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% \VignetteIndexEntry{Random Effects Design Document}

\begin{document}

\title{Design Document for Random Effects Aster Models}

\author{Charles J. Geyer}

\maketitle

\begin{abstract}
This design document works out details of approximate maximum likelihood
estimation for aster models with random effects.
Fixed and random effects are estimated by penalized log likelihood.
Variance components are estimated by integrating out the random effects
in the Laplace approximation of the complete data likelihood
(this can be done analytically) and maximizing the resulting approximate
missing data likelihood.
A further approximation treats the second derivative matrix of the cumulant
function of the exponential family where it appears in the approximate
missing data log likelihood as a constant (not a function of parameters).
Then first and second derivatives of the approximate missing data log
likelihood can be done analytically.  Minus the second derivative matrix
of the approximate missing data log likelihood is treated as approximate
Fisher information and used to estimate standard errors.
\end{abstract}

\section{Theory}

Aster models \citep*{gws,aster2} have attracted much recent attention.
Several researchers have raised the issue of incorporating random effects
in aster models, and we do so here.

\subsection{Complete Data Log Likelihood} \label{sec:complete}

Although we are particularly interested in aster models \citep{gws}, our
theory works for any exponential family model.  The log likelihood can
be written
$$
   l(\varphi) = y^T \varphi - c(\varphi),
$$
where $y$ is the canonical statistic vector,
$\varphi$ is the canonical parameter vector,
and the cumulant function $c$ satisfies
\begin{align}
   \mu(\varphi) & = E_{\varphi}(y) = c'(\varphi)
   \label{eq:moo}
   \\
   W(\varphi) & = \var_{\varphi}(y) = c''(\varphi)
   \label{eq:w}
\end{align}
where $c'(\varphi)$ denotes the vector of first partial derivatives
and $c''(\varphi)$ denotes the matrix of second partial derivatives.

We assume a canonical affine submodel with random effects determined by
\begin{equation} \label{eq:can-aff-sub}
   \varphi = a + M \alpha + Z b,
\end{equation}
where $a$ is a known vector, $M$ and $Z$ are known matrices, $b$
is a normal random vector with mean vector zero and
variance matrix $D$.
The vector $a$ is called the \emph{offset vector} and the matrices
$M$ and $Z$ are called the \emph{model matrices} for fixed and random
effects, respectively, in the terminology of the R function \texttt{glm}.
(The vector $a$ is called the \emph{origin} in the terminology of the
R function \texttt{aster}.  \emph{Design matrix} is alternative
terminology for \texttt{model matrix}.)
The matrix $D$ is assumed to be diagonal,
so the random effects are independent random variables.
The diagonal components of $D$ are called \emph{variance components}
in the classical terminology of random effects models \citep{searle}.
Typically the components of $b$ are divided into blocks having the same
variance \citep[Section~6.1]{searle}, so there are only
a few variance components but many random effects, but nothing in this
document uses this fact.

The unknown parameter vectors are $\alpha$ and $\nu$, where $\nu$ is
the vector of variance components.  Thus $D$ is a function of $\nu$,
although this is not indicated by the notation.

The ``complete data log likelihood'' (i.~e., what the log likelihood would
be if the random effect vector $b$ were observed) is
\begin{equation} \label{eq:foo}
   l_c(\alpha, b, \nu)
   =
   l(a + M \alpha + Z b)
   - \tfrac{1}{2} b^T D^{- 1} b
   - \tfrac{1}{2} \log \det(D)
\end{equation}
in case none of the variance components are zero.
We deal with the case of zero variance components in Sections~\ref{sec:zero},
\ref{sec:raw}, and~\ref{sec:roots} below.

\subsection{Missing Data Likelihood}

Ideally, inference about the parameters should be based on
the \emph{missing data likelihood}, which is the complete data likelihood with
random effects $b$ integrated out
\begin{equation} \label{eq:bar}
   L_m(\alpha, \nu) = \int e^{l_c(\alpha, b, \nu)} \, d b
\end{equation}
Maximum likelihood estimates (MLE) of $\alpha$ and $\nu$ are the values that
maximize \eqref{eq:bar}.  However MLE are hard to find.  The integral in
\eqref{eq:bar} cannot be done analytically, nor can it be done by numerical
integration except in very simple cases.
There does exist a large literature on doing such integrals
by ordinary or Markov chain Monte Carlo
\citep*{thompson,g+t,mcmcml,promislow,shaw-geyer-shaw,sung},
but these methods take a great
deal of computing time and are difficult for ordinary users to apply.
We wish to avoid that route if at all possible.

\subsection{A Digression on Minimization} \label{sec:minimize}

The theory of constrained optimization (Section~\ref{sec:raw} below)
has a bias in favor of minimization rather than maximization.
The explication below will be simpler if we switch now from maximization
to minimization (minimizing minus the log likelihood) rather than switch
later.

\subsection{Laplace Approximation} \label{sec:laplace}

\citet{b+c} proposed to replace the integrand in \eqref{eq:bar} by
its Laplace approximation, which is a normal probability density function
so the random effects can be integrated out analytically.
Let $b^*$ denote the result of maximizing \eqref{eq:foo} considered
as a function of $b$ for fixed $\alpha$ and $\nu$.
Then $- \log L_m(\alpha, \nu)$ is approximated by
$$
   q(\alpha, \nu)
   =
   \tfrac{1}{2} \log \det[\kappa''(b^*)] + \kappa(b^*)
$$
where
\begin{align*}
   \kappa(b)
   & =
   - l_c(a + M \alpha + Z b)
   \\
   \kappa'(b)
   & =
   - Z^T [ y + \mu(a + M \alpha + Z b) ] + D^{-1} b
   \\
   \kappa''(b)
   & =
   Z^T W(a + M \alpha + Z b) Z + D^{-1}
\end{align*}
Hence
\begin{equation} \label{eq:log-pickle}
\begin{split}
   q(\alpha, \nu)
   & =
   - l_c(\alpha, b^*, \nu)
   + \tfrac{1}{2} \log
   \det\bigl[ \kappa''(b^*) \bigr]
   \\
   & =
   - l(a + M \alpha + Z b^*)
   + \tfrac{1}{2} (b^*)^T D^{-1} b^*
   + \tfrac{1}{2} \log \det(D)
   \\
   & \qquad
   + \tfrac{1}{2} \log
   \det\bigl[ Z^T W(a + M \alpha + Z b^*) Z + D^{-1} \bigr]
   \\
   & =
   - l(a + M \alpha + Z b^*)
   + \tfrac{1}{2} (b^*)^T D^{-1} b^*
   \\
   & \qquad
   + \tfrac{1}{2} \log
   \det\bigl[ Z^T W(a + M \alpha + Z b^*) Z D + I \bigr]
\end{split}
\end{equation}
where $I$ denotes the identity matrix of the appropriate dimension
(which must be the same as the dimension of $D$ for the expression it
appears in to make sense),
where $b^*$ is a function of $\alpha$ and $\nu$
and $D$ is a function of $\nu$, although this is not indicated by the
notation, and where
the last equality uses the rule sum of logs is log of product and
the rule product of determinants is determinant of matrix product
\citep[Theorem~13.3.4]{harville}.

Since minus the log likelihood of an exponential family is a convex function
\citep[Theorem~9.1]{barndorff} and the middle term on the right-hand side
of \eqref{eq:foo} is a strictly convex function, it follows that
\eqref{eq:foo} considered as a function of $b$ for fixed $\alpha$ and $\nu$
is a strictly convex function.
Moreover, this function has bounded level sets, because the middle
term on the right-hand side of \eqref{eq:foo} does.
It follows that there is unique global minimizer
\citep[Theorems~1.9 and~2.6]{raw}.
Thus $b^*(\alpha, \nu)$ is well defined for all values
of $\alpha$ and $\nu$.

The key idea is to use \eqref{eq:log-pickle} as if it were the log likelihood
for the unknown parameters ($\alpha$ and $\nu$), although it is only an
approximation.  However, this is also problematic.  In doing likelihood
inference using \eqref{eq:log-pickle} we need first and second derivatives
of it (to calculate Fisher information), but $W$ is already
the second derivative matrix of the cumulant function, so
first derivatives of \eqref{eq:log-pickle} would involve third derivatives
of the cumulant function and
second derivatives of \eqref{eq:log-pickle} would involve fourth derivatives
of the cumulant function.  For aster models there are no published
formulas for derivatives higher than second of the aster model cumulant
function nor does
software (the R package \texttt{aster},
\citealp{package-aster}) provide such ---
the derivatives do, of course, exist because every cumulant function of
a regular exponential family is infinitely differentiable at every
point of the canonical parameter space \citep[Theorem~8.1]{barndorff} ---
they are just not readily available.
\citet{b+c} noted the same problem in the context of GLMM, and proceeded as
if $W$ were a constant function of its argument, so all derivatives of $W$
were zero.  This is not a bad approximation because ``in asymptopia'' the
aster model log likelihood is exactly quadratic and $W$ is a constant function,
this being a general property of likelihoods \citep{simple}.  Hence we adopt
this idea too, more because we are forced to by the difficulty of
differentiating $W$ than by our belief that we are ``in asymptopia.''

