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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 for Fixed Effect Values

theta
     lambda
     beta
     theta
Value Constraints
     theta_k
     L_k^v
     U_k^v
Age Difference Limits
     a_i(k)
     Delta^a
     L_k^a
     U_k^a
Time Difference Limits
Capital Theta
Normalization Constant, C_theta
p(theta)

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 ) @]@
Input File: omh/model/fixed_prior.omh