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lines 5-99 of file: example/user/sample_asy.py
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# {xrst_begin user_sample_asy.py}
# {xrst_comment_ch #}
#
# Sample from Asymptotic Distribution for Model Variables
# #######################################################
#
# Purpose
# *******
# The command
#
# | |tab| ``dismod_at`` *database* ``sample asymptotic both`` *number_sample*
# | |tab| ``dismod_at`` *database* ``sample asymptotic fixed`` *number_sample*
#
# samples form an asymptotic approximation for the posterior distribution
# of the :ref:`model_variables-name` .
#
# Notation
# ********
#
# .. csv-table::
#       :widths: auto
#
#       :math:`\iota_n`,model incidence for ``north_america``
#       *u_m*,incidence random effect for ``mexico``
#       *u_c*,incidence random effect for ``canada``
#       *y_n*,measured incidence for ``north_america``
#       *y_m*,measured incidence for ``mexico``
#       *y_c*,measured incidence for ``canada``
#       *s_n*,standard deviation for *y_n*
#       *s_m*,standard deviation for *y_m*
#       *s_c*,standard deviation for *y_c*
#       *s_r*,standard deviation for random effects
#
# The only fixed effect in this model is :math:`\iota_n`.
# The random effects are :math:`u_m` and :math:`u_c`.
#
# Likelihood
# **********
# We define :math:`h( y, \mu , \delta )`
# to be the log-density for a :math:`\B{N}( \mu, \delta^2 )` distribution;
# i.e.,
#
# .. math::
#
#   h( y, \mu, \delta ) =
#       - \frac{ ( y - \mu )^2 }{ \delta^2 }
#       - \log \left( \delta \sqrt{ 2 \pi } \right)
#
# The total log-likelihood for
# :ref:`fit fixed<fit_command@variables@fixed>` is
#
# .. math::
#
#   h[ y_n, \iota_n, s_n ] +
#   h[ y_m, \iota_n, s_m ] +
#   h[ y_c, \iota_n, s_c ]
#
# The total log-likelihood for
# :ref:`fit both<fit_command@variables@both>` is
#
# .. math::
#
#   h[ y_n, \iota_n, s_n ] +
#   h[ y_m, \exp( u_m ) \iota_n, s_m ] +
#   h[ y_c, \exp( u_c ) \iota_n, s_c ] +
#   h( u_m, 0, s_r ) + h( u_c , 0 , s_r )
#
# Problem Parameters
# ******************
# {xrst_spell_off}
# {xrst_code py}
measure = {
    'north_america' : 1.0e-2 , # y_n
    'mexico'        : 2.0e-2 , # y_m
    'canada'        : 0.5e-2   # y_c
}
standard = {
    'north_america' : 1.0e-3 , # s_n
    'mexico'        : 2.0e-3 , # s_m
    'canada'        : 0.5e-3   # s_c
}
standard_random_effect = 1.0   # s_r
number_sample          = 500   # number of posterior samples to generate
number_metropolis      = 10 * number_sample
# {xrst_code}
# {xrst_spell_on}
#
# Source Code
# ***********
# {xrst_literal
#       BEGIN PYTHON
#       END PYTHON
# }
#
# {xrst_end user_sample_asy.py}