Hedonic House Price ModelingBased on Multilevel Structured Additive Regression - Computational Approaches for Urban Environments - page 103

5.4.1 General Form of Basic Priors

In a frequentist setting, overfitting of a particular function f is avoided

by defining a roughness penalty on the regression coefficients; see, for instance,
Fahrmeir et al. ( 2013 ) in the context of structured additive regression. In a Bayesian
framework, a standard smoothness prior is a (possibly improper) Gaussian prior of
the form

D Z ˇ

1
2

rk . K / =2

exp

1
2 2 ˇ 0 K

p. ˇj 2 / / ˇ I. A ˇ D 0 /; (5.8)

where I. / is the indicator function. The key components of the prior are the penalty
matrix K , the variance parameter j , and the constraint A . Usually the

penalty matrix is rank deficient, that is, rk. K /<K, resulting in a partially improper
prior.
The amount of smoothness is governed by the variance parameter 2 . A conjugate
inverse Gamma prior is employed for 2 (as well as for the error variance parameter
2 in models with Gaussian responses), that is, 2

ˇ D 0

IG.a;b/ with small values
such as a D b D 0:001 for the hyperparameters a and b resulting in an
uninformative prior on the log scale.
The term I. A

ˇ D 0 / imposes required identifiability constraints on the
parameter vector. A straightforward choice is A D .1;:::;1/, that is, the regression
coefficients are centered around zero.

5.4.2 Continuous Covariate Effects

For a continuous covariate z , our basic approach for modeling, a smooth function
f is using P-splines introduced in a frequentist setting by Eilers and Marx ( 1996 )
and in a Bayesian version by Lang and Brezger ( 2004 ). P-splines assume that the
unknown functions can be approximated by a polynomial spline which can be
written in terms of a linear combination of B-spline basis functions. Hence, the
columns of the design matrix Z are given by the B-spline basis functions evaluated
at the observations z i . Lang and Brezger ( 2004 ) propose to use first- or second-order
random walks as smoothness priors for the regression coefficients, that is,

ˇ k D ˇ k1 C u k ; or ˇ k D 2ˇ k1 ˇ k2 C u k ; (5.9)

with Gaussian errors u k N.0; 2 / and diffuse priors p.ˇ 1 / / const, or p.ˇ 1 / and
p.ˇ 2 / / const, for initial values. This prior is of the form ( 5.8 ) with penalty matrix
given by K D D 0 D ,where D is a first- or second-order difference matrix.