This leads to the following idea.  Rather than basing inference on
\eqref{eq:log-pickle}, we actually use
\begin{equation} \label{eq:log-pickle-too}
   q(\alpha, \nu)
   =
   - l(a + M \alpha + Z b^*)
   + \tfrac{1}{2} (b^*)^T D^{-1} b^*
   + \tfrac{1}{2} \log
   \det\bigl[ Z^T \widehat{W} Z D + I \bigr]
\end{equation}
where $\widehat{W}$ is a constant matrix (not a function of $\alpha$ and
$\nu$).  This makes sense for any
choice of $\widehat{W}$ that is symmetric and positive semidefinite,
but we will choose $\widehat{W}$ that are close to
$W(a + M \hat{\alpha} + Z \hat{b})$, where
$\hat{\alpha}$ and $\hat{\nu}$ are the joint minimizers
of \eqref{eq:log-pickle} and $\hat{b} = b^*(\hat{\alpha}, \hat{\nu})$.
Note that \eqref{eq:log-pickle-too} is a redefinition of $q(\alpha, \nu)$.
Hereafter we will no longer use the definition \eqref{eq:log-pickle}.

\subsection{A Key Concept}

Introduce
\begin{equation} \label{eq:key}
   p(\alpha, b, \nu)
   =
   - l(a + M \alpha + Z b)
   + \tfrac{1}{2} b^T D^{-1} b
   + \tfrac{1}{2} \log
   \det\bigl[ Z^T \widehat{W} Z D + I \bigr]
\end{equation}
where, as the left-hand side says, $\alpha$, $b$, and $\nu$ are all
free variables and, as usual, $D$ is a function of $\nu$, although
the notation does not indicate this.
Since the terms that contain $b$ are the same in both \eqref{eq:foo}
and \eqref{eq:key}, $b^*$ can also be defined at the result of
minimizing \eqref{eq:key} considered
as a function of $b$ for fixed $\alpha$ and $\nu$.
Thus \eqref{eq:log-pickle-too} is a profile of \eqref{eq:key}
and $(\hat{\alpha}, \hat{b}, \hat{\nu})$ is the joint minimizer
of \eqref{eq:key}.

Since $p(\alpha, b, \nu)$ is a much simpler function than $q(\alpha, \nu)$,
the latter having no closed form expression and requiring an optimization
as part of each evaluation, it is much simpler to find
$(\hat{\alpha}, \hat{b}, \hat{\nu})$
by minimizing the former rather than the latter.

\subsection{A Digression on Partial Derivatives}

Let $f(\alpha, b, \nu)$ be a scalar-valued function of three vector
variables.  We write partial derivative vectors using subscripts:
$f_\alpha(\alpha, b, \nu)$ denotes the vector of partial derivatives
with respect to components of $\alpha$.  Our convention is that we take
this to be a column vector.  Similarly for $f_b(\alpha, b, \nu)$.
We also use this convention for partial derivatives with respect to
single variables: $f_{\nu_k}(\alpha, b, \nu)$, which are, of course,
scalars.  We use this convention for any scalar-valued function
of any number of vector variables.

We continue this convention for second partial derivatives:
$f_{\alpha b}(\alpha, b, \nu)$ denotes the matrix of partial derivatives
having $i, j$ component that is the (mixed) second partial derivative of
$f$ with respect to $\alpha_i$ and $b_j$.  Thus the row dimension of
$f_{\alpha b}(\alpha, b, \nu)$  is the dimension of $\alpha$,
the column dimension is the dimension of $b$, and
$f_{b \alpha}(\alpha, b, \nu)$  is the transpose of
$f_{\alpha b}(\alpha, b, \nu)$.

This convention allows easy indication of points at which partial derivatives
are evaluated.  For example, $f_{\alpha b}(\alpha, b^*, \nu)$ indicates
that $b^*$ is plugged in for $b$ in the expression for
$f_{\alpha b}(\alpha, b, \nu)$.

We also use this convention of subscripts denoting partial derivatives
with vector-valued functions.
If $f(\alpha, b, \nu)$ is a column-vector-valued function of vector
variables, then $f_\alpha(\alpha, b, \nu)$ denotes the matrix of
partial derivatives having $i, j$ component that is the partial derivative
of the $i$-th component of $f_\alpha(\alpha, b, \nu)$ with respect to
$\alpha_j$.
Thus the row dimension of
$f_{\alpha}(\alpha, b, \nu)$  is the dimension of
$f(\alpha, b, \nu)$ and
the column dimension is the dimension of $\alpha$.

\subsection{First Derivatives}

Start with \eqref{eq:key}.  Its derivatives are
\begin{equation} \label{eq:p-alpha}
   p_\alpha(\alpha, b, \nu)
   =
   - M^T \bigl[ y - \mu(a + M \alpha + Z b) \bigr]
\end{equation}
\begin{equation} \label{eq:p-c}
   p_b(\alpha, b, \nu)
   =
   - Z^T \bigl[ y - \mu(a + M \alpha + Z b) \bigr] + D^{- 1} b
\end{equation}
and
\begin{equation} \label{eq:p-theta}
   p_{\nu_k}(\alpha, b, \nu)
   =
   - \tfrac{1}{2} b^T D^{- 1} E_k D^{- 1} b
   + \tfrac{1}{2} \tr \Bigl(
   \bigl[ Z^T \widehat{W} Z D + I \bigr]^{- 1}
   Z^T \widehat{W} Z E_k
   \Bigr)
\end{equation}
where
\begin{equation} \label{eq:eek}
   E_k = A_{\nu_k}(\nu)
\end{equation}
is the diagonal matrix whose components are equal to one
if the corresponding components of $D$ are equal to $\nu_k$ by
definition (rather than by accident when some other component of $\nu$
also has the same value) and whose components are otherwise zero.
The formula for the derivative of a matrix inverse
comes from \citet[Chapter~15, Equation~8.15]{harville}.
The formula for the derivative of the log of a determinant
comes from \citet[Chapter~15, Equation~8.6]{harville}.

The estimating equation for $b^*$ can be written
\begin{equation} \label{eq:estimating-c-star}
   p_b\bigl(\alpha, b^*, \nu\bigr) = 0
\end{equation}
and by the multivariate chain rule \citep[Theorem~8.15]{browder}
we have
\begin{equation} \label{eq:q-alpha}
\begin{split}
   q_\alpha(\alpha, \nu)
   & =
   p_\alpha(\alpha, b^*, \nu)
   +
   b^*_\alpha(\alpha, \nu)^T p_b(\alpha, b^*, \nu)
   \\
   & =
   p_\alpha(\alpha, b^*, \nu)
\end{split}
\end{equation}
by \eqref{eq:estimating-c-star}, and
\begin{equation} \label{eq:q-theta}
\begin{split}
   q_{\nu_k}(\alpha, \nu)
   & =
   b^*_{\nu_k}(\alpha, \nu)^T
   p_b(\alpha, b^*, \nu)
   +
   p_{\nu_k}(\alpha, b^*, \nu)
   \\
   & =
   p_{\nu_k}(\alpha, b^*, \nu)
\end{split}
\end{equation}
again by \eqref{eq:estimating-c-star}.

\subsection{Second Derivatives} \label{sec:second}

We will proceed in the opposite direction from the preceding section,
calculating abstract derivatives before particular formulas for random
effects aster models, because we need to see what work needs to be done
before doing it (we may not need all second derivatives).

