## Revision 877

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PROCEDURE_SECTION

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  int k=0;

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  int k=1;

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  L(1,1) = 1.0;

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  for(i=2;i<=4;i++)

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  {

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\section{Built-in data likelihoods}

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In the simple \texttt{simple.tpl}, the mathematical expressions for all

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log-likehood contributions where written out in full detail. You may have hoped

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log-likelihood contributions where written out in full detail. You may have hoped

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that for the most common probability distributions, there were functions written

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so that you would not have to remember or look up their log-likelihood

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expressions. If your density is among those given in

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\index{Gauss-Hermite quadrature}

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In the situation where the model is separable of type Block diagonal

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Hessian,'' with only a single random effect in each block (see

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Section~\ref{separability}), Gauss-Hermite quadrature is available as an option

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Hessian,'' (see Section~\ref{separability}),

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Gauss-Hermite quadrature is available as an option

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to the Laplace approximation and to the \texttt{-is} (importance sampling)

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option. It is invoked with command line option \texttt{-gh N}, where \texttt{N}

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is the number of quadrature points.

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\subsection{Frequency weighting for multinomial likelihoods}

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%\subsection{Frequency weighting for multinomial likelihoods}

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%

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%In situations where the response variable only can take on a finite number of

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%different values, it is possibly to reduce the computational burden enormously.

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%As an example, consider a situation where observation $y_{i}$ is binomially

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%distributed with parameters $N=2$ and $p_{i}$. Assume that

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%\begin{equation*}

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%  p_{i}=\frac{\exp (\mu +u_{i})}{1+\exp (\mu +u_{i})},

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%\end{equation*}

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%where $\mu$ is a parameter and $u_{i}\sim N(0,\sigma ^{2})$ is a random effect.

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%For independent observations $y_1,\ldots,y_n$, the log-likelihood function for

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%the parameter $\theta =(\mu ,\sigma )$ can be written as

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%

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%  l(\theta )=\sum_{i=1}^{n}\log \bigl[\, p(x_{i};\theta )\bigr] .

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%

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%In \scAR, $p(x_{i};\theta)$ is approximated using the Laplace approximation.

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%However, since $y_i$ only can take the values~$0$, $1$, and~$2$, we can rewrite

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%the log-likelihood~as

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%$$ 1530 %l(\theta )=\sum_{j=0}^{2}n_{j}\log \left[ p(j;\theta )\right],  1531 %$$

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%where $n_j$ is the number $y_i$s being equal to $j$. Still, the Laplace

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%approximation must be used to approximate $p(j;\theta )$, but now only for

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%$j=0,1,2$, as opposed to $n$~times above. For large~$n$, this can give large a

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%large reduction in computing time.

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%

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%To implement the weighted log-likelihood~(\ref{l_w}), we define a weight vector

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%$(w_1,w_2,w_3)=(n_{0},n_{1},n_{2})$. To read the weights from file, and to tell

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%\scAR\ that~\texttt{w} is a weights vector, the following code is used:

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%\begin{lstlisting}

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%DATA_SECTION

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% init_vector w(1,3)

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%

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%PARAMETER_SECTION

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% !! set_multinomial_weights(w);

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%\end{lstlisting}

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%In addition, it is necessary to explicitly multiply the likelihood contributions

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%in~(\ref{l_w}) by~$w$. The program must be written with

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%\texttt{SEPARABLE\_FUNCTION}, as explained in Section~\ref{sec:nested}. For the

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%likelihood~(\ref{l_w}), the \texttt{SEPARABLE\_FUNCTION} will be invoked three

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%times.

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%

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%See a full example

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%\href{http://www.otter-rsch.com/admbre/examples/weights/weights.html}{here}.

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In situations where the response variable only can take on a finite number of

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different values, it is possibly to reduce the computational burden enormously.

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As an example, consider a situation where observation $y_{i}$ is binomially

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distributed with parameters $N=2$ and $p_{i}$. Assume that

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\begin{equation*}

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  p_{i}=\frac{\exp (\mu +u_{i})}{1+\exp (\mu +u_{i})},

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\end{equation*}

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where $\mu$ is a parameter and $u_{i}\sim N(0,\sigma ^{2})$ is a random effect.

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For independent observations $y_1,\ldots,y_n$, the log-likelihood function for

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the parameter $\theta =(\mu ,\sigma )$ can be written as

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  l(\theta )=\sum_{i=1}^{n}\log \bigl[\, p(x_{i};\theta )\bigr] .

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In \scAR, $p(x_{i};\theta)$ is approximated using the Laplace approximation.

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However, since $y_i$ only can take the values~$0$, $1$, and~$2$, we can rewrite

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the log-likelihood~as

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$$ 1530 l(\theta )=\sum_{j=0}^{2}n_{j}\log \left[ p(j;\theta )\right],  1531 $$

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where $n_j$ is the number $y_i$s being equal to $j$. Still, the Laplace

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approximation must be used to approximate $p(j;\theta )$, but now only for

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