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...likelihood. The goal is to get parameter estimates for $\beta$. The integral cannot be easily evaluated so I approximate it as: \begin{equation} \label{eqn:marginal2} L(\beta) \approx \prod_{s=1}^N \sum_{q=1}^Q \prod_{i=1}^K\frac{e^{x_{is}(\theta_{q}-\beta_i)}} {x_{is}!e^{e^(\theta_{q}-\beta_i)}} w_q \end{equation} \noindent where $\theta_{q}$ is the node at quadrature point $q = 1, \ldots, Q$ and $w_q$ is the weight at quadrature point $q$. For now, I am assuming $f(\theta)$ is Uniform but this may change and that is not a major issue. Now, I have written a function using both nlminb and opt...

...$i$ and $\beta_j$ is the location parameter for item $j$. The integral in Equation~(\ref{eqn:mml}) has no closed form expression, so it is approximated using Gauss-Hermite quadrature as: \begin{equation} \label{eqn:mml:approx} f(x) \approx \prod^N_{i=1}\sum_{q=1}^{Q}\prod^K_{j=1}p(x|\theta_q,\beta)w_q \end{equation} \noindent where $q$ indexes the node at quadrature point $q$ and $w$ is the weight at quadrature point $q$. With Equation~(\ref{eqn:mml:approx}), the remaining challenge is to find $\underset{x}{\operatorname{arg\,max}} \, f(x)$. \end{document} [[alternative HTML version deleted]]