By the multivariate chain rule \citep[Theorem~8.15]{browder}
\begin{align*}
   q_{\alpha \alpha}(\alpha, \nu)
   & =
   p_{\alpha \alpha}(\alpha, b^*, \nu)
   +
   p_{\alpha b}(\alpha, b^*, \nu) b^*_\alpha(\alpha, \nu)
   \\
   q_{\alpha \nu}(\alpha, \nu)
   & =
   p_{\alpha \nu}(\alpha, b^*, \nu)
   +
   p_{\alpha b}(\alpha, b^*, \nu) b^*_{\nu}(\alpha, \nu)
   \\
   q_{\nu \nu}(\alpha, \nu)
   & =
   p_{\nu \nu}(\alpha, b^*, \nu)
   +
   p_{\nu b}(\alpha, b^*, \nu) b^*_{\nu}(\alpha, \nu)
\end{align*}
The estimating equation \eqref{eq:estimating-c-star}
defines $b^*$ implicitly.  Thus derivatives of $b^*$ are computed using
the implicit function theorem \citep[Theorem~8.29]{browder}
\begin{align}
   b^*_\alpha(\alpha, \nu)
   & =
   -
   p_{b b}(\alpha, b^*, \nu)^{-1} p_{b \alpha}(\alpha, b^*, \nu)
   \label{eq:c-star-alpha-rewrite}
   \\
   b^*_{\nu}(\alpha, \nu)
   & =
   -
   p_{b b}(\alpha, b^*, \nu)^{-1} p_{b \nu}(\alpha, b^*, \nu)
   \label{eq:c-star-theta-rewrite}
\end{align}
This theorem requires that $p_{b b}(\alpha, b^*, \nu)$ be invertible,
and we shall see below that it is.
Then the second derivatives above can be rewritten
\begin{align*}
   q_{\alpha \alpha}(\alpha, \nu)
   & =
   p_{\alpha \alpha}(\alpha, b^*, \nu)
   -
   p_{\alpha b}(\alpha, b^*, \nu)
   p_{b b}(\alpha, b^*, \nu)^{-1} p_{b \alpha}(\alpha, b^*, \nu)
   \\
   q_{\alpha \nu}(\alpha, \nu)
   & =
   p_{\alpha \nu}(\alpha, b^*, \nu)
   -
   p_{\alpha b}(\alpha, b^*, \nu)
   p_{b b}(\alpha, b^*, \nu)^{-1} p_{b \nu}(\alpha, b^*, \nu)
   \\
   q_{\nu \nu}(\alpha, \nu)
   & =
   p_{\nu \nu}(\alpha, b^*, \nu)
   -
   p_{\nu b}(\alpha, b^*, \nu)
   p_{b b}(\alpha, b^*, \nu)^{-1} p_{b \nu}(\alpha, b^*, \nu)
\end{align*}
a particularly simple and symmetric form.  If we combine all the parameters
in one vector $\psi = (\alpha, \nu)$ and write $p(\psi, b)$ instead
of $p(\alpha, b, \nu)$ we have
\begin{equation} \label{eq:psi-psi}
   q_{\psi \psi}(\psi)
   =
   p_{\psi \psi}(\psi, b^*)
   -
   p_{\psi b}\bigl(\psi, b^*\bigr)
   p_{b b}\bigl(\psi, b^*\bigr)^{- 1}
   p_{b \psi}\bigl(\psi, b^*\bigr)
\end{equation}
This form is familiar from the conditional variance formula
for normal distributions if
\begin{equation} \label{eq:fat}
   \begin{pmatrix} \Sigma_{1 1} & \Sigma_{1 2}
   \\ \Sigma_{2 1} & \Sigma_{2 2} \end{pmatrix}
\end{equation}
is the partitioned variance matrix of a partitioned normal random vector
with components $X_1$ and $X_2$, then the variance matrix of the conditional
distribution of $X_1$ given $X_2$ is
\begin{equation} \label{eq:thin}
   \Sigma_{1 1} - \Sigma_{1 2} \Sigma_{2 2}^{- 1} \Sigma_{2 1}
\end{equation}
assuming that $X_2$ is nondegenerate \citep[Theorem~2.5.1]{anderson}.
Moreover, if the conditional distribution is degenerate, that is, if there
exists a nonrandom vector $v$ such that $\var(v^T X_1 \mid X_2) = 0$, then
$$
   v^T X_1 = v^T \Sigma_{1 2} \Sigma_{2 2}^{- 1} X_2
$$
with probability one, assuming $X_1$ and $X_2$ have mean zero
\citep[also by][Theorem~2.5.1]{anderson}, and the joint distribution
of $X_1$ and $X_2$ is also degenerate.  Thus we conclude that if
the (joint) Hessian matrix of $p$ is nonsingular, then so is the (joint)
Hessian matrix of $q$ given by \eqref{eq:psi-psi}.

The remaining work for this section is deriving the second derivatives
of $p$ that we need (it has turned out that we need all of them)
\begin{align*}
   p_{\alpha \alpha}(\alpha, b, \nu)
   & =
   M^T W(a + M \alpha + Z b) M
   \\
   p_{\alpha b}(\alpha, b, \nu)
   & =
   M^T W(a + M \alpha + Z b) Z
   \\
   p_{b b}(\alpha, b, \nu)
   & =
   Z^T W(a + M \alpha + Z b) Z + D^{- 1}
   \\
   p_{\alpha \nu_k}(\alpha, b, \nu)
   & =
   0
   \\
   p_{b \nu_k}(\alpha, b, \nu)
   & =
   - D^{- 1} E_k D^{- 1} b
   \\
   p_{\nu_j \nu_k}(\alpha, b, \nu)
   & =
   b^T D^{- 1} E_j D^{- 1} E_k D^{- 1} b
   \\
   & \qquad
   -
   \tfrac{1}{2} \tr \Bigl(
   \bigl[ Z^T \widehat{W} Z D + I \bigr]^{- 1}
   Z^T \widehat{W} Z E_j
   \\
   & \qquad \qquad
   \bigl[ Z^T \widehat{W} Z D + I \bigr]^{- 1}
   Z^T \widehat{W} Z E_k
   \Bigr)
\end{align*}
This finishes the derivation of all the derivatives we need.
Recall that in our use of the implicit function theorem we needed
$p_{b b}(\alpha, b^*, \nu)$ to be invertible.  From the explicit form
given above we see that it is actually negative definite, because
$W(a + M \alpha + Z b)$ is positive semidefinite by \eqref{eq:w}.

\subsection{Zero Variance Components} \label{sec:zero}

When some variance components are zero, the corresponding diagonal components
of $D$ are zero, and the corresponding components of $b$ are zero almost surely.
The order of the components of $b$ does not matter, so long as the rows of $Z$
and the rows and columns of $D$ are reordered in the same way.  So suppose
these objects are partitioned as
$$
   b = \begin{pmatrix} b_1 \\ b_2 \end{pmatrix}
   \qquad
   Z = \begin{pmatrix} Z_1 \\ Z_2 \end{pmatrix}
   \qquad
   D = \begin{pmatrix} D_1 & 0 \\ 0 & D_2 \end{pmatrix}
$$
where $D_2 = 0$ and the diagonal components of $D_1$ are all strictly positive,
so the components of $b_2$ are all zero almost surely and the components of
$b_1$ are all nonzero almost surely.  Since $Z b = Z_1 b_1$ almost
surely, the value of $Z_2$ is irrelevant.
In the expression for $D$ we are using the convention that $0$ denotes
the zero matrix of the dimension needed for the expression it appears in to
make sense, so the two appearances of 0 in the expression for $D$ as
a partitioned matrix denote different submatrices having all components zero
(they are transposes of each other).

