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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