text:
\part*{
\section{Introduction}

Semiparametric regression has achieved an impressive dissemination over the last
years. Its central idea is to replace parametric regression functions by smooth,
semiparametric components. Following \citet{HastieTibshirani90}, suppose we have
$p$ continuous covariates $x_{1}, \dotsc, x_{p}$ and use the additive model
\begin{equation}
  \label{eq:additive-model}
  y = \beta_{0} + \sum_{j=1}^{p} m_{j} (x_{j}) + \epsilon,
\end{equation}
where $m_{j}$ are smooth but otherwise unspecified functions and $\epsilon \sim
\Nor(0, \sigma^{2})$. For identifiability purposes we further assume that
$\E\{m_{j}(X_{j})\} = 0$ with respect to the marginal distribution of covariate
$X_{j}$. Estimation of the smooth terms in \eqref{eq:additive-model} can be
carried out in different ways, where we here make use of penalised splines, see
\eg \citet{eilers.marx2010} or \citet{Wood2006b}.
A general introduction to penalised spline smoothing has been provided by
\citet{Ruppert:Wand:2003} and the approach has become a popular smoothing
technique since then, see \citet{ruppert.etal2009}. The general idea is to
decompose the function $m_{j}$ into a linear and a nonlinear part, where the
latter is represented through a spline basis, that is
\begin{equation}
  \label{eq:spline-function}
  m_{j}(x_{j}) = x_{j}\beta_{j} +
  \boldsymbol{Z}_{j}(x_{j})^{T}\boldsymbol{u}_{j}.
\end{equation}
Here $\boldsymbol{Z}_{j}(x_{j})$ is a $K \times 1$ spline basis vector at
position $x_{j}$ and $\boldsymbol{u}_{j}$ is the corresponding coefficient
vector. Conveniently one may choose a truncated polynomial basis for
$\boldsymbol{Z}_{j}(\cdot)$ but representation \eqref{eq:spline-function} holds
in general as well, see \citet{wand.ormerod2008}. To achieve a smooth fit one
imposes a (quadratic) penalty on the spline coefficient vector
$\boldsymbol{u}_{j}$ which is formulated as the normal prior
\begin{equation}
  \label{eq:penalty-prior}
  \boldsymbol{u}_{j} \sim
  \Nor_{K}(\boldsymbol{0}_{K}, \sigma^{2}\rho_{j}\boldsymbol{I}_{K}),
\end{equation}
where $\boldsymbol{0}_{K}$ is the all-zeros vector and $\boldsymbol{I}_{K}$ is
the identity matrix of dimension $K$.
Here the variance factor $\rho_{j}$ steers the amount of penalisation (relative
to the regression variance $\sigma^{2}$). A larger $\rho_{j}$ leads to a higher
prior variance of the spline coefficients and hence a more wiggly function
$m_{j}$, while a smaller $\rho_{j}$ leads to a stronger penalty on
$\norm{\boldsymbol{u}_{j}}$ and thus a smoother function $m_{j}$. Setting
$\rho_{j}$ to zero imposes $\boldsymbol{u}_{j}\equiv \boldsymbol{0}_{K}$ so that
$m_{j}(x_{j})$ collapses to a linear term $m_{j}(x_{j})=x_{j} \beta_{j}$. Hence
the role of $\rho_{j}$ ($j=1, \dotsc, p$) extends to the selection of
(generalised) additive models, which will be the focus of this paper. Variable
selection will be treated by allowing the alternative $m_{j}(x_{j}) \equiv 0$.

Variable selection in generalised additive models is important to reduce the
variance of effect estimates due to uninformative covariates. The field is wide
and many different approaches have been proposed in the last years.
\citet{friedman2001} and \citet{tutz.binder2006} describe boosting algorithms,
which are extended by \citet{kneib.etal2009} to geoadditive regression models
\citep{fahrmeir.etal2004}. For the same model class, \citet{belitz.lang2008}
propose to use information-criteria or cross-validation, while
\citet{fahrmeir.etal2010} and \citet{panagiotelis.smith2008} use spike-and-slab
priors for variable and function selection. \citet{brezger.lang2008} adopt the
concept of Bayesian contour probabilities \citep{held2004} to decide on the
inclusion and form of covariate effects. \citet{CottetKohnNott2008}
generalise earlier work by \citet{YauKohnWood2003} to Bayesian
double-exponential regression models, which comprise generalised additive models
as a special case. Shrinkage approaches are proposed by \citet{wood2011} and
\citet{marra.wood2011}. \citet{zhang.lin2006} use a lasso-type penalised
likelihood approach, and \citet{ravikumar.etal2008} and \citet{meier.etal2009}
use penalties favouring both sparsity and smoothness of high-dimensional models.
Likelihood-ratio testing methods are described by \citet{kauermann.tutz2001} and
\citet{cantoni.hastie2002}. This list mirrors the multitude as well as the
variety of the different approaches and the enumeration is, of course, in no way
exhaustive.

In this paper we propose a novel Bayesian variable and function selection
approach based on mixtures of (generalised) $g$-priors. This type of prior for
the parameters in the generalised additive model traces back to the $g$-prior in
the linear model \citep{Zellner1986}. Its hyper-parameter $g$ acts as an inverse
relative prior sample size, and assigning it a hyper-prior solves the
information paradox \citep[section~4.1]{LiangPauloMolinaClydeBerger2008} of the
fixed-$g$ case \citep[p.~148]{BergerPericchi2001} in the linear model. We will
subsequently refer to such mixtures of $g$-priors generically as hyper-$g$
priors. One specific example are the hyper-$g$ priors of
\citet[section~3.2]{LiangPauloMolinaClydeBerger2008}, which enjoy a closed form
for the marginal likelihood and lead to consistent model selection and
model-averaged prediction. We will proceed to use these, because they have been
well studied and have shown good frequentist properties in the Gaussian linear
model, and have already been extended to generalised linear models by
\citet{sabanesbove.held2011}. We follow the conventional prior approach
\citep[section~2.1]{BergerPericchi2001} by using non-informative improper priors
for parameters which are common to all models, and default proper hyper-$g$
priors for model-specific parameters.

While hyper-$g$ priors have been discussed extensively in the Bayesian variable
selection literature, \eg by \citet{CuiGeorge2008},
\citet{LiangPauloMolinaClydeBerger2008}, \citet{forte2011} and
\citet{celeux.etal2012}, this is the first paper to our knowledge that applies
hyper-$g$ priors to generalised additive models. The general idea of applying
default priors (as hyper-$g$ priors) which have originally been developed for
linear models to generalised additive models is new. The methodology presented
here is straightforward to use with other default priors. The rationale is that
default priors have carefully and exhaustively been constructed for the linear
model, so their advantages should be used when inferring about generalised
additive models. Moreover, this paper is one of the few Bayesian papers
considering automatic and simultaneous variable selection and transformation.

The current work generalises the hyper-$g$ priors for generalised linear models
\citep{sabanesbove.held2011}. In the same paper, we showed how fractional
polynomials \citep[FPs, ][]{sabanesbove.held2011a}, which extend ordinary polynomials by
square roots, reciprocals and the logarithm, can be used to model nonlinear
covariate effects. However, FPs have the disadvantage of
being not invariant to linear transformations of the covariates.
For variable and function selection, \citet{fahrmeir.etal2010} and
\citet{scheipl.etal2011} use a mixture of two inverse-gamma distributions with a very
small (``spike'') and a larger mean (``slab'') as a hyper-prior for the
variances of the regression coefficients' independent normal priors. The
posterior probability for inclusion of a coefficient is then estimated from the
proportion of Markov chain Monte Carlo (MCMC) variance samples in the ``slab''.
While this prior structure eases the MCMC algorithm, it does not take into
account the correlation structure of the covariates, and depends on the
specification of the prior means in the two mixture components.
\citet{CottetKohnNott2008} also use independent normal inverse-gamma priors for
the regression coefficients, but they explicitly exclude coefficients from the
model. For nonlinear effects they utilise low-rank approximations of smoothing
splines, which require the choice of a threshold on the eigenvalue scale.

The paper is organised as follows. We first apply the hyper-$g$ priors of
\citet{LiangPauloMolinaClydeBerger2008} to additive models in
Section~\ref{sec:normal-models}. The methodology is extended to generalised
additive models in Section~\ref{sec:non-normal-models}. A
multiplicity-correction prior on the model space and a stochastic search
procedure are described in Section~\ref{sec:model-prior-and-search}. We apply our
approach to simulated and real data in Section~\ref{sec:applications} and suggest
postprocessing techniques in Section~\ref{sec:postprocessing}.
Section~\ref{sec:discussion} closes the paper with a discussion.

\section{Hyper-$g$ Priors for Additive Models}
\label{sec:normal-models}

Assume we have observed independent responses $y_{i}$ at covariate values
$x_{i1}, \dotsc, x_{ip}$, $i=1, \dotsc, n$, from the additive normal
model~\eqref{eq:additive-model}. For each covariate $j=1, \dotsc, p$, we stack
the covariate values into the $n \times 1$ vector $\tilde{\boldsymbol{x}}_{j} =
(x_{1j}, \dotsc, x_{nj})^{T}$ and the spline basis vectors into the $n \times K$
matrix $\tilde{\boldsymbol{Z}}_{j} = (\boldsymbol{Z}_{j}(x_{1j}), \dotsc,
\boldsymbol{Z}_{j}(x_{nj}))^{T}$. The subsequent Gram-Schmidt process
\citep[see][]{bjoerck1967}
\begin{align}
  \boldsymbol{x}_{j} &=
  \tilde{\boldsymbol{x}}_{j}
  - \boldsymbol{1}_{n}
  \frac{\boldsymbol{1}_{n}^{T}\tilde{\boldsymbol{x}}_{j}}
  {\boldsymbol{1}_{n}^{T}\boldsymbol{1}_{n}}
  = \tilde{\boldsymbol{x}}_{j}
  - \boldsymbol{1}_{n}\bar{x}_{j},
  \label{eq:normal-models:center-vector}
  \\
  \boldsymbol{Z}_{j} &=
  \tilde{\boldsymbol{Z}}_{j}
  - \boldsymbol{1}_{n}
  \frac{\boldsymbol{1}_{n}^{T}\tilde{\boldsymbol{Z}}_{j}}
  {\boldsymbol{1}_{n}^{T}\boldsymbol{1}_{n}}
  - \boldsymbol{x}_{j}
  \frac{\boldsymbol{x}_{j}^{T}\tilde{\boldsymbol{Z}}_{j}}
  {\boldsymbol{x}_{j}^{T}\boldsymbol{x}_{j}},
  \label{eq:normal-models:center-matrix}
\end{align}
where $\boldsymbol{1}_{n}$ denotes the all-ones vector of dimension $n$,
ensures that $\boldsymbol{1}_{n}$, $\boldsymbol{x}_{j}$ and the columns of
$\boldsymbol{Z}_{j}$ are orthogonal to each other, \abk{\latin{i.\,e}}
$\boldsymbol{1}_{n}^{T}\boldsymbol{x}_{j} = 0$ and
$\boldsymbol{1}_{n}^{T}\boldsymbol{Z}_{j} =
\boldsymbol{x}_{j}^{T}\boldsymbol{Z}_{j} = \boldsymbol{0}_{K}$.

A central measure of model complexity is the degrees of freedom. While in
parametric models this is just the number of parameters, for smoothing and
mixed models \citet[section~2.2]{aerts.etal2002} translate the variance factor
$\rho_{j}$ into the corresponding degrees of freedom
\begin{equation}
  \label{eq:normal-models:degrees-of-freedom}
  d_{j}(\rho_{j}) =
  \trace \{
  (\boldsymbol{Z}_{j}^{T}
  \boldsymbol{Z}_{j} + \rho_{j}^{-1} \boldsymbol{I})^{-1} \boldsymbol{Z}_{j}^{T}
  \boldsymbol{Z}_{j}
  \}
  + 1
  \in (1, K+1)
\end{equation}
for a smoothly modelled covariate effect $m_{j}$. Note that $d_{j}(\rho_{j}) =
\sum_{k=1}^{K} \lambda_{jk} / (\lambda_{jk} + \rho_{j}^{-1})$ is easy to
calculate via the (positive) eigenvalues $\lambda_{jk}$ of
$\boldsymbol{Z}_{j}^{T}\boldsymbol{Z}_{j}$. This also shows that
$d_{j}(\rho_{j})$ is strictly increasing with derivative
$\sum_{k=1}^{K}\lambda_{jk}/(\rho_{j}\lambda_{jk} + 1)^{2} > 0$, which implies
that we may (numerically) invert the function to $\rho_{j}(d_{j})$. In fact, by
fixing the degrees of freedom $d_{j}$ for function $m_{j}(x_{j})$ we define the
variance factor $\rho_{j}$. Subsequently we will restrict the degrees of freedom
to take values in a finite set $\mathcal{D} \subset \{0\} \cup [1, K+1)$, say
$\mathcal{D} = \{0, 1, 2, 3, \dotsc, K\}$. For $d_{j}=0$ we set
$m_{j}(x_{j})\equiv 0$ while for $d_{j}=1$ we have the linear model
$m_{j}(x_{j})=x_{j}\beta_{j}$. In general, model \eqref{eq:additive-model} is
indexed by $\boldsymbol{d}=(d_{1}, \dotsc, d_{p})$ giving the degrees of freedom
for each functional component and hence the structure of the model.