Then the correct expression for the complete data log likelihood is
\begin{equation} \label{eq:foo-baz}
   l_c(\alpha, b, \nu)
   =
   l(a + M \alpha + Z_1 b_1)
   - \tfrac{1}{2} b_1^T D_1^{- 1} b_1
   - \tfrac{1}{2} \log \det(D_1)
\end{equation}
that is, the same as \eqref{eq:foo} except with subscripts 1
on $b$, $Z$, and $D$.  And this leads to the correct expression for
the approximate log likelihood
\begin{equation} \label{eq:log-pickle-too-baz}
\begin{split}
   q(\alpha, \nu)
   & =
   - l(a + M \alpha + Z_1 b_1^*)
   + \tfrac{1}{2} (b_1^*)^T D_1^{-1} b_1^*
   \\
   & \qquad
   + \tfrac{1}{2} \log
   \det\bigl[ Z_1^T \widehat{W} Z_1 D_1 + I \bigr]
\end{split}
\end{equation}
where again $I$ denotes the identity matrix of the appropriate dimension
(which now must be the dimension of $D_1$ for the expression it appears
in to make sense) and where $b_1^*$ denotes the maximizer of
\eqref{eq:foo-baz} considered as a function of $b_1$ with $\alpha$ and $\nu$
fixed, so it is actually a function of $\alpha$ and $\nu$ although the
notation does not indicate this.
Since
\begin{align*}
   Z^T \widehat{W} Z D + I
   & =
   \begin{pmatrix}
   Z_1^T \widehat{W} Z_1 D_1 + I
   &
   Z_1^T \widehat{W} Z_2 D_2
   \\
   Z_2^T \widehat{W} Z_1 D_1
   &
   Z_2^T \widehat{W} Z_2 D_2 + I
   \end{pmatrix}
   \\
   & =
   \begin{pmatrix}
   Z_1^T \widehat{W} Z_1 D_1 + I
   &
   0
   \\
   Z_2^T \widehat{W} Z_1 D_1
   &
   I
   \end{pmatrix}
\end{align*}
where again we are using the convention that $I$ denotes the identity matrix
of the appropriate dimension and 0 denotes the zero matrix of the appropriate
dimension, so $I$ denotes different identity matrices in
different parts of this equation, having the dimension of $D$ on the left-hand
side, the dimension of $D_1$ in the first column of both partitioned matrices,
and the dimension of $D_2$ in the second column of both partitioned matrices,
\begin{align*}
   \det(Z^T \widehat{W} Z D + I)
   & =
   \det(Z_1^T \widehat{W} Z_1 D_1 + I) \det(I)
   \\
   & =
   \det(Z_1^T \widehat{W} Z_1 D_1 + I)
\end{align*}
by the rule that the determinant of a blockwise lower triangular
partitioned matrix
is the product of the determinants of the blocks on the diagonal
\citep[Theorem~13.3.1]{harville}.  And since $Z_1 b_1 = Z b$ almost surely,
\begin{equation} \label{eq:log-pickle-too-qux}
\begin{split}
   q(\alpha, \nu)
   & =
   - l(a + M \alpha + Z b^*)
   + \tfrac{1}{2} (b_1^*)^T D_1^{-1} b_1^*
   \\
   & \qquad
   + \tfrac{1}{2} \log
   \det\bigl[ Z^T \widehat{W} Z D + I \bigr]
\end{split}
\end{equation}
that is, the subscripts 1 are only needed in the term where the matrix inverse
appears and are necessary there because $D^{- 1}$ does not exist.
\citet[Section 2.3]{b+c} suggest using the Moore-Penrose pseudoinverse
\citep[Chapter~20]{harville}
$$
   D^+ = \begin{pmatrix} D_1^{- 1} & 0 \\ 0 & 0 \end{pmatrix}
$$
which gives
\begin{equation} \label{eq:log-pickle-too-too}
   q(\alpha, \nu)
   =
   - l(a + M \alpha + Z b^*)
   + \tfrac{1}{2} (b^*)^T D^+ b^*
   + \tfrac{1}{2} \log
   \det\bigl[ Z^T \widehat{W} Z D + I \bigr]
\end{equation}
for the approximate log likelihood.  This hides but does not eliminate
the partitioning.  Although there is no explicit partitioning in
\eqref{eq:log-pickle-too-too}, it is still there in the definition
of $b^*$.

Although this proposal \citep[Section 2.3]{b+c} does deal with the
situation where the zero variance components are somehow known, it
does not adequately deal with estimating which variance components are
zero.  That is the subject of the following two sections.

\subsection{The Theory of Constrained Optimization} \label{sec:raw}

\subsubsection{Incorporating Constraints in the Objective Function}

When zero variance components arise, optimization of \eqref{eq:key}
puts us in the
realm of constrained optimization.  The theory of constrained optimization
\citep{raw} has a notational bias towards minimization \citep[p.~5]{raw}.
One can, of course, straightforwardly translate every result in \citet{raw}
from the context of minimization to the context of maximization, because
for any objective function $f$, maximizing $f$ is the same as minimizing
$- f$, and \citeauthor{raw} give infrequent hints and discussions of
alternative terminology in aid of this.  But since the theory of
constrained optimization is strange to most statisticians, especially
the abstract theory that is needed here
(Karush-Kuhn-Tucker theory is not helpful
here, as we shall see), it is much simpler to switch from maximization to
minimization so we can use all of the theory in \citet{raw} without
modification.  And we have done so.

The theory of constrained optimization incorporates constraints in the
objective function by the simple device of defining the objective function
(for a minimization problem) to have the value $+ \infty$ off the constraint
set \citep[Section~1A]{raw}.
Since no point where the objective function has the value $+ \infty$
can minimize it, unless the the objective function has the value $+ \infty$
everywhere, which is not the case in any application, the unconstrained
minimizer of this sort of objective function is the same as the constrained
minimizer.

Thus we need to impose constraints on our key function
\eqref{eq:key}, requiring that
each component of $\nu$ be nonnegative and when any component of $\nu$ is
zero the corresponding components of $b$ are also zero.  However,
the formula \eqref{eq:key} does not make sense when components of $\nu$
are zero, so we will have to proceed differently.

\subsubsection{Lower Semicontinuous Regularization}

Since all but the middle term on the right-hand side of
\eqref{eq:key} are actually defined on some neighborhood of each point
of the constraint set and differentiable at each point
of the constraint set, we only need to deal with the middle term.
It is the sum of terms of the form $b_i^2 / \nu_k$, where $\nu_k$ is
the variance of $b_i$.  Thus we investigate functions of this form
\begin{equation} \label{eq:h}
   h(b, \nu)
   =
   b^2 / \nu
\end{equation}
where, temporarily, $b$ and $\nu$ are scalars rather than vectors
(representing single components of the vectors).  In case $\nu > 0$
we have derivatives
\begin{align*}
   h_b(b, \nu) & = 2 b / \nu
   \\
   h_\nu(b, \nu) & = - b^2 / \nu^2
   \\
   h_{b b}(b, \nu) & = 2 / \nu
   \\
   h_{b \nu}(b, \nu) & = - 2 b / \nu^2
   \\
   h_{\nu \nu}(b, \nu) & = 2 b^2 / \nu^3
\end{align*}
The Hessian matrix
$$
   h''(b, \nu)
   =
   \begin{pmatrix}
   2 / \nu & - 2 b / \nu^2
   \\
   - 2 b / \nu^2 & 2 b^2 / \nu^3
   \end{pmatrix}
$$
has nonnegative determinants of its principal submatrices, since the
diagonal components are positive and $\det\bigl(h''(b, \nu)\bigr)$ is zero.
Thus the Hessian matrix is nonnegative definite
\citep[Theorem~14.9.11]{harville}, which implies that $h$ itself is
convex \citep[Theorem~2.14]{raw} on the set where $\nu > 0$.

We then extend $h$ to the whole of the constraint set (this just adds
the origin to the points already considered) in two steps.
First we define it to have the value $+ \infty$ at all points not
yet considered (those where any component of $\nu$ is nonpositive).
This gives us an extended-real-valued convex function defined on
all of $\real^2$.
Second we take it to be the
lower semicontinuous (LSC) regularization \citep[p.~14]{raw} of the
function just defined.
The LSC regularization of a convex function is convex
\citep[Proposition~2.32]{raw}.  For any sequences $b_n \to b \neq 0$
and $\nu_n \searrow 0$ we have $h(b_n, \nu_n) \to \infty$.  Thus the LSC
regularization has the value $+ \infty$ for $\nu = 0$ but $b \neq 0$.
If $b_n = 0$ and $\nu_n \searrow 0$ we have $h(b_n, \nu_n) = 0$ for all
$n$.  Since $h(b, \nu) \ge 0$ for all $b$ and $\nu \ge 0$, we conclude
$$
   \liminf_{\substack{b \to 0\\\nu \searrow 0}} h(b, \nu) = 0
$$
Thus the LSC regularization has the value 0 for $b = \nu = 0$.
In summary
\begin{equation} \label{eq:h-lsc}
   h(b, \nu)
   =
   \begin{cases}
   b^2 / \nu, & \nu > 0
   \\
   0, & \nu = 0 \opand b = 0
   \\
   + \infty, & \text{otherwise}
   \end{cases}
\end{equation}
is an LSC convex function, which agrees with our original definition in
case $\nu > 0$.  Note that $h(b, 0)$ considered as a function of $b$
is minimized at $b = 0$ because that is the only point where this function
is finite.

Let $k$ denote the map from indices for $b$ to indices for $\nu$ that gives
corresponding components: $\nu_{k(i)}$ is the variance of $b_i$.
Let $\dim(b)$ denote the number of random effects.  Then our objective function
can be written
\begin{equation} \label{eq:key-lsc}
   p(\alpha, b, \nu)
   =
   - l(a + M \alpha + Z b)
   + \tfrac{1}{2} \sum_{i = 1}^{\dim(b)} h(b_i, \nu_{k(i)})
   + \tfrac{1}{2} \log
   \det\bigl[ Z^T \widehat{W} Z D + I \bigr]
\end{equation}
where $h$ is given by \eqref{eq:h-lsc}, provided all of the components of
$\nu$ are nonnegative.  The proviso is necessary because the third term
on the right-hand side is not defined for all values of $\nu$, only those
such that the argument of the determinant is a positive definite matrix.
Hence, we must separately define $p(\alpha, b, \nu) = + \infty$
whenever any component of $\nu$ is negative.

