
There are similarities for example to gauge fields in physics.
In particular, a gauge theory-like formalism can be constructed
by decomposing =
,
so that the inverse covariance
|
(466) |
can be expressed in terms of a ``covariant derivative''
=
.
Next, one may choose as hyperprior for
|
(467) |
which can be expressed in terms of
a ``field strength tensor'' (for Abelian fields),
|
(468) |
like, for example,
the Maxwell tensor in quantum electrodynamics.
(To relate this, as in electrodynamics,
to a local gauge symmetry
one can consider complex functions ,
with the restriction that their phase cannot be measured.)
Notice, that, due to the interpretation of the prior as
product
, an additional -dependent
normalization term
enters the energy functional.
Such a term is not present in quantum field theory,
where one relates the prior functional directly to
, so the norm is independent of and .
Next: Invariant determinants
Up: Adapting prior covariances
Previous: Automatic relevance detection
  Contents
Joerg_Lemm
2001-01-21