Revision 877 trunk/docs/manuals/admbre/admbre.tex
admbre.tex (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{Builtin data likelihoods} 
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In the simple \texttt{simple.tpl}, the mathematical expressions for all 
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loglikehood contributions where written out in full detail. You may have hoped 

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loglikelihood 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 loglikelihood 
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expressions. If your density is among those given in 
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\index{GaussHermite 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}), GaussHermite quadrature is available as an option


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


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GaussHermite 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 loglikelihood function for 

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

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%\begin{equation} 

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

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%\end{equation} 

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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 loglikelihood~as 

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%$$ 

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

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%$$ 

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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 loglikelihood~(\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.otterrsch.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 loglikelihood function for 

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

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\begin{equation} 

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

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\end{equation} 

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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 loglikelihood~as 

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$$ 

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

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$$ 

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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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