After combining the $I = \sum_{j=1}^{p}\mathbb{I}(d_{j} \geq 1)$
vectors $\boldsymbol{x}_{j}$ to the $n \times I$ linear design
matrix $\boldsymbol{X}_{\boldsymbol{d}} = (\boldsymbol{x}_{j} : d_{j} \geq 1)$
and the $J = \sum_{j=1}^{p}\mathbb{I}(d_{j} > 1)$ matrices
$\boldsymbol{Z}_{j}$ to the $n \times JK$ spline design
matrix $\boldsymbol{Z}_{\boldsymbol{d}} = (\boldsymbol{Z}_{j} : d_{j} > 1)$, and
analogously constructing the respective coefficient vectors
$\boldsymbol{\beta}_{\boldsymbol{d}}$ and $\boldsymbol{u}_{\boldsymbol{d}}$, the
conditional additive model for the response vector $\boldsymbol{y} = (y_{1},
\dotsc, y_{n})^{T}$ is
\begin{equation}
  \boldsymbol{y} \given
  \beta_{0}, \boldsymbol{\beta}_{\boldsymbol{d}},
  \boldsymbol{u}_{\boldsymbol{d}}, \sigma^{2}
  \sim
  \Nor_{n} \left(
    \boldsymbol{1}_{n} \beta_{0} +
    \boldsymbol{X}_{\boldsymbol{d}}\boldsymbol{\beta}_{\boldsymbol{d}} +
    \boldsymbol{Z}_{\boldsymbol{d}}\boldsymbol{u}_{\boldsymbol{d}},
    \,
    \sigma^{2}\boldsymbol{I}_{n}
  \right).
  \label{eq:normal-models:conditional}
\end{equation}
Integrating out the the spline coefficient vector
$\boldsymbol{u}_{\boldsymbol{d}} \sim \Nor_{JK} (\boldsymbol{0}_{JK},
\sigma^{2}\boldsymbol{D}_{\boldsymbol{d}})$, where
$\boldsymbol{D}_{\boldsymbol{d}}$ is block-diagonal
with $J$ blocks $\rho_{j}\boldsymbol{I}_{K}$ ($d_{j} > 1$), yields the marginal
model
\begin{equation}
  \boldsymbol{y} \given
  \beta_{0}, \boldsymbol{\beta}_{\boldsymbol{d}}, \sigma^{2}
  \sim
  \Nor_{n} \left(
    \boldsymbol{1}_{n} \beta_{0} +
    \boldsymbol{X}_{\boldsymbol{d}}\boldsymbol{\beta}_{\boldsymbol{d}},
    \,
    \sigma^{2}\boldsymbol{V}_{\boldsymbol{d}}
  \right)
  \label{eq:normal-models:marginal}
\end{equation}
with $\boldsymbol{V}_{\boldsymbol{d}} = \boldsymbol{I}_{n} +
\boldsymbol{Z}_{\boldsymbol{d}}\boldsymbol{D}_{\boldsymbol{d}}
\boldsymbol{Z}_{\boldsymbol{d}}^{T}$. This general linear model can be
decorrelated into a standard linear model by using the Cholesky decomposition
$\boldsymbol{V}_{\boldsymbol{d}} = \boldsymbol{V}_{\boldsymbol{d}}^{T/2}
\boldsymbol{V}_{\boldsymbol{d}}^{1/2}$: For the transformed response vector
$\tilde{\boldsymbol{y}} = \boldsymbol{V}_{\boldsymbol{d}}^{-T/2}\boldsymbol{y}$
we have
\begin{equation}
  \tilde{\boldsymbol{y}} \given
  \beta_{0}, \boldsymbol{\beta}_{\boldsymbol{d}}, \sigma^{2}
  \sim
  \Nor_{n}
  \left(
    \tilde{\boldsymbol{1}}_{n}\beta_{0} +
    \tilde{\boldsymbol{X}}_{\boldsymbol{d}}\boldsymbol{\beta}_{\boldsymbol{d}},
    \,
    \sigma^{2} \boldsymbol{I}_{n}
  \right)
  \label{eq:normal-models:decorrelated}
\end{equation}
with analogously transformed all-ones vector $\tilde{\boldsymbol{1}}_{n}=
\boldsymbol{V}_{\boldsymbol{d}}^{-T/2}\boldsymbol{1}_{n}$ and design matrix
$\tilde{\boldsymbol{X}}_{\boldsymbol{d}} =
\boldsymbol{V}_{\boldsymbol{d}}^{-T/2}\boldsymbol{X}_{\boldsymbol{d}}$.
Note that now also $\tilde{\boldsymbol{y}}$ and $\tilde{\boldsymbol{1}}_{n}$
depend on the model $\boldsymbol{d}$, but we suppress this dependence for ease
of notation.

We will now show how to use the hyper-$g$ priors of
\citet{LiangPauloMolinaClydeBerger2008} for the parameters
$\beta_{0}$, $\boldsymbol{\beta}_{\boldsymbol{d}}$ and $\sigma^{2}$ in the
decorrelated model~\eqref{eq:normal-models:decorrelated}. The hyper-$g$ priors
comprise a locally uniform prior $f(\beta_{0}) \propto 1$ on the intercept,
Jeffreys' prior $f(\sigma^{2}) \propto (\sigma^{2})^{-1}$ on the regression
variance and the $g$-prior \citep{Zellner1986}
\begin{equation}
  \boldsymbol{\beta}_{\boldsymbol{d}} \given g, \sigma^{2}
  \sim
  \Nor_{I}\left(
    \boldsymbol{0}_{I},
    \,
    g\sigma^{2}
    (\tilde{\boldsymbol{X}}_{\boldsymbol{d}}^{T}
    \tilde{\boldsymbol{X}}_{\boldsymbol{d}})^{-1}
  \right)
  \label{eq:normal-models:g-prior}
\end{equation}
on the linear coefficient vector. Note that the prior precision matrix
in~\eqref{eq:normal-models:g-prior} is proportional to
$\sigma^{-2}\tilde{\boldsymbol{X}}_{\boldsymbol{d}}^{T}
\tilde{\boldsymbol{X}}_{\boldsymbol{d}} =
\sigma^{-2}\boldsymbol{X}_{\boldsymbol{d}}^{T}
\boldsymbol{V}_{\boldsymbol{d}}^{-1}\boldsymbol{X}_{\boldsymbol{d}}$, which is
the Fisher information matrix of $\boldsymbol{\beta}_{\boldsymbol{d}}$ in
model~\eqref{eq:normal-models:marginal}. The prior construction is completed
with either a uniform hyper-prior on the shrinkage coefficient $g/(1 + g)$,
\[
\frac{g}{1 + g} \sim \Unif(0, 1),
\]
leading to the hyper-$g$ prior, or with
\[
\frac{g/n}{1 + g/n} \sim \Unif(0, 1),
\]
leading to the hyper-$g/n$ prior. We recommend to use the latter, because it
also leads to consistent posterior model probabilities if the true model is the
null model (see Table~\ref{tab:med-post-prob-true-model} in
Section~\ref{sec:applications:sim-study} for an illustration of
this).

Basically all formulae given by \citet{LiangPauloMolinaClydeBerger2008} carry
over to our setting, since inner products of the response vector
$\boldsymbol{y}$, the all-ones vector $\boldsymbol{1}_{n}$ and the design matrix
$\boldsymbol{X}_{\boldsymbol{d}}$ in model~\eqref{eq:normal-models:marginal}
carry over to their transformed counterparts $\tilde{\boldsymbol{y}}$,
$\tilde{\boldsymbol{1}}_{n}$ and $\tilde{\boldsymbol{X}}_{\boldsymbol{d}}$ in
model~\eqref{eq:normal-models:decorrelated}. This is due to
\begin{equation}
  \label{eq:normal-models:marginal-precision}
  \boldsymbol{V}_{\boldsymbol{d}}^{-1}
  =
  (\boldsymbol{I}_{n} +
  \boldsymbol{Z}_{\boldsymbol{d}}
  \boldsymbol{D}_{\boldsymbol{d}}
  \boldsymbol{Z}_{\boldsymbol{d}}^{T})^{-1}
  =
  \boldsymbol{I}_{n} -
  \boldsymbol{Z}_{\boldsymbol{d}}
  (\boldsymbol{Z}_{\boldsymbol{d}}^{T}\boldsymbol{Z}_{\boldsymbol{d}}
  + \boldsymbol{D}_{\boldsymbol{d}}^{-1})^{-1}
  \boldsymbol{Z}_{\boldsymbol{d}}^{T},
\end{equation}
which follows from the matrix inversion lemma
\citep[see][]{henderson.searle1981} and leads to
$\tilde{\boldsymbol{1}}_{n}^{T}\tilde{\boldsymbol{1}}_{n} =
\boldsymbol{1}_{n}^{T}\boldsymbol{1}_{n} = n$,
$\tilde{\boldsymbol{1}}_{n}^{T}\tilde{\boldsymbol{X}}_{\boldsymbol{d}} =
\boldsymbol{1}_{n}^{T}\boldsymbol{X}_{\boldsymbol{d}} = \boldsymbol{0}_{I}$ and
$\tilde{\boldsymbol{1}}_{n}^{T}\tilde{\boldsymbol{y}} =
\boldsymbol{1}_{n}^{T}\boldsymbol{y}$ by straightforward calculations. A most
convenient property of the hyper-$g$ priors is that they yield closed form
marginal likelihoods, which need to be computed on the original response scale
via the change of variables formula:
\begin{equation}
  f(\boldsymbol{y} \given \boldsymbol{d})
  \propto
  f(\tilde{\boldsymbol{y}} \given \boldsymbol{d})
  \abs{\boldsymbol{V}_{\boldsymbol{d}}^{1/2}}^{-1},
  \label{eq:normal-models:marginal-likelihood}
\end{equation}
where $f(\tilde{\boldsymbol{y}} \given \boldsymbol{d})$ is the marginal
likelihood of the transformed response vector $\tilde{\boldsymbol{y}}$ in the
standard linear model~\eqref{eq:normal-models:decorrelated}. The closed forms
for $f(\tilde{\boldsymbol{y}} \given \boldsymbol{d})$ under the hyper-$g$ priors
are given in Appendix~\ref{sec:impl-deta:marg-lik-comp}, along with other
implementation details.

Other hyper-priors could be assigned to $g$, but will typically not lead to a
closed form of the marginal likelihood. Examples are the incomplete
inverse-gamma prior on $1 + g$ \citep[p.~891]{CuiGeorge2008}, which generalises
the above uniform prior on $g/(1 + g)$, and an inverse-gamma prior on $g$, which
corresponds to the Cauchy prior of \citet{ZellnerSiow1980}. The hyper-$g/n$
prior is a special case of the conventional robust prior proposed by
\citet{forte2011}, for which a closed form of the marginal likelihood exists.
An overview of mixtures of $g$-priors is given by \citet{ley.steel2011}.

It is not clear that the good properties of hyper-$g$ priors (or other default
priors in the Gaussian linear model) would be retained if we based them on the
conditional model~\eqref{eq:normal-models:conditional} without integrating out
the random effects. We followed the natural idea of transforming the mixed model
into a standard model, where default priors have already been studied
extensively. Moreover, computation would be harder if we proceeded otherwise.
Hence we prefer to keep the good properties of the default priors by integrating
out the random effects.