\subsubsection{Subderivatives}

In calculus we learn that the first derivative is zero at a local minimum
and, therefore, to check points where the first derivative is zero.
This is called Fermat's rule.  This
rule no longer works for nonsmooth functions, including those that incorporate
constraints, such as \eqref{eq:key-lsc}.  It does, of course, still work
at points in the interior of the constraint set where \eqref{eq:key-lsc}
is differentiable.  It does not work to check points on the boundary.
There we need what \citet[Theorem~10.1]{raw}
call \emph{Fermat's rule, generalized:} at a local minimum the
\emph{subderivative function} is nonnegative.

For any extended-real-valued function $f$ on $\real^d$,
the \emph{subderivative function}, denoted $d f(x)$
is also an extended-real-valued function on $\real^d$ defined by
$$
   d f(x)(\bar{w}) = \liminf_{\substack{\tau \searrow 0 \\ w \to \bar{w}}}
   \frac{f(x + \tau w) - f(x)}{\tau}
$$
\citep[Definition~8.1]{raw}.  The notation on the left-hand side is read
the subderivative of $f$ at the point $x$ in the direction $\bar{w}$.
Fortunately, we do not have to use this
definition to calculate subderivatives we want, because the calculus
of subderivatives allows us to use simpler formulas in special cases.
Firstly, there is the notion of subdifferential regularity
\citep[Definition~7.25]{raw}, which we can use without knowing the definition.
The sum of regular functions is regular and the subderivative of a sum
is the sum of the subderivatives \citep[Corollary~10.9]{raw}.
A smooth function is regular and the subderivative is given by
\begin{equation} \label{eq:subderivative-smooth}
   d f(x)(w) = w^T f'(x),
\end{equation}
where, as in Sections~\ref{sec:complete} and~\ref{sec:laplace} above,
$f'(x)$ denotes the gradient vector (the vector of partial derivatives)
of $f$ at the point $x$ \citep[Exercise~8.20]{raw}.  Every LSC convex function
is regular \citep[Example~7.27]{raw}.  Thus in computing subderivatives
of \eqref{eq:key-lsc} we may compute them term by term, and for the
first and last terms, they are given in terms of the partial derivatives
already computed by \eqref{eq:subderivative-smooth}.
For an LSC convex function $f$, we have the following characterization
of the subderivative \citep[Proposition~8.21]{raw}.  At any point $x$
where $f(x)$ is finite, the limit
$$
   g(w) = \lim_{\tau \searrow 0} \frac{f(x + \tau w) - f(x)}{\tau}
$$
exists and defines a sublinear function $g$, and then $d f(x)$
is the LSC regularization of $g$.
An extended-real-valued
function $g$ is \emph{sublinear} if $g(0) = 0$ and
$$
   g(a_1 x_1 + a_2 x_2) \le a_1 g(x_1) + a_2 g(x_2)
$$
for all vectors $x_1$ and $x_2$ and positive scalars $a_1$ and $a_2$
\citep[Definition~3.18]{raw}.
The subderivative function of every regular LSC function
is sublinear \citep[Theorem~7.26]{raw}.

So let us proceed to calculate the subderivative of \eqref{eq:h-lsc}.
In the interior of the constraint set, where this function is smooth,
we can use the partial derivatives already calculated
$$
   d h(b, \nu)(u, v) = \frac{2 b u}{\nu} - \frac{b^2 v}{\nu^2}
$$
where the notation on the left-hand side means the subderivative of $h$
at the point $(b, \nu)$ in the direction $(u, v)$.
On the boundary of the constraint set, which consists of the single point
$(0, 0)$, we take limits.  In case $v > 0$, we have
$$
   \lim_{\tau \searrow 0}
   \frac{h(\tau u, \tau v) - h(0, 0)}{\tau}
   =
   \lim_{\tau \searrow 0}
   \frac{\tau^2 u^2 / (\tau v)}{\tau}
   =
   \lim_{\tau \searrow 0}
   \frac{u^2}{v}
   =
   \frac{u^2}{v}
$$
In case $v \le 0$ and $u \neq 0$, we have
$$
   \lim_{\tau \searrow 0}
   \frac{h(\tau u, \tau v) - h(0, 0)}{\tau}
   =
   \lim_{\tau \searrow 0} (+ \infty)
   =
   + \infty
$$
In case $v = 0$ and $u = 0$, we have
$$
   \lim_{\tau \searrow 0}
   \frac{h(\tau u, \tau v) - h(0, 0)}{\tau}
   =
   0
$$
Thus if we define
$$
   g(u, v) = \begin{cases} u^2 / v, & v > 0 \\ 0, & u = v = 0 \\ + \infty,
   & \text{otherwise} \end{cases}
$$
The theorem says $d h(0, 0)$ is the LSC regularization of $g$.
But we recognize $g = h$, so $g$ is already LSC, and we have
$$
   d h(0, 0)(u, v) = h(u, v)
$$

\subsubsection{Applying the Generalization of Fermat's Rule}

The theory of constrained optimization tells us nothing we did not already
know (from Fermat's rule) about smooth functions.  The only way we can
have $d f(x)(w) = w^T f'(x) \ge 0$ for all vectors $w$ is if $f'(x) = 0$.
It is only at points where the function is nonsmooth, in the cases of
interest to us, points on the boundary of the constraint set, where
the theory of constrained optimization tells us things we did not know and
need to know.

Even on the boundary, the conclusions of the theory about components
of the state that are not on the boundary agree with what we already knew.
We have
$$
   d p(\alpha, b, \nu)(s, u, v)
   =
   s^T p_\alpha(\alpha, b, \nu)
   +
   \text{terms not containing $s$}
$$
and the only way this can be nonnegative for all $s$ is if
\begin{equation} \label{eq:descent-alpha}
   p_\alpha(\alpha, b, \nu) = 0
\end{equation}
in which case
$d p(\alpha, b, \nu)(s, u, v)$ is a constant function of $s$, or, what
is the same thing in other words, the terms of
$d p(\alpha, b, \nu)(s, u, v)$ that appear to involve $s$ are all zero
(and so do not actually involve $s$).

Similarly, $d p(\alpha, b, \nu)(s, u, v) \ge 0$ for all $u_i$ and $v_j$
such that $\nu_j > 0$ and $k(i) = j$ only if
\begin{equation} \label{eq:descent-other-smooth}
\begin{split}
   p_{\nu_j}(\alpha, b, \nu) & = 0, \qquad \text{$j$ such that $\nu_j > 0$}
   \\
   p_{b_i}(\alpha, b, \nu) & = 0, \qquad \text{$i$ such that $\nu_{k(i)} > 0$}
\end{split}
\end{equation}
in which case we conclude that
$d p(\alpha, b, \nu)(s, u, v)$ is a constant function of such $u_i$ and $v_j$.

Thus, assuming that we are at a point $(\alpha, b, \nu$)
where \eqref{eq:descent-alpha}
and \eqref{eq:descent-other-smooth} hold,
and we do assume this throughout the rest of this section,
$d p(\alpha, b, \nu)(s, u, v)$ actually involves only $v_j$ and $u_i$ such
that $\nu_j = 0$ and $k(i) = j$.  Define
\begin{equation} \label{eq:key-smooth}
   \bar{p}(\alpha, b, \nu)
   =
   - l(a + M \alpha + Z b)
   + \tfrac{1}{2} \log
   \det\bigl[ Z^T \widehat{W} Z D + I \bigr]
\end{equation}
(the part of \eqref{eq:key-lsc} consisting of the smooth terms).
Then
\begin{equation} \label{eq:descent-other-nonsmooth}
\begin{split}
   d p(\alpha, b, \nu)(s, u, v)
   & =
   \sum_{j \in J}
   \Biggl[
   v_j \bar{p}_{\nu_j}(\alpha, b, \nu)
   \\
   & \qquad
   +
   \sum_{i \in k^{- 1}(j)}
   \Biggl(
   u_i \bar{p}_{b_i}(\alpha, b, \nu)
   +
   h(u_i, v_j)
   \Biggr)
   \Biggr]
\end{split}
\end{equation}
where $J$ is the set of $j$ such that $\nu_j = 0$,
where $k^{- 1}(j)$ denotes the set of $i$ such that $k(i) = j$,
and where $h$ is defined by \eqref{eq:h-lsc}.
Fermat's rule generalized says we must consider all of the terms
of \eqref{eq:descent-other-nonsmooth} together.
We cannot consider partial derivatives, because the partial derivatives
do not exist.  To check that we are at a local minimum we need to show
that \eqref{eq:descent-other-nonsmooth} is nonnegative
for all vectors $u$ and $v$.
Conversely, to verify that we are not at a local minimum, we need to find
one pair of vectors $u$ and $v$ such that \eqref{eq:descent-other-nonsmooth}
is negative.  Such a pair $(u, v)$ we call a \emph{descent direction}.
Since Fermat's rule generalized is a necessary but not sufficient condition
(like the ordinary Fermat's rule), the check that we are at a local minimum
is not definitive, but the check that we are not is.  If a descent direction
is found, then moving in that direction away from
the current value of $(\alpha, b, \nu)$ will decrease the objective
function \eqref{eq:key-lsc}.

