smooth_dtime#

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Prior Density Function for Smoothing Time Difference#

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_dtime_prior_id multiplier for the smoothing.

prior_ij#

For \(i = 0, \ldots , I-1\), \(j = 0, \ldots , J-2\) we use prior_ij to denote the dtime_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 \(J-1\) in this notation, but not the other notation above.

T^s#

The time difference density \(T^s (s, v)\) is defined by

\[\log T^s (s, v, \theta ) = \sum_{i=0}^{I-1} \sum_{j=0}^{J-2} D \left( v_{i,j+1} \W{,} v_{i,j} \W{,} \mu_{i,j} \W{,} \lambda \sigma_{i,j} \W{,} d_{i,j} \right)\]

where \(D\) is the log-density function . Note that we include \(\theta\) as an argument because \(\lambda\) is a component of \(\theta\).