Posterior inference in a given model $\boldsymbol{d}$ is based on Monte Carlo
estimation of the parameters in model~\eqref{eq:normal-models:conditional},
using the factorisation
\begin{equation}
  \label{eq:normal-models:sampling}
  f(\beta_{0}, \boldsymbol{\beta}_{\boldsymbol{d}},
  \boldsymbol{u}_{\boldsymbol{d}}, \sigma^{2}, g \given \boldsymbol{y})
  =
  f(\boldsymbol{u}_{\boldsymbol{d}} \given \beta_{0},
  \boldsymbol{\beta}_{\boldsymbol{d}}, \sigma^{2}, \boldsymbol{y})
  f(\beta_{0}, \boldsymbol{\beta}_{\boldsymbol{d}} \given \sigma^{2}, g,
  \boldsymbol{y})
  f(\sigma^{2} \given \boldsymbol{y})
  f(g \given \boldsymbol{y}).
\end{equation}
Sampling of $g$, $\sigma^{2}$ and subsequently $\beta_{0},
\boldsymbol{\beta}_{\boldsymbol{d}}$ can be done along the lines of
\citet[section~2.3]{sabanesbove.held2011a}, by adapting their algorithm to the
transformations in model~\eqref{eq:normal-models:decorrelated}. Finally, the
spline coefficient vector $\boldsymbol{u}_{\boldsymbol{d}}$ is sampled from
\begin{align}
  f(\boldsymbol{u}_{\boldsymbol{d}} \given \beta_{0},
  \boldsymbol{\beta}_{\boldsymbol{d}}, \sigma^{2}, \boldsymbol{y})
  &\propto
  f(\boldsymbol{u}_{\boldsymbol{d}} \given \sigma^{2})
  f(\boldsymbol{y} \given \beta_{0}, \boldsymbol{\beta}_{\boldsymbol{d}},
  \boldsymbol{u}_{\boldsymbol{d}}, \sigma^{2})
  \notag \\
  &\propto
  \exp
  \left\{
    - \frac{1}{2\sigma^{2}}
    \left[
      \boldsymbol{u}_{\boldsymbol{d}}^{T}
      \boldsymbol{D}_{\boldsymbol{d}}^{-1}
    \boldsymbol{u}_{\boldsymbol{d}}
    + \norm{\boldsymbol{y} -
      \boldsymbol{1}_{n}\beta_{0} -
      \boldsymbol{X}_{\boldsymbol{d}}\boldsymbol{\beta}_{\boldsymbol{d}} -
      \boldsymbol{Z}_{\boldsymbol{d}}\boldsymbol{u}_{\boldsymbol{d}}}^{2}
    \right]
  \right\}
  \notag \\
  &\propto
  \Nor_{JK}
  \left(
    \boldsymbol{u}_{\boldsymbol{d}} \given
    \boldsymbol{\Sigma}_{\boldsymbol{d}}
    \boldsymbol{Z}_{\boldsymbol{d}}^{T}
    (\boldsymbol{y} -
    \boldsymbol{X}_{\boldsymbol{d}}\boldsymbol{\beta}_{\boldsymbol{d}}),
    \,
    \sigma^{2}\boldsymbol{\Sigma}_{\boldsymbol{d}}
  \right),
  \label{eq:normal-models:spline-coefs-conditional}
\end{align}
where $\boldsymbol{\Sigma}_{\boldsymbol{d}} =
(\boldsymbol{Z}_{\boldsymbol{d}}^{T}\boldsymbol{Z}_{\boldsymbol{d}} +
\boldsymbol{D}_{\boldsymbol{d}}^{-1})^{-1}$ and $\beta_{0}$ disappears because
$\boldsymbol{Z}_{\boldsymbol{d}}^{T}\boldsymbol{1}_{n} =
\boldsymbol{0}_{JK}$.

Given posterior samples for the linear coefficient $\beta_{j}$ and the spline
coefficient vector $\boldsymbol{u}_{j}$ for covariate $j$ ($d_{j} > 1$), we would
like to transform these into samples for the function $m_{j}(x_{j})$, along a
grid vector $\tilde{\boldsymbol{x}}_{j}^{*}$ of $n^{*}$ points (on the same
scale as the original $\tilde{\boldsymbol{x}}_{j}$ used for the model fitting).
This is in principle straightforward, but one has to carefully apply
analogous transformations as in~\eqref{eq:normal-models:center-vector}
and~\eqref{eq:normal-models:center-matrix} to $\tilde{\boldsymbol{x}}_{j}^{*}$
and the corresponding spline basis matrix $\tilde{\boldsymbol{Z}}_{j}^{*}$:
\begin{align}
  \boldsymbol{x}_{j}^{*} &=
  \tilde{\boldsymbol{x}}_{j}^{*}
  - \boldsymbol{1}_{n^{*}}
  \frac{\boldsymbol{1}_{n}^{T}\tilde{\boldsymbol{x}}_{j}}
  {\boldsymbol{1}_{n}^{T}\boldsymbol{1}_{n}},
  \label{eq:normal-models:center-grid-vector}
  \\
  \boldsymbol{Z}_{j}^{*} &=
  \tilde{\boldsymbol{Z}}_{j}^{*}
  - \boldsymbol{1}_{n^{*}}
  \frac{\boldsymbol{1}_{n}^{T}\tilde{\boldsymbol{Z}}_{j}}
  {\boldsymbol{1}_{n}^{T}\boldsymbol{1}_{n}}
  - \boldsymbol{x}_{j}^{*}
  \frac{\boldsymbol{x}_{j}^{T}\tilde{\boldsymbol{Z}}_{j}}
  {\boldsymbol{x}_{j}^{T}\boldsymbol{x}_{j}}.
  \label{eq:normal-models:center-grid-matrix}
\end{align}
Afterwards, for each coefficient sample one can compute the corresponding
vector of function values $m_{j}(\tilde{\boldsymbol{x}}_{j}^{*}) =
\boldsymbol{x}_{j}^{*}\beta_{j} + \boldsymbol{Z}_{j}^{*}\boldsymbol{u}_{j}$.
Similarly, prediction samples for the corresponding response vector
$\boldsymbol{y}^{*}$ can be extracted from the sampling output.

\section{Hyper-$g$ Priors for Generalised Additive Models}
\label{sec:non-normal-models}

Now we extend the above setting and assume that the covariate effects
$m_{j}(x_{j})$ enter additively into the linear predictor
\begin{equation}
  \label{eq:generalised-additive-model}
  \eta = \beta_{0} + \sum_{j=1}^{p} m_{j}(x_{j})
\end{equation}
of an exponential family distribution with canonical parameter $\theta$, mean
$\E(y) = h(\eta) = db(\theta)/d\theta$ and variance $\Var(y) = \phi/w\cdot
d^{2}b(\theta)/d\theta^{2}$ \citep[see][]{McCullaghNelder1989}. We restrict our
attention to non-normal distributions with fixed dispersion~$\phi$ (as $\phi =
1$ for the Bernoulli and Poisson distribution) and known weight~$w$. For $n$
observations, the linear predictor vector $\boldsymbol{\eta} = (\eta_{1},
\dotsc, \eta_{n})^{T}$ is
\begin{equation}
  \label{eq:lin-pred-vector}
  \boldsymbol{\eta} = \boldsymbol{1}_{n} \beta_{0} +
  \boldsymbol{X}_{\boldsymbol{d}}\boldsymbol{\beta}_{\boldsymbol{d}} +
  \boldsymbol{Z}_{\boldsymbol{d}}\boldsymbol{u}_{\boldsymbol{d}}
\end{equation}
and the likelihood is
\begin{equation}
  \label{eq:non-normal-models:conditional}
  f(\boldsymbol{y} \given \beta_{0}, \boldsymbol{\beta}_{\boldsymbol{d}},
  \boldsymbol{u}_{\boldsymbol{d}})
  \propto
  \exp\left\{
    \sum_{i=1}^{n}
    \frac{y_{i}\theta_{i} - b(\theta_{i})}{\phi/w_{i}}
  \right\}.
\end{equation}
The main challenge for the derivation of a generalised $g$-prior is that the
marginal density $f(\boldsymbol{y} \given \beta_{0},
\boldsymbol{\beta}_{\boldsymbol{d}})$, which results from integrating out the
spline coefficient vector
\begin{equation}
  \label{eq:non-normal-models:spline-coef-prior}
  \boldsymbol{u}_{\boldsymbol{d}} \sim \Nor_{JK}
  (\boldsymbol{0}_{JK}, \boldsymbol{D}_{\boldsymbol{d}})
\end{equation}
from~\eqref{eq:non-normal-models:conditional}, has no closed form. In
particular, it is not Gaussian, in contrast
to~\eqref{eq:normal-models:marginal}.

Before addressing this problem we first consider appropriate
construction of the design matrices $\boldsymbol{X}_{\boldsymbol{d}}$ and
$\boldsymbol{Z}_{\boldsymbol{d}}$ and calculation of the degrees of freedom
$d_{j}(\rho_{j})$ for a smoothly modelled term $m_{j}$. Starting with the
latter, a reasonable generalisation
of~\eqref{eq:normal-models:degrees-of-freedom} is
\citep[see][section~11.4]{Ruppert:Wand:2003}
\begin{equation}
  \label{eq:non-normal-models:degrees-of-freedom}
  d_{j}(\rho_{j}) =
  \trace \{
  (\boldsymbol{Z}_{j}^{T} \widehat{\boldsymbol{W}} \boldsymbol{Z}_{j} +
  \rho_{j}^{-1} \boldsymbol{I})^{-1}
  \boldsymbol{Z}_{j}^{T} \widehat{\boldsymbol{W}} \boldsymbol{Z}_{j}
  \}
  + 1,
\end{equation}
which uses a fixed weight matrix $\widehat{\boldsymbol{W}} =
\boldsymbol{W}(\boldsymbol{1}_{n}\widehat{\beta}_{0})$, where
$\boldsymbol{W}(\boldsymbol{\eta}) = \diag\{ (dh(\eta_{i})/d\eta)^{2} /
\Var(y_{i})\}_{i=1}^{n}$ is the usual generalised linear model weight matrix and
$\widehat{\beta}_{0}$ is the intercept estimate from the null model
$\boldsymbol{d}=\boldsymbol{0}_{p}$. This definition avoids dependence of
$\rho_{j}(d_{j})$ on the model $\boldsymbol{d}$ under consideration.
As a consequence, we need to generalise the orthogonalisation of the
original covariate vector $\tilde{\boldsymbol{x}}_{j}$ and spline basis matrix
$\tilde{\boldsymbol{Z}}_{j}$ from \eqref{eq:normal-models:center-vector} and
\eqref{eq:normal-models:center-matrix} to
\begin{align}
  \boldsymbol{x}_{j} &=
  \tilde{\boldsymbol{x}}_{j}
  - \boldsymbol{1}_{n}
  \frac{\boldsymbol{1}_{n}^{T}\widehat{\boldsymbol{W}}\tilde{\boldsymbol{x}}_{j}}
  {\boldsymbol{1}_{n}^{T}\widehat{\boldsymbol{W}}\boldsymbol{1}_{n}}
  \label{eq:non-normal-models:center-vector}
  \\
  \text{and}\quad
  \boldsymbol{Z}_{j} &=
  \tilde{\boldsymbol{Z}}_{j}
  - \boldsymbol{1}_{n}
  \frac{\boldsymbol{1}_{n}^{T}\widehat{\boldsymbol{W}}\tilde{\boldsymbol{Z}}_{j}}
  {\boldsymbol{1}_{n}^{T}\widehat{\boldsymbol{W}}\boldsymbol{1}_{n}}
  - \boldsymbol{x}_{j}
  \frac{\boldsymbol{x}_{j}^{T}\widehat{\boldsymbol{W}}\tilde{\boldsymbol{Z}}_{j}}
  {\boldsymbol{x}_{j}^{T}\widehat{\boldsymbol{W}}\boldsymbol{x}_{j}}
  \label{eq:non-normal-models:center-matrix},
\end{align}
implying that $\boldsymbol{1}_{n}$, $\boldsymbol{x}_{j}$ and the columns of
$\boldsymbol{Z}_{j}$ are orthogonal to each other with respect to the inner
product in terms of $\widehat{\boldsymbol{W}}$. This ensures
that~\eqref{eq:non-normal-models:degrees-of-freedom} correctly captures only the
degrees of freedom associated with the nonlinear part of $m_{j}$. Note that
\eqref{eq:normal-models:center-grid-vector} and
\eqref{eq:normal-models:center-grid-matrix} are adapted analogously.