So how do we find a descent direction?  We want to minimize
\eqref{eq:descent-other-nonsmooth} considered as a function of $u$ and $v$
for fixed $\alpha$, $b$, and $\nu$.
On further consideration, we can consider the terms of
\eqref{eq:descent-other-nonsmooth} for each $j$ separately.
If the minimum of
\begin{equation} \label{eq:descent-other-nonsmooth-j}
   v_j \bar{p}_{\nu_j}(\alpha, b, \nu)
   +
   \sum_{i \in k^{- 1}(j)}
   \Biggl(
   u_i \bar{p}_{b_i}(\alpha, b, \nu)
   +
   h(u_i, v_j)
   \Biggr)
\end{equation}
over all vectors $u$ and $v$ is nonnegative, then the minimum is zero,
because \eqref{eq:descent-other-nonsmooth-j} has the value zero
when $u = 0$ and $v = 0$.  Thus we can ignore this $j$ in calculating
the descent direction.

On the other hand, if the minimum is negative,
then the minimum does not occur at $v = 0$ and the minimum is actually
$- \infty$ by the sublinearity of the subderivative,
one consequence of sublinearity being positive homogeneity
$$
   d f(x)(\tau w) = \tau d f(x)(w), \qquad \tau \ge 0
$$
which holds for
for any subderivative.  Thus (as our terminology hints) we are only trying
to find a descent \emph{direction}, the length of the vector $(u, v)$ does
not matter, only its direction.  Thus to get a finite minimum we can
do a constrained minimization of \eqref{eq:descent-other-nonsmooth-j},
constraining $(u, v)$ to lie in a ball.  This is found by the well-known
Karush-Kuhn-Tucker theory of constrained optimization to be the minimum
of the Lagrangian function
\begin{equation} \label{eq:laggard}
   L(u, v)
   =
   \lambda v_j^2
   +
   v_j \bar{p}_{\nu_j}(\alpha, b, \nu)
   +
   \sum_{i \in k^{- 1}(j)}
   \Biggl(
   \lambda
   u_i^2
   +
   u_i \bar{p}_{b_i}(\alpha, b, \nu)
   +
   \frac{u_i^2}{v_j}
   \Biggr)
\end{equation}
where $\lambda > 0$ is the Lagrange multiplier, which would have to be
adjusted if we were interested in constraining $(u, v)$ to lie in a particular
ball.  Since we do not care about the length of $(u, v)$ we can use any
$\lambda$.  We have replaced $h(u_i, v_i)$ by $u_i^2 / v_j$ because we
know that if we are finding an actual descent direction, then we will
have $v_j > 0$. Now
\begin{align*}
   L_{u_i}(u, v)
   & =
   2 \lambda u_i
   +
   \bar{p}_{b_i}(\alpha, b, \nu)
   +
   \frac{2 u_i}{v_j},
   \qquad
   i \in k^{- 1}(j)
   \\
   L_{v_j}(u, v)
   & =
   2 \lambda v_j
   +
   \bar{p}_{\nu_j}(\alpha, b, \nu)
   -
   \sum_{i \in k^{- 1}(j)}
   \frac{u_i^2}{v_j^2}
\end{align*}
The minimum occurs where these are zero.
Setting the first equal to zero and solving for $u_i$ gives
$$
   \hat{u}_i(v_j)
   =
   - \frac{\bar{p}_{b_i}(\alpha, b, \nu)}{2 (\lambda + 1 / v_j)}
$$
plugging this back into the second gives
\begin{align*}
   L_{v_j}\bigl(\hat{u}(v), v\bigr)
   & =
   2 \lambda v_j
   +
   \bar{p}_{\nu_j}(\alpha, b, \nu)
   -
   \frac{1}{4 (\lambda v_j + 1)^2}
   \sum_{i \in k^{- 1}(j)}
   \bar{p}_{b_i}(\alpha, b, \nu)^2
\end{align*}
and we seek zeros of this.  The right-hand is clearly an increasing
function of $v_j$ so it is negative somewhere only if it is negative when
$v_j = 0$ where it has the value
\begin{equation} \label{eq:descent-test}
   \bar{p}_{\nu_j}(\alpha, b, \nu)
   -
   \frac{1}{4}
   \sum_{i \in k^{- 1}(j)}
   \bar{p}_{b_i}(\alpha, b, \nu)^2
\end{equation}
So that gives us a test for a descent direction: we have a descent
direction if and only if \eqref{eq:descent-test} is negative.
Conversely, we appear to have $\hat{\nu}_j = 0$ if \eqref{eq:descent-test}
is nonnegative.

That finishes our treatment of the theory of constrained optimization.
We have to ask is all of this complication really necessary?
It turns out that it is and it isn't.  We can partially avoid it
by a change of variables.  But the cure is worse than the disease
in some ways.  This is presented in the following section.

\subsection{Square Roots} \label{sec:roots}

We can avoid constrained optimization by the following change of parameter.
Introduce new parameter variables by
\begin{align*}
   \nu_j & = \sigma_j^2
   \\
   b & = A c
\end{align*}
where $A$ is diagonal and $A^2 = D$, so the $i$-th diagonal component of
$A$ is $\sigma_{k(i)}$.
Then the objective function \eqref{eq:key} becomes
\begin{equation} \label{eq:key-rooted}
   \tilde{p}(\alpha, c, \sigma)
   =
   - l(a + M \alpha + Z A c)
   + \tfrac{1}{2} c^T c
   + \tfrac{1}{2} \log
   \det\bigl[ Z^T \widehat{W} Z A^2 + I \bigr]
\end{equation}
There are now no constraints and \eqref{eq:key-rooted} is a continuous
function of all variables.

The drawback is that by symmetry we must have
$\tilde{p}_{\sigma_j}(\alpha, c, \sigma)$ equal to zero when $\sigma_j = 0$.
Thus first derivatives become useless for checking for descent directions,
and second derivative information is necessary.
However, that is not the way unconstrained optimizers like the R functions
\texttt{optim} and \texttt{nlminb} work.  They do not expect such pathological
behavior and do not deal with it correctly.  If we want to use such optimizers
to find local minima of \eqref{eq:key-rooted}, then we must provide starting
points that have no component of $\nu$ equal to zero,
and hope that the optimizer will never get any component of $\nu$ close to
zero unless zero actually is a solution.  But this is only a hope.
The theory that guided the design of these optimizers does not provide
any guarantees for this kind of objective function.

Moreover, optimizer algorithms stop when close to but not exactly
at a solution, a consequence of inexactness of computer arithmetic.
Thus when the optimizer stops and declares convergence with one or more
components of $\nu$ close to zero, how do we know whether the true solution
is exactly zero or not?  We don't unless we return to the original
parameterization and apply the theory of the preceding section.
The question of whether the MLE of the variance components are exactly
zero or not is of scientific interest, so it seems that the device of this
section does not entirely avoid the theory of constrained optimization.
We must change back to the original parameters and use \eqref{eq:descent-test}
to determine whether or not we have $\nu_j = 0$.

Finally, there is another issue with this ``square root'' parameterization.
The analogs of the second derivative formulas derived
in Section~\ref{sec:second} above, for this new parameterization are
extraordinarily ill-behaved.  The Hessian matrices are badly conditioned
and sometimes turn out to be not positive definite when calculated by
the computer's arithmetic (which is inexact) even though theory says
they must be positive definite.  We know this because at one point we
thought that this ``square root'' parameterization was the answer to
everything and tried to use it everywhere.  Months of frustration ensued
where it mostly worked, but failed on a few problems.  It took us a long
time to see that it is fundamentally wrong-headed.  As we said above,
the cure is worse than the disease.

Thus we concluded that, while we may use this ``square root'' parameterization
to do unconstrained rather than constrained minimization,
we should only use it only for that.
The test \eqref{eq:descent-test} should be used to determine whether
variance components are exactly zero or not, and the formulas in
Section~\ref{sec:second} should be used to derive Fisher information.

