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@(@\newcommand{\B}[1]{ {\bf #1} } \newcommand{\R}[1]{ {\rm #1} } \newcommand{\W}[1]{ \; #1 \; }@)@ This is dismod_at-20221105 documentation: Here is a link to its current documentation .
Prior Density Function for Smoothing Age Difference

S
I
J
lambda
prior_ij
d_ij
sigma_ij
mu_ij
v_ij
A^s

S
We are given a smoothing, @(@ s @)@.

I
We use @(@ I @)@ to denote n_age the number of age points in the smoothing.

J
We use @(@ J @)@ to denote n_time the number of time points in the smoothing.

lambda
We use @(@ \lambda @)@ to denote the mulstd_dage_prior_id multiplier for the smoothing.

prior_ij
For @(@ i = 0, \ldots , I-2 @)@, @(@ j = 0, \ldots , J-1 @)@ we use prior_ij to denote the dage_prior corresponding to age index @(@ i @)@ and time index @(@ j @)@ in the smoothing.

d_ij
We use @(@ d_{i,j} @)@ to denote the density in prior_ij . In an abuse of notation, we include the value of eta and nu and in @(@ d_{i,j} @)@; see d .

sigma_ij
We use @(@ \sigma_{i,j} @)@ to denote the std in prior_ij .

mu_ij
We use @(@ \mu_{i,j} @)@ to denote the mean in prior_ij .

v_ij
We use @(@ v_{i,j} @)@ for the value of the model variable corresponding to the i-th age point and j-th time point in the smoothing. We include the index @(@ I-1 @)@ in this notation, but not the other notation above.

A^s
The age difference density @(@ A^s (s, v) @)@ is defined by @[@ \log A^s (s, v, \theta ) = \sum_{i=0}^{I-2} \sum_{j=0}^{J-1} D \left( v_{i+1,j} \W{,} v_{i,j} \W{,} \mu_{i,j} \W{,} \lambda \sigma_{i,j} \W{,} d_{i,j} \right) @]@ where @(@ D @)@ is the log-density function . Note that we include @(@ \theta @)@ as an argument because @(@ \lambda @)@ is a component of @(@ \theta @)@.
Input File: omh/model/smooth_dage.omh