We will now derive a generalised $g$-prior analogous
to~\eqref{eq:normal-models:g-prior} for the linear coefficient vector
$\boldsymbol{\beta}_{\boldsymbol{d}}$ in the generalised additive model. The
idea is to use the iterative weighted least squares (IWLS) approximation
to~\eqref{eq:non-normal-models:conditional} to obtain an approximate normal
model of the form \eqref{eq:normal-models:conditional} and then derive the
resulting $g$-prior~\eqref{eq:normal-models:g-prior}. With a slight abuse of
notation, \eg $h(\boldsymbol{\eta}) = (h(\eta_{1}), \dotsc, h(\eta_{n}))^{T}$,
let
\begin{equation}
  \label{eq:adjusted-response-variable}
  \boldsymbol{z}_{0} =
  \boldsymbol{\eta}_{0} +
  \diag\{dh(\boldsymbol{\eta}_{0})/d\boldsymbol{\eta}\}^{-1}
  (\boldsymbol{y} - h(\boldsymbol{\eta}_{0}))
\end{equation}
be the adjusted response vector resulting from a first-order approximation to
$h^{-1}(\boldsymbol{y})$ around $\boldsymbol{y} = h(\boldsymbol{\eta}_{0})$. Then
\begin{equation}
  \label{eq:iwls-model}
  \boldsymbol{z}_{0} \given
  \beta_{0}, \boldsymbol{\beta}_{\boldsymbol{d}},
  \boldsymbol{u}_{\boldsymbol{d}}
  \sima
  \Nor\bigl(\boldsymbol{1}_{n}\beta_{0} +
  \boldsymbol{X}_{\boldsymbol{d}}\boldsymbol{\beta}_{\boldsymbol{d}} +
  \boldsymbol{Z}_{\boldsymbol{d}}\boldsymbol{u}_{\boldsymbol{d}},
  \,
  \boldsymbol{W}_{0}^{-1}\bigr)
\end{equation}
with $\boldsymbol{W}_{0} = \boldsymbol{W}(\boldsymbol{\eta}_{0})$ is the working
normal model \citep[see {\eg}][p.~40]{McCullaghNelder1989}. Remember that the
IWLS algorithm iteratively updates $\boldsymbol{\eta}_{0}$ by weighted least
squares estimation of the coefficients in~\eqref{eq:iwls-model}. Here, we fix
$\boldsymbol{\eta}_{0} = \boldsymbol{0}_{n}$, which is the value expected
\latin{a priori}. Then we rewrite~\eqref{eq:iwls-model} using
$\tilde{\boldsymbol{z}}_{0} = \boldsymbol{W}_{0}^{1/2}\boldsymbol{z}_{0}$ \abk{\latin{etc}}
as
\begin{equation}
  \label{eq:iwls-model-rewritten}
  \tilde{\boldsymbol{z}}_{0} \given
  \beta_{0}, \boldsymbol{\beta}_{\boldsymbol{d}},
  \boldsymbol{u}_{\boldsymbol{d}}
  \sima
  \Nor(\tilde{\boldsymbol{1}}_{n}\beta_{0} +
  \tilde{\boldsymbol{X}}_{\boldsymbol{d}}\boldsymbol{\beta}_{\boldsymbol{d}} +
  \tilde{\boldsymbol{Z}}_{\boldsymbol{d}}\boldsymbol{u}_{\boldsymbol{d}},
  \,
  \boldsymbol{I}_{n}),
\end{equation}
which brings us back to a normal model of the form
in~\eqref{eq:normal-models:conditional}. By
computing the corresponding $g$-prior~\eqref{eq:normal-models:g-prior}, we
arrive at the generalised $g$-prior
\begin{equation}
  \label{eq:generalised-g-prior}
  \boldsymbol{\beta}_{\boldsymbol{d}} \given g
  \sim
  \Nor_{I}(\boldsymbol{0}_{I},
  g \boldsymbol{J}_{0}^{-1})
\end{equation}
with prior precision matrix proportional to
\begin{align}
  \boldsymbol{J}_{0}
  &=
  \tilde{\boldsymbol{X}}_{\boldsymbol{d}}^{T}
  (\boldsymbol{I}_{n} +
  \tilde{\boldsymbol{Z}}_{\boldsymbol{d}}
  \boldsymbol{D}_{\boldsymbol{d}}
  \tilde{\boldsymbol{Z}}_{\boldsymbol{d}}^{T})^{-1}
  \tilde{\boldsymbol{X}}_{\boldsymbol{d}}
  \notag \\
  &=
  \boldsymbol{X}_{\boldsymbol{d}}^{T}
  \boldsymbol{W}_{0}^{1/2}
  (\boldsymbol{I}_{n} +
  \boldsymbol{W}_{0}^{1/2}
  \boldsymbol{Z}_{\boldsymbol{d}}
  \boldsymbol{D}_{\boldsymbol{d}}
  \boldsymbol{Z}_{\boldsymbol{d}}^{T}
  \boldsymbol{W}_{0}^{1/2})^{-1}
  \boldsymbol{W}_{0}^{1/2}
  \boldsymbol{X}_{\boldsymbol{d}}.
  \label{eq:approx-fisher-info}
\end{align}
An appealing feature of this prior is that it directly generalises the $g$-prior
proposed by \citet{sabanesbove.held2011} for generalised linear models, to which
it reduces when there are no spline effects in the model, \abk{\latin{i.\,e}}
$\boldsymbol{J}_{0} = \boldsymbol{X}_{\boldsymbol{d}}^{T} \boldsymbol{W}_{0}
\boldsymbol{X}_{\boldsymbol{d}}$. An alternative and more rigorous derivation of
\eqref{eq:approx-fisher-info} as the Fisher information obtained from a Laplace
approximation to the marginal model $f(\boldsymbol{y} \given \beta_{0},
\boldsymbol{\beta}_{\boldsymbol{d}})$ is presented in the supplementary material
available at Biometrika online.

The generalised hyper-$g$ prior
\begin{equation}
  \label{eq:generalised-hyper-g-prior}
  f(\beta_{0}, \boldsymbol{\beta}_{\boldsymbol{d}},
  \boldsymbol{u}_{\boldsymbol{d}}, g) =
  f(\beta_{0}) f(\boldsymbol{\beta}_{\boldsymbol{d}} \given g) f(g)
  f(\boldsymbol{u}_{\boldsymbol{d}})
\end{equation}
is defined to comprise the locally uniform prior $f(\beta_{0}) \propto 1$ on the
intercept $\beta_{0}$, the generalised $g$-prior \eqref{eq:generalised-g-prior}
on the linear coefficient vector $\boldsymbol{\beta}_{\boldsymbol{d}}$, the
penalty prior \eqref{eq:non-normal-models:spline-coef-prior} on the spline
coefficient vector $\boldsymbol{u}_{\boldsymbol{d}}$, and some proper
hyper-prior $f(g)$ on the hyper-parameter $g$. Posterior inference under this
prior can be implemented by a straightforward extension of the approach of
\citet[section~3]{sabanesbove.held2011}, which is outlined in the following.
The efficient \texttt{R}-package ``\texttt{hypergsplines}'' for this and all
other computations in this paper is available from
\texttt{R}-Forge.\footnote{The website is
  \url{http://hypergsplines.r-forge.r-project.org/}. To install the
  \texttt{R}-package, just type
  \texttt{install.packages("hypergsplines",repos="http://r-forge.r-project.org")}
  into \texttt{R}.}

Let $\boldsymbol{X}_{a} = (\boldsymbol{1}_{n}, \boldsymbol{X}_{\boldsymbol{d}},
\boldsymbol{Z}_{\boldsymbol{d}})$ and $\boldsymbol{\beta}_{a} = (\beta_{0},
\boldsymbol{\beta}_{\boldsymbol{d}}^{T},
\boldsymbol{u}_{\boldsymbol{d}}^{T})^{T}$ denote the grand design matrix and
regression coefficient vector, respectively, such that $\boldsymbol{\eta} =
\boldsymbol{X}_{a}\boldsymbol{\beta}_{a}$. The prior for
$\boldsymbol{\beta}_{a}$ conditional on $g$ has a Gaussian form with mean zero and
singular precision matrix $\diag\{0, g^{-1} \boldsymbol{J}_{0},
\boldsymbol{D}_{\boldsymbol{d}}^{-1}\}$. Thus, the Gaussian approximation of
$f(\boldsymbol{\beta}_{a} \given \boldsymbol{y}, g, \boldsymbol{d})$, which is
necessary for the Laplace approximation of $f(\boldsymbol{y} \given g,
\boldsymbol{d})$, can be obtained by the Bayesian IWLS algorithm
\citep{West1985}. Afterwards, an approximation of the marginal likelihood of
model $\boldsymbol{d}$,
\begin{equation}
  \label{eq:marginal-likelihood}
  f(\boldsymbol{y} \given \boldsymbol{d}) =
  \int_{0}^{\infty} f(\boldsymbol{y} \given g, \boldsymbol{d})
  f(g)\, dg,
\end{equation}
is obtained by numerical integration of the Laplace approximation
$\tilde{f}(\boldsymbol{y} \given g, \boldsymbol{d})$. Note that recently
integrated Laplace approximations have successfully been applied in a more
general context \citep{RueMartinoChopin2009}. Finally, we can use a tuning-free
Metropolis-Hastings algorithm to sample from the joint posterior of
$\boldsymbol{\beta}_{a}$ and $g$ in a specific model $\boldsymbol{d}$.

\section{Model Prior and Stochastic Search}
\label{sec:model-prior-and-search}

We propose a prior $f(\boldsymbol{d})$ on the model space $\mathcal{D}^{p}$
which explicitly corrects for the multiplicity of testing inherent in the
simultaneous analysis of the $p$ covariates \citep[see][]{scott.berger2010}:
\latin{A priori}, the number of covariates included in the model
($I$) is uniformly distributed on $\{0, 1, \dotsc, p\}$.
The choice of the $I$ covariates is then uniformly distributed on all possible
configurations, and their degrees of freedom are independent and uniformly
distributed on $\mathcal{D} \setminus \{0\} = \{1, 2, 3, \dotsc, K\}$.
Altogether, this gives
\begin{equation}
  1 / f(\boldsymbol{d}) =
  (p+1)
  \binom{p}{I}
  K^I.
  \label{eq:model-prior}
\end{equation}
A nice property of this prior is that it leads to marginal prior probabilities
$\P(d_{j} = 0) = \P(d_{j} > 0) = 1/2$. Elsewhere this is often achieved by
assigning independent priors to the $p$ covariates, which implies that averaged
over all models, $I \sim \Bin(p, 1/2)$.
It is clear that our uniform prior on $I$ allows the data $\boldsymbol{y}$ to
have a maximum effect on the posterior of $I$ because it is the reference prior
\citep{bernardo1979}. Note that this prior actually favours models with high or
low numbers of covariates, as there are fewer such models. This or similar model
priors have been used in a number of previous papers, including \eg
\citet{GeorgeMcCulloch1993}, \citet{panagiotelis.smith2008} and
\citet{LeySteel2009}.

Alternatively, one might also use a fixed (independent of $K$) prior probability
for a linear effect ($d_{j} = 1$). This is appropriate for the
situation where one explicitly wants to test linearity versus nonlinearity of
each effect. Furthermore, a multiplicity correction for these tests can be
implemented by assuming that the number of smoothly included covariates ($J$) is
uniformly distributed on $\{0, 1, \dotsc, I\}$ and their choice is uniform on
all possible choices. This would add one level to the prior hierarchy.

As the model space $\mathcal{D}^{p}$ grows exponentially in the number of
covariates $p$, only for small values of $p$ all possible models can be
evaluated. Otherwise the marginal likelihoods $f(\boldsymbol{y} \given
\boldsymbol{d})$ and posterior model probabilities $f(\boldsymbol{d}
\given \boldsymbol{y}) \propto f(\boldsymbol{y} \given \boldsymbol{d})
f(\boldsymbol{d})$
can be computed only for a subset of the model space. Usually
this subset is determined by stochastic search procedures. Here we propose to
use a simple Metropolis-Hastings algorithm with two possible move types in the
proposal kernel:
\begin{description}
\item[Move] Sample a covariate index $j \sim \Unif\{1, 2, \dotsc, p\}$ and
  decrease or increase $d_{j}$ to the next adjacent value in $\mathcal{D}$ (with
  probability $1/2$ each, or deterministically if $d_{j} = 0$ or
  $d_{j} = K$, respectively).
\item[Swap] Sample a pair $(i, j) \sim \Unif\{(1, 1), (1, 2), \dotsc, (p,
  p)\}$ of covariate indices ($i \leq j$) and swap $d_{i}$ and $d_{j}$.
\end{description}
The `Swap' move is designed to efficiently trace models with high posterior
probability even in situations where covariates are almost collinear. For each
Metropolis-Hastings iteration, a `Move' is chosen with some fixed probability
(we use $3/4$), and otherwise a `Swap'. Denote the current model by
$\boldsymbol{d}$, then the proposed model $\boldsymbol{d}'$ is accepted with
probability
\begin{equation*}
  \alpha(\boldsymbol{d}'\given \boldsymbol{d}) =
  1
  \wedge
  \frac
  {f(\boldsymbol{y}\given \boldsymbol{d}')
    f(\boldsymbol{d}')
    q(\boldsymbol{d}'\given  \boldsymbol{d})}
  {f(\boldsymbol{y}\given \boldsymbol{d})
    f(\boldsymbol{d})
    q(\boldsymbol{d}\given  \boldsymbol{d}')}
\end{equation*}
where the calculation of the proposal probability ratio $q(\boldsymbol{d}'\given
\boldsymbol{d}) / q(\boldsymbol{d}\given \boldsymbol{d}')$ is
straightforward, see Appendix~\ref{sec:impl-deta:proposal-probs}.

