11likelihoods.tex

\section{Likelihood components}

The objective function $L$ is the weighted sum of the individual components
indexed by kind of data $i$, and fishery/survey $f$ as appropriate:

\begin{equation}
	L = \sum_{i=1}^{I}\sum_{f=1}^{A_f}\lambda_{i,f} L_{i,f}+\lambda_R L_R + \sum_{\theta}^{}\lambda_\theta L_\theta + \sum_{P}^{}\lambda_P L_P
\end{equation}

where $L$ is the total objective function, $i$ is the index of the objective
function component, $A_f$ is the number of fleets, $L_{i,f}$ is the
objective function for data kind $i$ for the fishery/survey $f$,
$\lambda_{i,f}$ is a weighting factor for each objective function component,
$\theta$ is the parameter priors, and $P$ is the random parameter deviations.

The components of the objective function based on the model set-up and data are: 

\begin{longtable}{p{1cm} p{3.5cm} p{5.5cm} Rp{5cm}}
	\hline
	Index & Source & Data/Parameter & Error structure\Tstrut\Bstrut\\
	\hline	
	$i$ & fishery/survey $f$ & CPUE or Abundance index & user choice \Tstrut\\
	$i$ & fishery $f$        & Discard biomass         & user choice \Tstrut\\
	$i$ & fishery/survey $f$ & Mean body weight        & normal \Tstrut\\
	$i$ & fishery/survey $f$ & Length composition      & multinomial or Dirichlet-multinomial\Tstrut\\
	$i$ & fishery/survey $f$ & Age composition         & multinomial or Dirichlet-multinomial\Tstrut\\
%	$i$ & fishery/survey $f$ & Dirichlet data weighting& gamma in log-space\Tstrut\\
	$i$ & fishery/survey $f$ & Mean size-at-age        & normal \Tstrut\\
	$i$ & fishery/survey $f$ & Tag-recapture 1         & multinomial \Tstrut\\
	$i$ & fishery/survey $f$ & Tag-recapture 2         & negative binomial\Tstrut\\
	$i$ & fishery $f$        & Initial equilibrium catch & normal \Tstrut\\
	$R$ & 					 & Recruitment deviations  & lognormal \Tstrut\\
	$P$ & 					 & Random parameter devs.  & normal \Tstrut\\
	$\theta$ & 				 & Parameter priors	       & user choice\Tstrut\\
	$F_B$ & 				 & F ballpark penalty	   &  \Tstrut\\
	$C_P$ &				     & Crash penalty           &  \Tstrut\Bstrut\\
	\hline
\end{longtable}
%Full description of likelihood distributions by source will be added in the future.

%\subsection{CPUE or abundance index}

%The distribution for CPUE or abundance index data can be lognormal, student-t, or normal.  The %lognormal likelihood formulation is:  

%\begin{equation} L_{i,f} = N(ln(\sigma)) + \sum_{y=1}^{N_y}\frac{(ln(I_{y,f})-ln(Q_f B_{y,f}))^2}{2\sigma^2}
%\end{equation}

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