\subsubsection{First Derivatives}

Some of R's optimization routines can use first derivative information,
thus we derive first derivatives in this parameterization.
\begin{align}
   \tilde{p}_\alpha(\alpha, c, \sigma)
   & =
   - M^T [ y - \mu(a + M \alpha + Z A c) ]
   \label{eq:key-rooted-alpha}
   \\
   \tilde{p}_c(\alpha, c, \sigma)
   & =
   - A Z^T [ y - \mu(a + M \alpha + Z A c) ]
   + c
   \label{eq:key-rooted-c}
   \\
   \tilde{p}_{\sigma_j}(\alpha, c, \sigma)
   & =
   - c^T E_j Z^T [ y - \mu(a + M \alpha + Z A c) ]
   \nonumber
   \\
   & \qquad
   +
   \tr\left( [ Z^T \widehat{W} Z A^2 + I \bigr]^{- 1}
   Z^T \widehat{W} Z A E_j
   \right)
   \label{eq:key-rooted-sigma}
\end{align}
where $E_j$ is given by \eqref{eq:eek}.

\subsection{Fisher Information} \label{sec:fisher}

The observed Fisher information matrix is minus the second derivative matrix
of the log likelihood.  As we said above, we want to do this in the original
parameterization.

Assembling stuff derived in preceding sections and introducing
\begin{align*}
   \mu^*
   & =
   \mu\bigl(a + M \alpha + Z b^*(\alpha, \nu) \bigr)
   \\
   W^*
   & =
   W\bigl(a + M \alpha + Z b^*(\alpha, \nu) \bigr)
   \\
   H^* & = Z^T W^* Z + D^{- 1}
   \\
   \widehat{H} & = Z^T \widehat{W} Z D + I
\end{align*}
we obtain
\begin{align*}
   q_{\alpha \alpha}(\alpha, \nu)
   & =
   M^T W^* M - M^T W^* Z (H^*)^{- 1} Z^T W^* M
   \\
   q_{\alpha \nu_j}(\alpha, \nu)
   & =
   M^T W^* Z (H^*)^{- 1} D^{- 1} E_j D^{-1} b^*
   \\
   q_{\nu_j \nu_k}(\alpha, \nu)
   & =
   (b^*)^T D^{- 1} E_j D^{- 1} E_k D^{- 1} b^*
   \\
   & \qquad
   -
   \tfrac{1}{2} \tr \Bigl( \widehat{H}^{- 1} Z^T \widehat{W} Z E_j
   \widehat{H}^{- 1} Z^T \widehat{W} Z E_k \Bigr)
   \\
   & \qquad
   -
   (b^*)^T D^{- 1} E_j D^{- 1}
   (H^*)^{- 1}
   D^{- 1} E_k D^{- 1} b^*
\end{align*}
In all of these $b^*$, $\mu^*$, $W^*$, and $H^*$ are functions of $\alpha$
and $\nu$ even though the notation does not indicate this.

It is tempting to think expected Fisher information simplifies things
because we ``know'' $E(y) = \mu$ and $\var(y) = W$, except we don't know
that!  What we do know is
$$
   E(y \mid b) = \mu(a + M \alpha + Z b)
$$
but we don't know how to take the expectation of the right hand side
(and similarly for the variance).  Rather than introduce further approximations
of dubious validity, it seems best to just use (approximate)
observed Fisher information.

\subsection{Standard Errors for Random Effects}

Suppose that the approximate Fisher information derived
in Section~\ref{sec:fisher} can be used to give an approximate
asymptotic variance for the parameter vector $\psi = (\alpha, \nu)$.
This estimate of the asymptotic variance is $q_{\psi \psi}(\hat{\psi})^{- 1}$,
where $q_{\psi \psi}(\psi)$ is given by \eqref{eq:psi-psi} and
$\hat{\psi} = (\hat{\alpha}, \hat{\nu})$.

To apply the delta method to get asymptotic standard errors for
$\hat{b}$ we need the derivatives
\eqref{eq:c-star-alpha-rewrite} and \eqref{eq:c-star-theta-rewrite}.
Stacking these we obtain
$$
   b^*_\psi(\hat{\psi})
   =
   \begin{pmatrix}
   -
   p_{b b}(\hat{\alpha}, \hat{b}, \hat{\nu})^{-1}
   p_{b \alpha}(\hat{\alpha}, \hat{b}, \hat{\nu})
   \\
   -
   p_{b b}(\hat{\alpha}, \hat{b}, \hat{\nu})^{-1}
   p_{b \nu}(\hat{\alpha}, \hat{b}, \hat{\nu})
   \end{pmatrix}
$$
and the delta method gives
\begin{equation} \label{eq:b-ass-var}
   b^*_\psi(\hat{\psi})^T q_{\psi, \psi}(\hat{\psi})^{- 1}
   b^*_\psi(\hat{\psi})
\end{equation}
for the asymptotic variance of the estimator $\hat{b}$.

It must be conceded that in this section
we are living what true believers in random effects
models would consider a state of sin.  The random effects
vector $b$ is not a parameter, yet $b^*(\hat{\psi})$ treats it
as a function of parameters (which is thus a parameter) and the
``asymptotic variance'' \eqref{eq:b-ass-var} is derived by considering
$\hat{b}$ just such a parameter estimate.  So \eqref{eq:b-ass-var} is
correct in what it does, so long as we buy the assumption that
$q_{\psi \psi}(\hat{\psi})$ is approximate Fisher information
for $\psi$, but it fails to treat random effects as actually random.
Since any attempt to actually treat random effects as random would lead
us to integrals that we cannot do, we leave the subject at this point.
The asymptotic variance \eqref{eq:b-ass-var} may be philosophically
incorrect in some circles, but it seems to be the best we can do.

\subsection{REML?}

\citet{b+c} do not maximize the approximate log likelihood
\eqref{eq:log-pickle}, but make further
approximations to give estimators motivated by REML (restricted maximum
likelihood) estimators for linear mixed models (LMM).
\citet{b+c} concede that
the argument that justifies REML estimators for LMM does not carry over to
their REML-like estimators for generalized linear mixed models (GLMM).
Hence these REML-like estimators have no mathematical justification.
Even in LMM the widely used procedure of following REML estimates of the
variance components with so-called BLUE estimates of fixed effects and BLUP
estimates of random effects, which are actually only BLUE and BLUP if the
variance components are assumed known rather than estimated, is obviously
wrong: ignoring the fact that the variance components are estimated cannot be
justified (and \citeauthor{b+c} say this in their discussion section).
Hence REML is not justified even in LMM when fixed effects
are the parameters of interest.  In aster models, because components of the
response vector are dependent and have distributions in different families,
it is very unclear what REML-like estimators in the style of \citet{b+c}
might be.  The analogy just breaks down.
Hence, we do not pursue this REML analogy and stick with what we have
described above.

\section{Practice} \label{sec:reaster}

Our goal is to minimize \eqref{eq:log-pickle}.
We replace \eqref{eq:log-pickle} with \eqref{eq:log-pickle-too}
in some steps because of our inability to differentiate \eqref{eq:log-pickle},
but our whole procedure must minimize \eqref{eq:log-pickle}.

\subsection{Step 1}

To get close to $(\hat{\alpha}, \hat{c}, \hat{\sigma})$ starting from far away
we minimize
\begin{equation} \label{eq:r}
\begin{split}
   r(\sigma)
   & =
   - l(a + M \tilde{\alpha} + Z A \tilde{c})
   + \tfrac{1}{2} \tilde{c}^T \tilde{c}
   \\
   & \qquad
   + \tfrac{1}{2} \log
   \det\bigl[ Z^T W(a + M \tilde{\alpha} + Z A \tilde{c}) Z A^2 + I \bigr]
\end{split}
\end{equation}
where $\tilde{\alpha}$ and $\tilde{c}$ are the joint minimizers of
\eqref{eq:key-rooted} considered as a function of $\alpha$ and $c$ for
fixed $\sigma$.  In \eqref{eq:r}, $\tilde{\alpha}$, $\tilde{c}$, and $A$
are all functions of $\sigma$ although the notation does not indicate
this.

Because we cannot calculate derivatives of \eqref{eq:r}
we minimize using by the R function \texttt{optim}
with \texttt{method = "Nelder-Mead"}, the so-called Nelder-Mead simplex
algorithm, a no-derivative method nonlinear optimization,
not to be confused with the simplex algorithm
for linear programming.

\subsection{Step 2}

Having found $\alpha$, $c$, and $\sigma$ close to the MLE values
via the preceding step, we then switch to minimization of
\eqref{eq:key-rooted} for which we have the derivative formulas
\eqref{eq:key-rooted-alpha}, \eqref{eq:key-rooted-c},
and \eqref{eq:key-rooted-sigma}.
In this step we can use one of R's optimization functions that uses
first derivative information: \texttt{nlm} or \texttt{nlminb} or
\texttt{optim} with optional argument \texttt{method = "BFGS"}
or \texttt{method = "CG"} or \texttt{method = "L-BFGS-B"}.