The advantage of such an MCMC based model exploration compared to more elaborate
stochastic search algorithms \citep[\eg][]{HansDobraWest2007,clyde.etal2011} is
that it does not preclude estimation of posterior model probabilities via
sampling frequencies, as it was originally proposed for MCMC model composition
by \citet{MadiganYork1995}. Recently reported problems with renormalized
probability estimates \citep{clyde.ghosh2010,garcia-donato.martinez-beneito2011}
can be avoided by using the model sampling frequencies instead.
Nevertheless, other search procedures might be beneficial when only the
\latin{maximum a posteriori} (MAP) model and not \eg the marginal posterior
inclusion probabilities for the covariates are of interest.

\section{Applications}
\label{sec:applications}

We examine the performance of the proposed additive model selection methodology
with a simulation study in Section~\ref{sec:applications:sim-study}, and
illustrate logistic regression using the Pima Indian data set in
Section~\ref{sec:applications:pima}.

\subsection{Simulation Study in Additive Models}
\label{sec:applications:sim-study}

In order to study the frequentist properties of our approach, we performed a
simulation study. The full details are provided as supplementary material which
is available at Biometrika online. Here we summarise the main results.

Three different true models were simulated: The first model (``null'') was the
null model with $p=20$ nuisance covariates. The second model (``small'') also
had $p=20$ covariates of which 3 had a linear effect and 3 had a nonlinear
(quadratic, sine, and skew-normal density) effect. Correlations of different
strength were generated between some the covariates. The third model (``large'')
was identical to the second model, but included additional 80 nuisance
covariates, which were independent of the first 20 covariates. For the ``small''
and ``large'' model, one covariate was chosen to be a surrogate for the true,
quadratic, effect of another covariate. It masks the quadratic effect if only
linear effects can be fitted by a variable selection algorithm. For three
different sample sizes $n=50, 100, 1000$, and for the three different true
models, we simulated $n$ observations from the Gaussian additive
model~\eqref{eq:additive-model} with $\beta_{0} = 0$ and $\sigma^{2} = 0.2^{2}$.
This was repeated 50 times for each combination of model and sample size, in
order to assess the sampling variability.

We applied the proposed additive model selection approaches to each data set,
using the hyper-$g$ and hyper-$g/n$ priors. As the computational complexity of
the marginal likelihood \eqref{eq:normal-models:marginal-likelihood} is cubic in
the spline basis dimension $K$ (see Appendix~\ref{sec:impl-deta:marg-lik-comp}),
we want to use splines with few, quantile-based knots. Therefore, we choose
cubic O'Sullivan splines \citep{wand.ormerod2008}. Here, we got basis matrices
$\boldsymbol{Z}_{j}$ with $K=8$ columns from 6~inner knots at the septiles. We
applied the stochastic search algorithm described in
Section~\ref{sec:model-prior-and-search} with $10^{6}$~iterations.

We compared the results with those from pure variable selection including only
linear functions, Bayesian FPs \citep{sabanesbove.held2011a}, spike-and-slab
function selection \citep{scheipl.etal2011} and splines knot selection
(\citeauthor{denison.etal1998}, \citeyear{denison.etal1998}, using code from
chapters 3 and 4 in \citeauthor{DenisonHolmes02}, \citeyear{DenisonHolmes02}).

Concerning discovery of the true set of influential covariates, the additive
model selection procedures introduced in this paper were very competitive with
the considered alternative methods, as is illustrated in
Table~\ref{tab:med-post-prob-true-model}. In particular, they showed clear
advantages in the case of small and moderate sample sizes. Using splines instead
of only linear functions proved essential for the discovery of the masked
quadratic effect and hence convergence to the true model.

\begin{table}
  \centering
  {\footnotesize
\begin{tabular}{lrrrcrrrcrrr}
\toprule
\multicolumn{1}{l}{\small }&
\multicolumn{3}{c}{\small null}&
\multicolumn{1}{c}{\small }&
\multicolumn{3}{c}{\small small}&
\multicolumn{1}{c}{\small }&
\multicolumn{3}{c}{\small large}
\tabularnewline
\cline{2-4} \cline{6-8} \cline{10-12}
\multicolumn{1}{l}{}&\multicolumn{1}{c}{$n=50,$}&\multicolumn{1}{c}{$100,$}&\multicolumn{1}{c}{$1000$}&\multicolumn{1}{c}{}&\multicolumn{1}{c}{$n=50,$}&\multicolumn{1}{c}{$100,$}&\multicolumn{1}{c}{$1000$}&\multicolumn{1}{c}{}&\multicolumn{1}{c}{$n=50,$}&\multicolumn{1}{c}{$100,$}&\multicolumn{1}{c}{$1000$}\tabularnewline
\midrule
Hyper-$g$ splines&$83$&$84$&$84$&&$49$&$65$&$86$&&$2$&$74$&$87$\tabularnewline
Hyper-$g/n$ splines&$86$&$91$&$97$&&$47$&$68$&$87$&&$0$&$75$&$89$\tabularnewline
Hyper-$g$ linear&$20$&$21$&$23$&&$ 0$&$ 0$&$ 0$&&$0$&$ 0$&$ 0$\tabularnewline
Hyper-$g/n$ linear&$50$&$64$&$90$&&$ 0$&$ 0$&$ 0$&&$0$&$ 0$&$ 0$\tabularnewline
Bayesian FPs&$37$&$37$&$37$&&$ 2$&$35$&$ 3$&&$0$&$47$&$37$\tabularnewline
Spike-and-slab&$89$&$93$&$98$&&$ 3$&$45$&$79$&&$0$&$10$&$71$\tabularnewline
Knot selection&$92$&$94$&$98$&&$ 0$&$34$&$95$&&$0$&$ 0$&$89$\tabularnewline
\bottomrule
\end{tabular}}
\caption{Median posterior probability of the true model in percentage, when the
  true model is defined by correct variable inclusion.}
\label{tab:med-post-prob-true-model}
\end{table}

Variable inclusion performance did not differ substantively with respect to
sensitivity, specificity and area under the ROC curve between the considered
methods, with the exception of a slightly worse performance of the two linear
methods. However, as shown in Table~\ref{tab:diff-inc-probs}, the hyper-$g$ and
hyper-$g/n$ spline methods were clearly better in distinguishing truly effective
covariates from highly correlated nuisance covariates. Moreover, for small
sample sizes, they outperformed the other nonlinear methodologies concerning
discovery of the masked quadratic effect. In this task the merely linear methods
obviously failed.

\begin{table}
  \centering
  {\footnotesize
\begin{tabular}{lrrrcrrr}
\toprule
\multicolumn{1}{l}{\small }&
\multicolumn{3}{c}{\small small}&
\multicolumn{1}{c}{\small }&
\multicolumn{3}{c}{\small large}
\tabularnewline
\cline{2-4} \cline{6-8}
\multicolumn{1}{l}{}&\multicolumn{1}{c}{$n=50,$}&\multicolumn{1}{c}{$100,$}&\multicolumn{1}{c}{$1000$}&\multicolumn{1}{c}{}&\multicolumn{1}{c}{$n=50,$}&\multicolumn{1}{c}{$100,$}&\multicolumn{1}{c}{$1000$}\tabularnewline
\midrule
Hyper-$g$ splines&$75$&$97$&$98$&&$26$&$100$&$100$\tabularnewline
Hyper-$g/n$ splines&$79$&$97$&$98$&&$20$&$100$&$100$\tabularnewline
Hyper-$g$ linear&$18$&$44$&$87$&&$ 6$&$ 26$&$ 98$\tabularnewline
Hyper-$g/n$ linear&$22$&$48$&$90$&&$17$&$ 26$&$ 98$\tabularnewline
Bayesian FPs&$41$&$89$&$68$&&$ 9$&$ 92$&$ 81$\tabularnewline
Spike-and-slab&$30$&$88$&$97$&&$ 1$&$ 60$&$ 97$\tabularnewline
Knot selection&$ 9$&$78$&$99$&&$ 4$&$ 13$&$ 99$\tabularnewline
\bottomrule
\end{tabular}}
\caption{Average difference $\frac{1}{2}(P_{16} + P_{17}) - \frac{1}{3}(P_{18} +
  P_{19} + P_{20})$ of inclusion probabilities $P_{j} = \P\{m_{j}(x_{j}) \neq 0
  \given \boldsymbol{y}\}$ (in percentage points) between the truly effective
  covariates $x_{16}$ and $x_{17}$ and the nuisance covariates $x_{18}, x_{19},
  x_{20}$, which had correlation $0.8$ with $x_{16}$ and $x_{17}$. (The optimal
  value is $100$, the worst value is $-100$.)}
\label{tab:diff-inc-probs}
\end{table}

Concerning the average mean squared errors of the model-averaged posterior mean
function estimates $\hat{m}_{j}(x_{j})$, the proposed additive model selection
procedures were very competitive. They performed well or better than the best
compared method each, as is shown in Table~\ref{tab:errors-bma-overall}. It is
interesting that the hyper-$g$ splines were slightly but consistently better
than the hyper-$g/n$ splines. We also investigated the coverage rates of
pointwise 95\% credible intervals for the functions, and found that the two
proposed methods were slightly conservative.

\begin{table}
  \centering
  {\footnotesize
\begin{tabular}{lrrrcrrrcrrr}
\toprule
\multicolumn{1}{l}{\small }&
\multicolumn{3}{c}{\small null}&
\multicolumn{1}{c}{\small }&
\multicolumn{3}{c}{\small small}&
\multicolumn{1}{c}{\small }&
\multicolumn{3}{c}{\small large}
\tabularnewline
\cline{2-4} \cline{6-8} \cline{10-12}
\multicolumn{1}{l}{}&\multicolumn{1}{c}{$n=50,$}&\multicolumn{1}{c}{$100,$}&\multicolumn{1}{c}{$1000$}&\multicolumn{1}{c}{}&\multicolumn{1}{c}{$n=50,$}&\multicolumn{1}{c}{$100,$}&\multicolumn{1}{c}{$1000$}&\multicolumn{1}{c}{}&\multicolumn{1}{c}{$n=50,$}&\multicolumn{1}{c}{$100,$}&\multicolumn{1}{c}{$1000$}\tabularnewline
\midrule
Hyper-$g$ splines&$0.03$&$0.01$&$0.00$&&$   39.15$&$  10.32$&$  1.68$&&$ 30.42$&$  1.88$&$ 0.33$\tabularnewline
Hyper-$g/n$ splines&$0.05$&$0.01$&$0.00$&&$   47.82$&$  18.33$&$  3.20$&&$784.44$&$  2.78$&$ 0.61$\tabularnewline
Hyper-$g$ linear&$0.76$&$0.14$&$0.01$&&$  158.10$&$ 133.55$&$121.97$&&$ 45.11$&$ 32.26$&$24.36$\tabularnewline
Hyper-$g/n$ linear&$0.22$&$0.02$&$0.00$&&$  189.57$&$ 169.00$&$120.96$&&$378.07$&$ 36.23$&$26.09$\tabularnewline
Bayesian FPs&$0.14$&$0.03$&$0.00$&&$16837.92$&$3026.61$&$ 29.51$&&$ 76.78$&$356.30$&$ 5.80$\tabularnewline
Spike-and-slab&$1.90$&$1.82$&$0.57$&&$   80.94$&$  14.00$&$  2.09$&&$ 45.45$&$  8.71$&$ 0.81$\tabularnewline
Knot selection&$0.03$&$0.00$&$0.00$&&$  180.03$&$  35.29$&$  2.07$&&$ 47.23$&$ 29.33$&$ 0.78$\tabularnewline
\bottomrule
\end{tabular}}
\caption{Average mean squared errors (in $10^{-4}$ units) of function estimates.
 Numbers are averaged over all covariates and the 50~replications.}
\label{tab:errors-bma-overall}
\end{table}

Finally, the average computational effort of the two proposed additive model
selection procedures ranged between one minute for $n=100$ in a ``null'' data
set to about 50~minutes for $n=50$ in a ``large'' data set.

