11likelihoods.tex

\hypertarget{LikeComp}{}
\section[Likelihood components]{\protect\hyperlink{LikeComp}{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 +
	  \lambda_{F_B} L_{F_B} + \lambda_{C_P} L_{C_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{2.75cm} p{4.75cm} p{6.5cm}}
	\hline
	Index & Source & Data/Parameter & Error structure \Tstrut\Bstrut\\
	\hline	
	$i$ & fishery/survey $f$ & \gls{cpue} or Abundance index & user choice \Tstrut\\
	$i$ & fishery $f$		 & Discard Biomass			 & user choice \Tstrut\\
	$i$ & fishery/survey $f$ & Mean body W or L (all ages) & normal \Tstrut\\
	$i$ & fishery/survey $f$ & Generalized size (W or L) composition & multinomial or Dirichlet-multinomial \Tstrut\\
	$i$ & fishery/survey $f$ & L 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 L (or W)-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 & log-normal \Tstrut\\
	$i$ & fishery $f$		 & catch					 & log-normal \Tstrut\\
	$R$ & 					 & Recruitment Deviations	 & log-normal \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 log-normal, student-t, or normal. The %log-normal 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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