Prior Density Function for Smoothing Age Difference

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smooth_dage

$\newcommand{\B}[1]{ {\bf #1} } \newcommand{\R}[1]{ {\rm #1} } \newcommand{\W}[1]{ \; #1 \; }$ 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