\subsection{Pima Indian Diabetes Data}
\label{sec:applications:pima}

We now apply the generalised additive model selection approach to the logistic
regression of $p=7$ potential risk factors on the presence of diabetes in
$n=532$ women of Pima Indian heritage
\citep{FrankAsuncion2010,Ripley1996}, see Table~\ref{tab:pima-description} for
details. We use cubic O'Sullivan splines with 4~inner knots at the quintiles and
the generalised hyper-$g/n$ prior, and explore the model space of dimension
$7^{7} = 823\,543$ with $10^{6}$
iterations of the stochastic search algorithm. The computational complexity is
higher than for the normal response case, with
95~minutes required for the evaluation
of 39\,081~models. We validated the
results with an exhaustive evaluation of all models, requiring
33~hours. Indeed, the stochastic
search found 99\% of the posterior
probability mass and the 733~top models.

\begin{table}[htbp]
  \centering
  \begin{tabular}{ll}
    \toprule
    Variable & Description \\
    \midrule
    $y$ & Signs of diabetes according to WHO criteria (Yes = $1$, No = $0$) \\
    $x_{1}$ & Number of pregnancies \\
    $x_{2}$ & Plasma glucose concentration in an oral glucose tolerance
    test [mg/dl]\\
    $x_{3}$ & Diastolic blood pressure [mm Hg] \\
    $x_{4}$ & Triceps skin fold thickness [mm] \\
    $x_{5}$ & Body mass index (BMI) [kg/m\textsuperscript{2}]\\
    $x_{6}$ & Diabetes pedigree function \\
    $x_{7}$ & Age [years] \\
    \bottomrule
  \end{tabular}
  \caption{Description of the variables in the Pima Indian diabetes data set.}
  \label{tab:pima-description}
\end{table}

In Table~\ref{tab:pima-inc-probs} the marginal posterior probabilities for
linear and smooth inclusion of the covariates are shown.
There is clear evidence for inclusion of the covariates $x_{2}$, $x_{5}$,
$x_{6}$ and $x_{7}$, which have posterior inclusion probabilities over 96\%. For
the other three covariates, the inclusion probability is below 30\%. Smooth
modelling of the effects of $x_{5}$, $x_{6}$ and $x_{7}$ seems to be necessary,
while this is not so clear for $x_{2}$.

\begin{table}
\centering
\begin{tabular}{llllllll}
\toprule
\multicolumn{1}{l}{}&\multicolumn{1}{c}{$x_{1}$}&\multicolumn{1}{c}{$x_{2}$}&\multicolumn{1}{c}{$x_{3}$}&\multicolumn{1}{c}{$x_{4}$}&\multicolumn{1}{c}{$x_{5}$}&\multicolumn{1}{c}{$x_{6}$}&\multicolumn{1}{c}{$x_{7}$}\tabularnewline
\midrule
not included ($d_{j} = 0$)&0.74&0.00&0.88&0.91&0.00&0.04&0.01\tabularnewline
linear ($d_{j} = 1$)&0.07&0.48&0.06&0.04&0.11&0.26&0.00\tabularnewline
smooth ($d_{j} \textgreater  1$)&0.19&0.52&0.06&0.05&0.89&0.70&0.99\tabularnewline
\bottomrule
\end{tabular}\caption{Marginal posterior inclusion probabilities in the Pima Indian diabetes
  data set.}
\label{tab:pima-inc-probs}
\end{table}

In order to examine the mixing properties of the stochastic search algorithm
proposed in Section~\ref{sec:model-prior-and-search}, we compared the results
based on starting the MCMC chain from the full model with $d_{j} = 4$ instead of
the previously used null model with $d_{j} = 0$ ($j=1, \dotsc, p$). The results
are very close: for example, the entries in Table~\ref{tab:pima-inc-probs}
differ by at most $2.28\cdot 10^{-4}$, and the top 500
models which were visited by the chains are identical. These results are an
indication that slow mixing is not a problem for the presented stochastic search
algorithm.

Figure~\ref{fig:pima-hpm} shows the estimated covariate effects in the MAP model
which features a linear term for $x_{2}$ and smooth terms for $x_{5}$, $x_{6}$
and $x_{7}$. The estimates are obtained from
10\,000~MCMC samples.\footnote{Every
  2nd sample was saved after burning the first
  1000 iterations, with acceptance
  rate 67\% using two
  IWLS steps per proposal.}
Note that for linear functions $m_{j}$, the pointwise credible intervals
coincide with the simultaneous credible intervals
\citep[p.~30]{BesagGreenHigdonMengersen1995}. This is because all straight lines
samples intersect in one point, which is due to the centring of the covariates
in \eqref{eq:non-normal-models:center-vector}.
Furthermore, we observe that the \citet{ChibJeliazkov2001} estimate
($-240.924$, MCMC standard
error $0.008$) of the log
marginal likelihood of the MAP model, which was also computed, is quite close to
the integrated Laplace approximation
($-241.01$). This
indicates that the integrated Laplace approximation is fairly accurate.

\begin{figure}
  \centering
  \begin{tabular}{cc}
{
\input{out-pima-plot-x2.tikz}
}
&
{
\input{out-pima-plot-x5.tikz}
}
\\
{
\input{out-pima-plot-x6.tikz}
}
&
{
\input{out-pima-plot-x7.tikz}
}
\end{tabular}
\caption{Estimated covariate effects in the MAP model for the Pima Indian
  diabetes data set, based on
  10\,000~MCMC samples: Posterior means
  (solid lines), pointwise (dashed lines) and simultaneous (dotted lines)
  95\%-credible intervals are shown.}
  \label{fig:pima-hpm}
\end{figure}

The results are qualitatively similar to those obtained with a FP modelling
approach by \citet[section~5]{sabanesbove.held2011} and with a cubic smoothing
spline approach by \citet[section~3.2]{CottetKohnNott2008}. It is interesting
that in the earlier work by \citet[section~5.2]{YauKohnWood2003}, a very low
posterior inclusion probability ($0.07$) for $x_{6}$ was reported for a
different subset of the original Pima Indian diabetes data set. If pure variable
selection without covariate transformation is considered, as in
\citet[section~2.6]{HolmesHeld2006} and \citet[section~4]{sabanesbove.held2011},
the strong nonlinear effect of $x_{7}$ is missed completely, and instead $x_{1}$
gets a higher posterior inclusion probability. This may be a case of a masked
nonlinear effect, as was simulated in Section~\ref{sec:applications:sim-study},
and highlights the importance of allowing for nonlinear covariate effects.

\section{Postprocessing}
\label{sec:postprocessing}

Given the list of all possible models $\boldsymbol{d} \in \mathcal{D}^{p}$, or a
subset found by the stochastic search procedure described in
Section~\ref{sec:model-prior-and-search}, one may consider postprocessing the
results.

First, when the main interest lies in variable selection, the models which
feature the same covariates can be summarised into a meta-model as follows: The
posterior probabilities of the sub-models are summed up to give the posterior
probability of the meta-model, and estimates in the meta-model are obtained by
averaging the sub-models with weights proportional to their posterior
probabilities \citep[see \eg][for model averaging]{hoeting.etal1999}.
For example, the best meta-model for the Pima Indian diabetes data includes
$x_{2}$, $x_{5}$, $x_{6}$ and $x_{7}$ and has posterior probability
$0.598$. The
corresponding estimates of the covariate effects are shown in
Figure~\ref{fig:pima-best-meta-model}. This best meta-model happens to be
identical with the median probability meta-model, which features all covariates
having marginal posterior inclusion probability greater than 50\%
\citep{BarbieriBerger2004}, \abk{\latin{cp}} Table~\ref{tab:pima-inc-probs}.
Similarly, it could be interesting to summarise models which only differ in the
degrees of freedom for smooth terms. This would correspond to the situation of
testing linearity versus nonlinearity of covariate effects (\abk{\latin{cp}}
Section~\ref{sec:model-prior-and-search}).

\begin{figure}
  \centering
  \begin{tabular}{cc}
{
\input{out-pima-topInc-plot-x2.tikz}
}
&
{
\input{out-pima-topInc-plot-x5.tikz}
}
\\
{
\input{out-pima-topInc-plot-x6.tikz}
}
&
{
\input{out-pima-topInc-plot-x7.tikz}
}
\end{tabular}
  \caption{Estimated covariate effects in the best meta-model (and median
    probability meta-model) for the Pima Indian diabetes data,
    based on
    20\,000
    samples:
    Posterior means (solid lines), pointwise (dashed lines) and simultaneous
    (dotted lines) 95\%-credible intervals are shown.}
  \label{fig:pima-best-meta-model}
\end{figure}

Second, in order to allow for \emph{continuous} degrees of freedom, one can optimise
the marginal likelihood of the MAP model with respect to the degrees of freedom
of the covariates included. That is, an optimisation of $f(\boldsymbol{y} \given
\boldsymbol{d})$ over the continuous range $1 < d_{j} < K + 1$ is
performed for all covariates included in the MAP model.
For example, the MAP configuration for the Pima Indian diabetes data is
$(0,1,0,0,3,2,4)$ and the resulting optimised
configuration is $(0,1,0,0,3.42,2.1,3.74)$,
which increases the log marginal likelihood from
$-241.01$ to
$-240.86$. Although $d_{5}$ and $d_{7}$
changed considerably in the optimisation, the resulting function estimates are
very similar to those from the MAP model in Figure~\ref{fig:pima-hpm} and are
hence omitted.

\section{Discussion}
\label{sec:discussion}

Our Bayesian approach to simultaneous variable and function selection in
generalised regression is based on fixed-dimensional spline bases and
penalty-parameter smoothness control. In this respect, it differs from
knot-selection approaches such as \citet{smith.kohn1996} and
\citet{denison.etal1998}. We found that fixed-dimensional spline bases are
flexible enough to capture the functional forms we expect \citep[see
\eg][]{abrahamowicz.etal1996}. Moreover, by using fixed-dimensional smooth
components we can constrain a covariate effect to be exactly linear. This
enables us to look at posterior probabilities of linear \latin{versus} smooth
inclusion of covariates. Approaches which use variable-dimensional smooth
components and select knots, as \citet{denison.etal1998}, cannot fit linear
functions.

We are only considering roughness penalties on a fixed grid of values, which
scales automatically for each covariate via the degrees of freedom
transformation. We found that it is a very useful approximation of a continuous
scale, and postprocessing is possible to remove the restriction to the grid
values. In the examples we have looked at, the resulting optimised models
yielded very similar results compared to the MAP model.
In this regard, our approach is close to many popular Lasso-type proposals,
which optimise the tuning-parameters on a fixed grid via cross-validation
\citep[\eg][]{zou.hastie2005}. \citet{cantoni.hastie2002} propose a
likelihood-ratio-type test statistic to compare additive models with different
degrees of freedom.
\citet{FongRueWakefield2010} use a similar scaling to examine the prior on the
degrees of freedom implied by the prior on the variance component in a
generalised linear mixed model. They also use O'Sullivan spline bases as we did
in our applications, but they do not consider variable selection.

In a frequentist setting, \citet[section~2.1]{marra.wood2011} propose to use an
additional penalty on the linear part of the spline function in order to shrink
it adaptively to zero. To include variable selection, a lower threshold for the
effective degrees of freedom must be chosen. Our generalised $g$-prior
\eqref{eq:generalised-g-prior} also shrinks the linear parts of the spline
functions to zero, where the prior covariance matrix takes the correlations
between the covariates into account. Incorporating the covariates correlation in
the coefficients prior allows for better discrimination between influential and
correlated nuisance covariates. Empirical results from our simulation study in
Section~\ref{sec:applications:sim-study} support this. Furthermore, we
explicitly ex- or include covariates and then compare the resulting models based
on their posterior probabilities.

We propose a conventional prior for the intercept and the linear coefficients,
which directly generalises the hyper-$g$ priors in the linear model
\citep{LiangPauloMolinaClydeBerger2008} and in the generalised linear model
\citep{sabanesbove.held2011}. \citet{pauler1998} proposes a related
unit-information prior for the fixed effects in linear mixed models, but fixes
$g = n$ in \eqref{eq:normal-models:g-prior}. \citet{OverstallForster2010}
propose a unit-information prior for the fixed effects in generalised linear
mixed models, but the information matrix is based on the first-stage likelihood
and not on the integrated likelihood as in our approach. Also, no hyper-prior on
the parameter $g$ is considered, because it is fixed at $g = n$. As they use an
inverse-Wishart prior on the covariance matrix of the random effects, their
approach is perhaps better suited to generic random effects models.
\citet{forster.etal2012} propose a novel reversible-jump MCMC algorithm to infer
the corresponding posterior model probabilities.

