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lines 5-89 of file: example/user/predict_mulcov.py
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# {xrst_begin user_predict_mulcov.py}
# {xrst_comment_ch #}
#
# Predict Covariate Multiplier Values
# ###################################
#
# Purpose
# *******
# This examples demonstrates covariate multiplier predictions.
#
# Problem Parameters
# ******************
# The following values are used to simulate the data and set the priors
# for fitting the data:
# {xrst_literal
#     begin problem parameters
#     end problem parameters
# }
#
# Age and Time Values
# *******************
# The reference value for *iota* is constant
# in age and time, but the value of the covariate multiplier that
# affects iota changes with age and time.
#
# Rate Table
# **********
# The :ref:`rate_table-name` specifies that the only
# :ref:`var_table@var_type@rate` variable is
# *iota* for north_america.
# In addition, it specifies the smoothing for this rate
# has one grid point.
#
# Mulcov Table
# ************
# The :ref:`mulcov_table-name` specifies that
# the covariate *alpha* is a bilinear function of age and time.
# In fact, it is equal to the function ``alpha_true`` defined
# as one of the problem parameters.
#
# Variables
# *********
# There are five model variables in this example:
#
# .. list-table::
#     :widths: auto
#
#     * - *iota_reference*
#       - There is one variable corresponding to the reference value for
#         :ref:`iota(a,t)<diff_eq@Incidence, iota(a, t)>` in north_america.
#     * - *alpha*
#       - There are four variables corresponding to the
#         :ref:`mulcov_table@mulcov_type@rate_value`
#         covariate multiplier that affect *iota* .
#
# Integrand Table
# ***************
# The :ref:`integrand_table-name` for this example includes
# :ref:`avg_integrand@Integrand, I_i(a,t)@Sincidence` and
# :ref:`avg_integrand@Integrand, I_i(a,t)@mulcov` .
# The ``mulcov_0`` integrand corresponds the value of *alpha* .
#
# Data Table
# **********
# There are four measurements of *Sincidence*
# in the :ref:`data_table-name` , one for (age, time) pair corresponding
# to an *alpha* model variable.
# No noise is added to the measurements, and the prior on
# *iota* constrains it to the *iota_reference* .
#
# Avgint Table
# ************
# There are four predictions of *alpha* requested by the
# :ref:`avgint_table-name` , one for each *alpha* model variable.
# The predictions are compared with the truth to see that the
# fit is prefect (there is no noise in the data).
#
# Source Code
# ***********
# {xrst_literal
#     BEGIN PYTHON
#     END PYTHON
# }
#
# {xrst_end user_predict_mulcov.py}