To define \eqref{eq:key-rooted} we also need a $\widehat{W}$, and we take the
value at the current values of $\alpha$, $c$, and $\sigma$.
Because $W$ is typically a very large matrix ($n \times n$, where $n$
is the number of nodes in complete aster graph, the number of nodes
in the subgraph for a single individual times the number of individuals),
we actually store $Z^T \widehat{W} Z$, which is only $r \times r$, where
$r$ is the number of random effects.  We set
\begin{equation} \label{eq:zwz}
   Z^T \widehat{W} Z
   =
   Z^T W(a + M \alpha + Z A c) Z
\end{equation}
where $\alpha$, $c$, and $A = A(\sigma)$ are the current values before
we start minimizing $\tilde{p}(\alpha, c, \sigma)$ and this value of
$Z^T \widehat{W} Z$ is fixed throughout the minimization,
as is required by the definition of $\tilde{p}(\alpha, c, \sigma)$.

Having minimized $\tilde{p}(\alpha, c, \sigma)$ we are still not done, because
now \eqref{eq:zwz} is wrong.  We held it fixed at the values of
$\alpha$, $c$, and $\sigma$ we had before the minimization, and now
those values have changed.  Thus we should re-evaluate \eqref{eq:zwz}
and re-minimize, and continue doing this until convergence.

We terminate this iteration when $\sigma$ values do not change
(to within some prespecified tolerance) because the $\alpha$ and $c$
values are, in theory, determined by $\sigma$, because $\tilde{p}$
considered as a function of $\alpha$ and $c$ for fixed $\sigma$ is
convex and hence has at most one local minimizer,
so we do not need to worry about them converging.

When this iteration terminates we are done with this step, and we have
our point estimates $\hat{\alpha}$, $\hat{c}$, and $\hat{\sigma}$.
We also have our point estimates $\hat{b}$ of the random effects
on the original scale given by $A(\hat{\nu}) \hat{c}$
and our point estimates $\nu_j = \sigma_j^2$ of the variance components.

\subsection{Step 3}

Having converted back to the original parameters, if any of the $\nu_j$
are close to zero we use the check \eqref{eq:descent-test} to determine
whether or not they are exactly zero.

\subsection{To Do}

A few issues that have not been settled.  Points~1 and~2 in the following
list are not specific to random effects models.  They arise in fixed effect
aster models too, even in generalized linear models and log-linear models
in categorical data analysis.
\begin{enumerate}
\item Verify no directions of recession of fixed-effects-only model.
\item Verify supposedly nested models are actually nested.
\item How about constrained optimization and hypothesis tests of
    variance components being zero?  How does the software automagically
    or educationally do the right thing?  That is, do we just do the
    Right Thing or somehow explain to lusers what the Right Thing is?
\end{enumerate}

\appendix

\section{Cholesky}

How do we calculate log determinants and derivatives thereof?
R has a function \texttt{determinant} that calculates the log determinant.
It uses $L U$ decomposition.

An alternative method is to use Cholesky decomposition, but that only works
when the given matrix is symmetric.  This may be better because there
is a sparse version (the \texttt{chol} function in the \texttt{Matrix}
package) that may enable us to do much larger problems (perhaps after some
other issues getting in the way of scaling are also fixed).

We need to calculate the log determinant that appears in \eqref{eq:key}
or \eqref{eq:key-rooted}, but the matrix is not symmetric.
It can, however, be rewritten so as to be symmetric.
Assuming $A$ is invertible \begin{align*}
   \det\bigl( Z^T \widehat{W} Z A^2 + I \bigr)
   & =
   \det\bigl( Z^T \widehat{W} Z A + A^{- 1} \bigr)
   \det\bigl( A \bigr)
   \\
   & =
   \det\bigl( A Z^T \widehat{W} Z A + I \bigr)
\end{align*}
If $A$ is singular, we can see by continuity
that the two sides must agree there too.
That takes care of \eqref{eq:key-rooted}.
The same trick works for \eqref{eq:key}; just replace $A$ by
$D^{1 / 2}$, which is the diagonal matrix whose diagonal components
are the nonnegative square roots of the corresponding diagonal
components of $D$.

Cholesky can also be used to efficiently calculate matrix inverses
(done by the \texttt{chol2inv} function in the \texttt{Matrix} package).
So we investigate whether we can use Cholesky to calculate derivatives.

\subsection{First Derivatives}

For the trace in the formula \eqref{eq:key-rooted-sigma}
for $\tilde{p}_{\sigma_j}(\alpha, c, \sigma)$
we have in case $A$ is invertible
\begin{multline*}
   \tr\left(
   [ Z^T \widehat{W} Z A^2 + I \bigr]^{- 1}
   Z^T \widehat{W} Z A E_j
   \right)
   \\
   =
   \tr\left(
   [ A^{- 1} ( A Z^T \widehat{W} Z A + I ) A \bigr]^{- 1}
   Z^T \widehat{W} Z A E_j
   \right)
   \\
   =
   \tr\left(
   A^{- 1}
   [ A Z^T \widehat{W} Z A + I \bigr]^{- 1}
   A
   Z^T \widehat{W} Z A E_j
   \right)
   \\
   =
   \tr\left(
   [ A Z^T \widehat{W} Z A + I \bigr]^{- 1}
   A
   Z^T \widehat{W} Z A E_j
   A^{- 1}
   \right)
   \\
   =
   \tr\left(
   [ A Z^T \widehat{W} Z A + I \bigr]^{- 1}
   A
   Z^T \widehat{W} Z E_j
   \right)
\end{multline*}
the next-to-last equality being $\tr(A B) = \tr(B A)$ and the
last equality using the fact that $A$, $E_j$, and $A^{- 1}$ are all
diagonal so they commute.
Again we see that we get the same identity of the first and last
expressions even when $A$ is singular by continuity.

For the trace in the formula \eqref{eq:p-theta}
for $p_{\nu_k}(\alpha, b, \nu)$
we have in case $D$ is invertible
\begin{multline*}
   \tr \Bigl(
   \bigl[ Z^T \widehat{W} Z D + I \bigr]^{- 1}
   Z^T \widehat{W} Z E_k
   \Bigr)
   \\
   =
   \tr \Bigl(
   D^{- 1 / 2} \bigl[ D^{1 / 2} Z^T \widehat{W} Z D^{1 / 2} + I \bigr]^{- 1}
   D^{1 / 2}
   Z^T \widehat{W} Z E_k
   \Bigr)
   \\
   =
   \tr \Bigl(
   \bigl[ D^{1 / 2} Z^T \widehat{W} Z D^{1 / 2} + I \bigr]^{- 1}
   D^{1 / 2}
   Z^T \widehat{W} Z
   D^{- 1 / 2} 
   E_k
   \Bigr)
\end{multline*}
This, of course, does not work when $D$ is singular.
We already knew we cannot differentiate $p(\alpha, b, \nu)$ on the
boundary of the constraint set.

\subsection{Second Derivatives}

For the trace in the formula in Section~\ref{sec:second}
for $p_{\nu_j \nu_k}(\alpha, b, \nu)$
we have in case $D$ is invertible
\begin{multline*}
   \tr \Bigl(
   \bigl[ Z^T \widehat{W} Z D + I \bigr]^{- 1}
   Z^T \widehat{W} Z E_j
   \bigl[ Z^T \widehat{W} Z D + I \bigr]^{- 1}
   Z^T \widehat{W} Z E_k
   \Bigr)
   \\
   =
   \tr \Bigl(
   D^{- 1 / 2}
   \bigl[ D^{1 / 2} Z^T \widehat{W} Z D^{1 / 2} + I \bigr]^{- 1}
   D^{1 / 2}
   Z^T \widehat{W} Z E_j
   \\
   D^{- 1 / 2}
   \bigl[ D^{1 / 2} Z^T \widehat{W} Z D^{1 / 2} + I \bigr]^{- 1}
   D^{1 / 2}
   Z^T \widehat{W} Z E_k
   \Bigr)
   \\
   =
   \tr \Bigl(
   \bigl[ D^{1 / 2} Z^T \widehat{W} Z D^{1 / 2} + I \bigr]^{- 1}
   D^{1 / 2}
   Z^T \widehat{W} Z E_j
   D^{- 1 / 2}
   \\
   \bigl[ D^{1 / 2} Z^T \widehat{W} Z D^{1 / 2} + I \bigr]^{- 1}
   D^{1 / 2}
   Z^T \widehat{W} Z E_k
   D^{- 1 / 2}
   \Bigr)
\end{multline*}
Again, this does not work when $D$ is singular.

The same trace occurs in the expression for $q_{\nu_j \nu_k}(\alpha, \nu)$
given in Section~\ref{sec:fisher} and can be calculated the same way.

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\end{thebibliography}

\end{document}