In future work, we would like to combine the semiparametric splines with the
parametric FPs \citep{sabanesbove.held2011a}. The idea is that a smooth term
$m_{j}(x_{j})$ could also be modelled by a FP instead of a spline. This
extension could be implemented coherently, because the prior formulations are
compatible. With such a general framework, the important question whether a
parsimonious FP (\eg $m_{7} = x_{7}\beta_{71} + x_{7}^{2}\beta_{72}$ in the Pima
Indian diabetes data example) is sufficient could be answered via posterior
probabilities (see \citet{strasak.etal2011} for a simulation study comparing the
stepwise FP approach by \citet{Royston:Sauerbrei:2008-Buch} with penalised
spline approaches). Moreover we will apply mixtures of $g$-priors to additive
Cox-type survival models.

\section*{Acknowledgments}}%
{}

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redpajama_set_name: RedPajamaArXiv


an instrument for measuring work done in muscular action.
Ergograph is an instrument used to assess the performance of the flexors of the finger of the hand.
8), work done using Mosso's ergograph (males 2792 [+ or -] 1445, female 2402 [+ or -] 896) and bicycle ergometer (male 3584 [+ or -] 1345, female 2948 [+ or -] 1156) were significantly higher in males when compared to females (P < 0.
Our results showed that work done assessed using bicycle ergometer and Mosso ergograph was significantly higher in males than females.


The International Food Policy Research Institute (IFPRI) said rural communities were in "a state of crisis", with high poverty rates and poor services driving hunger and malnutrition. One in five people, or more than 256-million, are hungry in Africa, according to the latest figures from the UN Food and Agriculture Organisation. But there are opportunities too, the IFPRI said in its annual report.
"They are not just asking for imported food, wine and cheese but to have traditional staples on the tables. But they don't want to eat them the traditional way," he said.
Turning opportunity into reality needs technology and financing that would allow locals to innovate and compete, he said. There should also be investment in rural areas and access to energy and telecommunications, he added. Nearly half the world's population live in rural areas but represent 70% of the extremely poor, according to the IFPRI.


PRICE JUST LOWERED TO $2,500 WAS $3,400. Use the savings towards your shipping cost. Original Unrestored 1950 Hudson Commodore 8. 4 door sedan. Original 8 cylinder engine/transmission, not running. Car was parked in 1962, and still has the 1962 license plates on it. Has good floors, good trunk floor. There is some rust out in the front edge of the hood. Interior is original and in need of restoration. Is missing the radio. There are some of the dash pieces lying on the floor. We even have the correct factory fender skirts too. Body has some damage to the right rear quarter panel, it is creased. Solid rockers. This is not a Rusty Car! If you are looking for a neat car to build, this is a good one. PRICE IS NOW ONLY $2,500 WAS $3,400. YES YOU WILL GET A CLEAR CLEAN TITLE TOO. Shipping is available. For more information call Bill at Motion Unlimited Museum and Old Car Lot, Rapid City, South Dakota. 605-348-7373.


How often articles will be reviewed?
I submitted one article yesterday and it was almost 24 hours. Is there any particular time where all the articles will be reviewed? Just want to know when my article will be reviewed?
Thank you and waiting for the response!
Articles are usually reviewed within a week. Our editors are doing the editing at their free time and editing is usually done during week-ends. So, depending on when you post the article, it may take 1 to 7 days to complete the reviews.
Sridhar, your anxiety about getting your article reviewed is understandable. But since our editors have their own professional and personal commitments and as such may find it difficult to review the section rather quickly.
You might have noticed that a couple of articles have been reviewed over the last weekend. Please have patience.
Articles are reviews very strictly and needs proper time to carefully read all the articles. And also the editors are busy with their schedule so they mostly review the articles at the weekends. So I will suggest you to just have patience all the time (though it is very tough for all).
I think the Webmasters have been trying to get more people on board the editorial panel for articles section. The problem is getting somebody who can understand SEO and link building to ensure good traffic to the section. It requires better knowledge as opposed to, say, editing an Ask Expert post. Let's hope we do find the requisite efficient people in due course of time. Hold that thought....and keep contributing, without worrying too much about delays in article reviews!


text:
Most Popular Original Series on Netflix
Terms like 'binge watch' and 'Netflix and Chill' are quite synonymous to this generation and has entered the cultural lexicon. Many OTT platforms are releasing their original productions and some of them turn out to be worth the wait and binge-watching experience. With tons of series being produced these days, it has become really confusing for people to choose which one to watch and which one to leave. We have taken the liberty to collate a list of series that are most popular and that you must watch.
One of the most anticipated releases of this year. The series is packed with thrill and action sequences that keep you gripped and looking for more throughout. The movie is about a mercenary who is sent to Bangladesh to rescue the son of an Indian drug lord because of the rivalry between the 2. Endless fight scenes, chase scenes, great direction, and background score are things that come along.
The series has 3 seasons with the first one being about a boy named Will who is captured by a creature. His mother then recruits the local sheriff to investigate her son's disappearance. Meanwhile, Will's friends befriend an alien-like girl who has telepathic. This is a good PG sci-fi/horror like blockbuster.
This is a British teen comedy that explores cringe worthy topics and taboo topics that are associated with sex. It showcases the journey of a mother-son pair who make their way through those uncomfortable talks. The mother is a sex therapist. Sex is a goldmine of comedy.
The series has been well written and brilliantly shot. This character of the Marvel universe lives way out of their complete universe that it is completely distinctive. It is darker and grittier than the movie franchise but with enough humour as well. The fight scenes in the series are brutal. The combination of great writing, direction, and acting make it one of the best Netflix originals

redpajama_set_name: RedPajamaCommonCrawl


Shams, the new Saudi tabloid which was banned for a period of time during the Danish cartoons controversy, has published one of my AFP photos but in a different context for another story. They have used a different caption than the one distributed by the agency. The caption here reads, "Ahmed Al-Omran, a young Saudi browses websites on his laptop." What a lame line is this?
Well, the size of the picture is almost the same, but it looks bigger because my brother scanned it with a high resolution. Choosing the color of shirt to wear was actually confusing, but I settled finally for green and it seems to be a good choice. I'm glad that people are liking it, because I did not expect they would.


text:
Q: Why does the axle need to be rotated in a turbine generator? Inside a turbine a wire rotates between the poles of a magnet, i.e. rotating inside the magnetic field, so that the field forces the electrons in the wire to move and generate current. But why does the conductor wire need to rotate between the two poles of the magnet? Why can't it generate current when the wire is stationary?

A: You are having some misconceptions

*

*Wire(more pricisely axle) is not rotated between magnetic field in a turbine generator.Wire(more pricisely axle) is used to rotate an coil (present between magnetic fields) present in generator.

*Magnetic field itself can not force electrons in a conductor to move untill it is at rest.

*We need an moving conductor so that a force can be applied on electrons by magnetic field. (F=qv*B here q is charge on electron, v is velocity of conductor and B is magnetic field)

*A turbine and generator are two different things.

*Turbine just rotates that wire but to produce electric current you have to connect the ends of that wire ( better to say it axle not wire )to an generator.

*A generator converts that rotating mechanical energy into electrical energy

*A generator have magnets not the turbine.
8.Axle ( you like the word wire) does not contain current.
Now why do you need to rotate that turbine is because generator requires mechanical energy
How does it do it ?
As follows
According to faraday's law of electromagnetic induction current will be induced in the circuit (coil) if magnetic flux through the circuit(coil) changes with time .
     If you do not rotate the turbine then there will be no flux change and hence no emf or current induced.
Remember you can also change the flux by
1.Keeping the circuit stationary and moving the magnets
2. Changing the shape of coil (circuit) so that area changes
3.changing magnitude of magnetic field
4.many more

redpajama_set_name: RedPajamaStackExchange


Virgil Beaston is an engineer, patent attorney, and business person focusing on the design and manufacture of scalable energy storage systems for Powin Energy Corporation. In March 2008, he resigned as a Director from a large USA intellectual property law firm and co-founded a start-up company to make energy storage products. Soon thereafter, he moved to Shenzhen China and setup and ran a small factory that designed and produced energy storage products until joining Powin Energy in 2011 as its Chief Technology Officer. His experiences as an engineer, lawyer and business person have given him a unique insight into what it takes to commercialize new clean-tech technologies and help fuel the green economy.


text:
An Act relative to the unlawful passing of a school bus
By Mr. Boldyga of Southwick, a petition (accompanied by bill, House, No. 1472) of Nicholas A. Boldyga and Tommy Vitolo relative to the unlawful passing of a school bus. The Judiciary.
Nicholas A. Boldyga
Referred to Joint Committee on the Judiciary
3/29/2021 House Referred to the committee on The Judiciary
12/7/2021 Joint Hearing scheduled for 12/14/2021 from 10:00 AM-02:00 PM in Virtual Hearing
An Act relative to seat belts on school buses
An Act relative to school bus safety
An Act relative to seatbelts on school buses
An Act relative to the safety of children in school buses


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Dal 18 marzo 1931 al 4 gennaio 1934 è stato Presidente del Soviet Supremo della Repubblica socialista sovietica tagica e rappresentante della Repubblica socialista sovietica tagica nel Comitato esecutivo centrale dell'Unione Sovietica. Maksum ha una posizione 'moderata' nei confronti dei kulaki, e quando questa posizione viene giudicata 'erronea' dal Politburo del CC del PCUS (5 dicembre 1933), Maksum viene destituito a fine mese dal Consiglio dei Commissarî del Popolo (per politiche "antipartitiche ed antisovietiche"), e pochi giorni dopo anche dalla presidenza del Comitato Esecutivo Centrale dell'URSS (sostituito da A. R. Rakhimbaev, poi vittima delle purghe staliniane a sua volta).

Venne arrestato a Mosca l'8 luglio 1937 e condannato alla pena capitale il 31 ottobre dello stesso anno (giustiziato da un plotone d'esecuzione il giorno successivo). Durante la destalinizzazione, il 15 dicembre 1957 il Collegio Militare della Corte Suprema, dando seguito alla decisione della Procura generale dell'URSS, annullò la sentenza contro Maksum per mancanza di corpus delicti, ma ci vorrà il 26 giugno del 1964 perché il Presidium del Soviet Supremo dell'URSS riabiliti Nusratullo Maksum. Dopo l'indipendenza del Tagikistan dall'Unione Sovietica, Nusratullo Maksum è stato proclamato eroe nazionale del Tagikistan e rappresentato sulla banconota di 200 Somoni tagichi.

Onorificenze

Note

redpajama_set_name: RedPajamaWikipedia


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Point 2.- new transfer arrangements for the United Kingdom. The UK`s new transmission system is set out in the Data Protection, Privacy and Electronic Communications Regulations 2019 (amendments, etc.) (leaving the EU) (SI 2019/419); These apply from the day of release (which may or may not be October 31 at 23 p.m.). The regulation assumes that the UK data protection regime will deviate from the GDPR. Anyone who transfers personal data from the EU/EEA to the UK will do so on a third-country basis, with or without an adequacy decision by the European Commission in relation to the UK. There are currently no changes in the way you send personal data to the EU/EEA, Gibraltar and other countries deemed appropriate by the EU. If this situation changes, we will update this page. We have also created an interactive tool on the use of standard contractual clauses for transfers to the UK to help you. Given that government departments are large controllers, there is a risk that these documents distort the delicate data protection balances between data subjects and the controller in favour of the interests of the controller and the processing of the Ministry (and the political interests of the competent Minister of Foreign Affairs, who is responsible for the actions of the department). There is no immediate need for action. Organisations can still transfer personal data to the UK until the end of 2020 without a transfer mechanism. In order to be able to transfer personal data to the UK after the end of the transition period, organisations must take the following steps: You may also consider changing your company`s data flows in relation to personal data from the EU so that it is transferred directly from an EU data exporter to a data importer outside the EEA/non-UK in the framework for an appropriate data transfer mechanism (e.B.
CSC). Within weeks, the EU has yet to make a decision on whether or not to accept whether the UK`s data protection regime is still appropriate. If your company currently relies on CCAs for some or all of its international transfers, please note that the EDDC recently updated its guidelines on the use of CCTs following the CJEU ruling in Schrems II. We anticipate that the transfer of data from the UK to the EU and Gibraltar may continue for the time being on the basis of new UK adequacy rules. For more information on how transfer rules work, see the International Transfers page of our Enforcement Guide. This can complicate international transfers for organizations that transfer sets of personal data to which the frozen GDPR may apply. Does the ICO govern the laws applicable in Article 71 or is the processing of this data subject to the regulation of EEA regulators? In the event of a personal data breach related to personal data subject to Article 71, the organisation must inform the supervisory authorities of the ICO or the EEA (or both?). The ICO regulates the laws applicable in Article 71. Depending on the circumstances of the data breach, the frozen GDPR and/or the UK GDPR may also apply with the EU GDPR, resulting in the obligation to notify the ICO as well as the eea competent authorities. .
By Werbeagentur Gerald Wiedermann| 2021-10-16T11:18:18+00:00	Oktober 16th, 2021|Allgemein|Kommentare deaktiviert für Withdrawal Agreement Data Transfer
About the Author: Werbeagentur Gerald Wiedermann


And he asked me to be a guest on his show sometime in the near future. Can't wait, sounds very exciting.
I see Marshon two or three times a week at the La Plata, Starbuck's. As you can tell he's a Millennial, which are typically children of us Baby Boomers. He always helps me with computer application questions. My latest lesson was about the Cloud.
Thanks guys for buying my book.


