Geoscience Reference
In-Depth Information
It then turns out that maximizing the corresponding posterior (for fixed
2
)
n
Y
p.
ˇ
j
y
;
2
/
/
p.y
i
j ˇ
;
2
/
i D1
/
exp
!
X
n
x
i
ˇ
j
2
w
.y
i
;
x
i
ˇ
/
j
y
i
;
i D1
with respect to
ˇ
, is equivalent to minimizing the optimization criterion (
5.7
).
While the asymmetric Laplace distribution allows to conveniently express quan-
tile regression in a Bayesian framework, it complicates inference based on Markov
chain Monte Carlo (MCMC) simulations due to the absolute value contained in
its definition. It is therefore advantageous to represent the asymmetric Laplace
distribution as a scale mixture of normal distributions as suggested in Yue and Rue
(
2011
): Let
u
i
j
2
Expo.1=
2
/, i
D
1;:::;n, be i.i.d. exponentially distributed
with rate parameter
2
and
y
i
j
u
i
;
ˇ
;
2
N.
x
i
ˇ
C
u
i
;
2
=
w
i
/
with
1
2
.1
/
;
1
ı
2
u
i
;
2
.1
/
:
ı
2
D
w
i
D
D
Then the marginal distribution y
i
j ˇ
;
2
is obtained by integrating out
u
i
and is
indeed an asymmetric Laplace distribution, that is,
y
i
j ˇ
;
2
ALD.
x
i
ˇ
;
2
;/:
Bayesian inference can now efficiently be implemented after imputing the scale
variables
u
i
as additional unknowns. Basically, the resulting model is a conditionally
Gaussian regression model with offsets
u
i
and weights
w
i
; see Waldmann et al.
(
2013
) and Fahrmeir et al. (
2013
) for details.
Finally note that the linear predictor
D
x
i
ˇ
in Bayesian quantile regression
can be replaced by a (hierarchical) structured additive predictor as in (
5.1
) without
any further difficulties. This is in contrast to frequentist quantile regression as, for
example, in Koenker and Mizera (
2004
). Here, statistical inference is considerably
complicated by replacing linear by additive predictors.
5.4
Effect Modeling and Priors
Effect modeling and priors depend on the covariate or term type. We first describe
the general form of priors. Sections
5.4.2
and
5.4.3
give specific examples for effect
modeling using specific design matrices and forms of the basic prior.
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