fixed_prior#

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Prior for Fixed Effect Values#

theta#

lambda#

We use \(\lambda\) to denote the sub-vector of the fixed effects that are standard deviation multipliers .

beta#

We use \(\beta\) to denote the sub-vector of the fixed effects that are Parent Rates or Group Covariate Multipliers .

theta#

We use \(\theta\) to denote the entire fixed effects vector; i.e., \(\theta = ( \lambda , \beta )\).

Value Constraints#

theta_k#

We use \(\theta_k\) to denote one component of \(\theta\).

L_k^v#

We use \(L_k^v\) to denote the lower limit corresponding to the value_prior_id that corresponds to the fixed effect \(\theta_k\).

U_k^v#

We use \(U_k^v\) to denote the upper limit corresponding to the value_prior_id that corresponds to the fixed effect \(\theta_k\).

Age Difference Limits#

The fixed effects corresponding to the standard deviation multipliers mulstd_value_prior_id , mulstd_dage_prior_id , and mulstd_dtime_prior_id are constant with respect to age and time. Hence the constraints below do not apply to the standard deviation multipliers.

a_i(k)#

We use \(a_{i(k)}\) to denote the age corresponding to the age_id for the fixed effect \(\theta_k\). If this is the maximum age for the corresponding smooth_id , then there is no age difference for this fixed effect. Otherwise, we use \(a_{i(k)+1}\) to denote the next larger age in the smoothing grid and \(\theta_{\ell(k)}\) denote the corresponding component of \(\theta\) corresponding to the next larger age.

Delta^a#

If \(a_{i(k)}\) is not the maximum age, we use the notation

\[\Delta^a_k \theta = \theta_{\ell(k)} - \theta_k\]

L_k^a#

We use \(L_k^a\) to denote the lower limit corresponding to the dage_prior_id that corresponds to the fixed effect \(\theta_k\).

U_k^a#

We use \(U_k^a\) to denote the upper limit corresponding to the dage_prior_id that corresponds to the fixed effect \(\theta_k\).

Time Difference Limits#

The time difference \(\Delta^t_k \theta\), the index \(j(k)\), and limits \(L_k^t\) and \(U_k^t\) are defined in a fashion similar as for the age differences.

Capital Theta#

The constraint set \(\Theta\) is defined as the set of \(\theta\) such that the following conditions hold:

  1. For all \(k\),

    \[L_k^v \leq \theta_k \leq U_k^v\]

  2. For \(k\), that are not standard deviation multipliers, and such that \(a_{i(k)}\) is not the maximum age for the corresponding smoothing,

    \[L_k^a \leq \Delta^a_k \theta \leq U_k^a\]

  3. For \(k\), that are not standard deviation multipliers, and such that \(t_{j(k)}\) is not the maximum time for the corresponding smoothing,

    \[L_k^t \leq \Delta^t_k \theta \leq U_k^t\]

Normalization Constant, C_theta#

The normalization constant for the fixed effects density is

\[C_\theta = \int_{\Theta} V^\theta ( \theta ) A^\theta ( \theta ) T^\theta ( \theta ) \; \B{d} \theta\]

See V^theta , A^theta , and T^theta for the definitions of \(V^\theta\), \(A^\theta\) and \(T^\theta\).

p(theta)#

We define the prior for the fixed effects by

\[C_\theta \; \B{p} ( \theta ) = V^\theta ( \theta ) A^\theta ( \theta ) T^\theta ( \theta )\]