The Original Mini now has an updated model called the Version 3. This brings the most anticipated update to the mini, a rerouting of the heated bed wiring. Specifically designed to improve more accurate movement of the Y-Axis and prolong the lifespan.


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There Is No Long and Winding Path
You are the cutting edge
Do you think of life as a long and winding path like this one leading to Evergreen? Think again!
We tend to regard life as a series of adventures or disasters as we make our way along a long and winding path that stretches temptingly before us.
We're always trying to get further along the path, so we spend our lives focussing on what's just ahead, or around the next bend, or on the other side of the mountain.
We plod on, one foot after the other. But each step doesn't really get us further along the path because we keep mentally throwing life forward, then struggling to catch up with what we've projected, while the path stretches even further ahead of us than it did before.
When we're young we want to grow up. Then we can't wait to graduate, become employed, have a relationship, get that promotion. Or we want to be on holiday, live in a maintenance-free house, enjoy a perfect garden, be the ideal weight, go viral and suddenly make lots of money. As we turn grey and begin to stoop, we long to be the recipient of accolades, make it to the board of directors, get a seat in the senate, land a king-sized single bed in the 5-star permanent residency facility, or be allocated the plot with the best views at Serenity Central Cemetery.
If something unexpected crops up that stops our progress towards where we think we should be already, we try to dodge around it, push it away, ignore it, or deny its existence. "This shouldn't be!" we shout to the gods inhabiting the sky. But none of them are listening. So we plod on.
When we view life this way we're never where we want to be.
And we wonder why we feel weary!
But what would happen if we changed how we perceive life?
What if we stopped thinking of life as a journey on a long and winding path?
What if it dawns on us that our current step is all there is? There is no path stretching before us. We are on the edge, the cusp, the cliff, every moment of our lives. What we do in this moment is all there is.
How would we feel if this was how we looked at life?
We'd be on high alert, curious, intrigued, energized, open to inspiration, free.
We'd be wide awake!
Pema Chödrön discusses this in her book When Things Fall Apart:
This path has one very distinct characteristic: it is not prefabricated. It doesn't already exist. The path that we're talking about is the moment-by-moment evolution of our experience, the moment-by-moment evolution of the world of phenomenon, the moment-by moment evolution of our thoughts and our emotions.
And further on:
When we realize that the path is the goal, there's a sense of workability.
When we realise that the path and the goal are one and the same, the "journey" is reduced to this moment only. Suddenly there's a lot less we have to deal with.
When we become aware that moment-by-moment we are the considerers of possibilities, the explorers, the choice-makers, we feel energized to take charge of our lives, now.
There is no long and winding path stretching before us.
There is just this step.
We are the cutting edge.


Vicki Benckert (b. 1960) voiced Shadow (Skuggan) in Nalle har ett Stort Blått Hus, as well as translating the original Bear in the Big Blue House episodes into Swedish. She also did the Swedish translations for Lek med mig i Sesam (Play with Me Sesame) and Räkna med Greven (Count TV).
Benckert got her breakthrough as a singer and composer when she had three songs in the Swedish qualification for the Eurovision Song Contest in 1984. She performed one of the songs herself and received fifth place. In 1986, she won the Swedish round and was placed 5th in the Eurovision Song Contest. She writes music for many Swedish artists and appears as a backup singer as well. She took part in Roxette's Joyride world tour as a backup singer.
On stage, she is mainly seen in musicals, including Beauty and the Beast and Chess.
Her voice work include Over the Hedge as Penny, Bob the Builder, Batman, The Animated Series, The Rugrats Movie and many others.


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10 Alan Moore Comics You Must Read! (Besides Watchmen)
Filed to: Alan MooreFiled to: Alan Moore
Captain Britain
The Ballad Of Halo Jones
With next week's movie coming out, everybody's rediscovering the awesomeness of Watchmen. But there are tons of other mind-expanding Alan Moore comics that you should also check out. Here are our favorites.
The Ballad Of Halo Jones. Moore was writing for 2000 A.D., Britain's long-running science fiction adventure comic best known for its Judge Dredd feature. Moore saw that most of England's "IPC girl comics were heading for that last great midnight feast in the dorm," and that 2000 A.D. had a bigger female readership than anyone realized. So he pitched a comic about an ordinary young woman — not "another Tough Bitch With A Disintegrator And An Extra Y Chromosome" — having adventures in space in the far future. The result is one of the most unique space operas of all time, featuring crazy adventures and silly humor and lots and lots of bleakness. (Jones' friends tend to drop dead on her on a regular basis, and in the final volume, she gets involved in a bloody space war.) The whole thing is available as a single hardcover volume from Titan Books.
Captain Britain. Moore started writing for Marvel Comics in the U.K., and took Captain Britain on a tour through alternate universes. This may be the first time that the Marvel Universe's "normal" version of Earth is referred to as Earth 616 in comics, and it also features an evil Prime Minister of England, who wants to round up all the superheroes and put them into concentration camps. I read these comics when they were reprinted as X-Men Archives Featuring Captain Britain a few years ago, and was amazed at how fresh and weird they still seem. They'll probably never be reprinted again, but those reissues can be tracked down, and there's also a collected edition. Also notable: Moore's work on Marvelman, aka Miracleman, which is even harder to find these days (and which I've never actually read!)
Captain Britain TPB
2 bought by readersGMG may get a commission
The Saga Of Swamp Thing. Swamp Thing was a low-selling horror comic when Moore took it over, and he transformed it into a supernatural/weird science epic that still helps to define all of DC's "mature" horror/supernatural comics, including all the Vertigo Comics line. My favorite is still the first volume, in which Swamp Thing discovers that he's not Alec Holland turned into a plant, after all — he's a plant that thinks its Alec Holland, thanks to a weird chemical accident. And he becomes the guardian of The Green, the spirit of all plant life on Earth, which is nearly usurped by the insane Jason Woodrue. All of a sudden the Swamp Thing has, not just pathos, but also a soul and real relationships, especially with the prematurely white-haired Abby.
V For Vendetta. This one, you've probably already read — but if you haven't, you should rush out and track down a copy. It's a dystopian future, and England has collapsed, giving rise to a new fascist regime run by a psychopath who's in love with his computer — literally. So it's up to the Guy Fawkes-masked anarchist vigilante known only as V to help topple the hateful oppressive regime, but his methods — especially his way of recruiting a successor — leave a lot to be desired. Just as much as Watchmen, V4V is a fantastic exploration of whether the ends justify the means, and the individual's relationship with a messed-up society.
Whatever Happened To The Man Of Tomorrow? Of all Moore's work on DC's main characters, this is my favorite. (Yes, more than "The Killing Joke" or "For The Man Who Has Everything.") DC was winding up its Superman stories, in preparation for John Byrne's classic reboot. So Moore had the opportunity to write the final Superman story, in which he shows how Superman's foes become darker and more horrifying, until finally Superman has to resort to the ultimate sanction. Superman disappears soon afterwards, and is presumed dead, but 10 years later, a reporter investigates. You'll have a hard time viewing other Superman stories the same way after reading this one. Luckily, it's collected in a single volume along with all of Moore's other DC Universe work — including the amazing Green Lantern short story about the aliens that don't have any concept of light or colors.
GMG may get a commission
1963. Moore (with regular collaborators like Steve Bissette, Dave Gibbons and Rick Veitch) put out a six-issue miniseries of pastiches of early 1960s Marvel comics, with titles like Tales From Beyond, Tomorrow Syndicate and Mystery Incorporated. They feature made-up superheroes like Horus, and even though they claim to be stand-alone issues of different comics, they have a continuing storyline of sorts. Plus hilarious fake ads and crazy letters to the editor. It's Moore at his most goofy and fun, and paying homage to superheroes instead of trying to recreate them or drag them into the "real world." (And it's more fun, for my money, than Moore's later Tom Strong's Terrific Tales and Tomorrow Stories anthologies.) There's no collected edition, as far as I know, but I used to see the individual issues in the dime bins at many comic book stores. Amazon now has them all for between $1.00 and $15.00 per issue.
From opengraph
From Hell. Moore and artist Eddie Campbell piece together all the clues about Jack The Ripper, in a huge, sprawling story of Victorian politics and Satanic rituals. The mystery isn't who killed those women — it's why, and as the graphic novel goes on, it peels back layer after layer of Victorian society to reveal more and more twisted reasons for the violence.
The League Of Extraordinary Gentlemen. If you've only seen the horrendous movie, you owe it to yourself to track down the comic. A set of famous literary figures, including Edward Hyde, Mina Harker, Allan Quartermain, the Invisible Man and Captain Nemo, team up to save the British Empire from a series of otherworldly threats. My absolute favorite is volume two, where our heroes face off against the Martian Tripods from War Of The Worlds... and this time it'll take a bit more than the common cold to put those alien scumbags out of action. As with 1963 and several other Moore works, the fake ads accompanying the comics are worth the price of admission all by themselves, and Moore also includes some amazing text pieces. It's a journey into retrofuturist Victoriana. And thank goodness there's a third full volume coming soon, after the slightly disappointing hardcover oneshot The Black Dossier.
Promethea. Okay, this is blasphemy, I know — but the first 12 issues of Promethea might actually be my favorite Moore work of all time. I'm not saying it's better than Watchmen, just that it holds a special place in my heart. With Promethea, we come full circle to Halo Jones — it's another tale of an ordinary girl in extraordinary circumstances, except this time around our heroine, Sophie Bangs, is inquisitive and curious. She pieces together the history of Promethea and figures out her own way of turning into the heroine, which requires an individual act of creativity. Promethea's not just your average superhero — she's an avatar of creativity and storytelling, and she may be destined to destroy the world instead of saving it. (In the end, it's actually a lot more complicated, confusing and — yes — rewarding than that binary implies.) The more Sophie discovers about magic and fables, the more powerful she gets and the closer Moore and artist J.H. Williams come to finally taking the comics medium apart altogether. (The first two volumes are my favorites, but the rest of the series is, at the very least, fascinating and memorable.) Oh, and did I mention it's an alternate 1999 with superheroes and weird cyberculture? And an android supervillain called the Painted Doll?
Top 10. Another comic which Moore created for his America's Best Comics imprint was Top 10, the story of a super-powered police squad in a world where pretty much everybody has weird powers. In contrast to Extraordinary Gentleman's literary exploration and Promethea's magical journey, Top 10 is mostly just hilarious wicked weird fun. At times, it really does read like a version of Hill Street Blues set in a world of flying people and superstrong blue men. My favorite character: the exoskeleton-wearing canine police sergeant, Caesar. Moore gave supercop Jeff Smax his own spin-off graphic novel, and later did an amazing prequel called Top 10: The 49ers. It's all pretty addictive stuff. Science fiction writer Paul Di Filippo later did a Top 10 miniseries, which captured the inventiveness of Moore's world pretty well but wasn't quite as magnetic.
Note: I know I'm leaving out his other big ABC series Tom Strong, which I like a lot, but not quite as much as these other series. Feel free to protest and throw sharp objects in comments.


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Laprade är en kommun i departementet Charente i regionen Nouvelle-Aquitaine i västra Frankrike. Kommunen ligger  i kantonen Aubeterre-sur-Dronne som ligger i arrondissementet Angoulême. År  hade Laprade  invånare.

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