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lines 5-93 of file: example/user/sample_asy_sim.py
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# {xrst_begin user_sample_asy_sim.py}
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# {xrst_comment_ch #}
#
# Sampling From The Asymptotic Distribution for a Simulated Data Fit
# ##################################################################
#
# Purpose
# *******
# This example demonstrates using the commands
#
# | |tab| ``dismod_at`` *database* ``set truth_var prior_mean``
# | |tab| ``dismod_at`` *database* ``simulate`` *number_sample*
# | |tab| ``dismod_at`` *database* ``sample simulate`` *number_sample*
# | |tab| ``dismod_at`` *database* ``sample asymptotic both`` *number_sample*
#
# To obtain the specified number of samples from the posterior distribution;
# see :ref:`posterior@Simulation` .
# This is very simple case because it has only one rate, no random effects,
# and no data.
#
# Discussion
# **********
# We use :math:`\bar{\omega}` to denote the prior mean for omega.
# The following describes the model and data for this example:
#
# #.  The :ref:`age_table-name` has values
#     ``0.0`` , ``100.0`` .
#     The :ref:`time_table-name` has values
#     ``1995.0`` , ``2015.0`` .
# #.  The only node is the world.
# #.  The only :ref:`model_variables-name` in this example are
#     :ref:`rate_table@rate_name@omega` for the world.
#     This rate is modeled as constant with respect to age and
#     linear between time 1995 and 2015.
# #.  There are no measurements for this example.
# #.  The prior for the world omega is a gaussian with mean :math:`\bar{\omega}`
#     and standard deviation :math:`\bar{\omega}`.
# #.  The prior for the difference in omega between time 1995
#     and time 2015 is a Gaussian
#     with mean zero and standard deviation :math:`\bar{\omega}`.
#
# Posterior Variance
# ******************
# We use :math:`\omega_0` and :math:`\omega_1` for the value
# of omega at times 1995 and 2015 respectively.
# Dropping terms that are constant with respect to :math:`\omega`
# the negative log likelihood :math:`\ell ( \omega )` for this case is
# given by
#
# .. math::
#
#   \ell ( \omega )
#   = \frac{1}{2} \bar{\omega}^{-2} \left[
#        ( \omega_0 - \bar{\omega} )^2
#        + ( \omega_1 - \omega_0 )^2
#        + ( \omega_1 - \bar{\omega} )^2
#   \right]
#
# The Hessian of this function is given by
#
# .. math::
#
#   \ell^{(2)} ( \omega ) = \bar{\omega}^{-2}
#   \left( \begin{array}{cc}
#    2 & -1 \\
#    -1 & 2
#   \end{array} \right)
#
# It follows that the variance of the maximum likelihood estimator is
#
# .. math::
#
#    [ \ell^{(2)} ( \omega ) ]^{-1} = \bar{\omega}^2
#   \left( \begin{array}{cc}
#    2/3 &  1/3 \\
#    1/3 &  2/3
#   \end{array} \right)
#
# Source Code
# ***********
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# {xrst_end user_sample_asy_sim.py}