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lines 5-168 of file: example/user/sample_sim.py
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# {xrst_begin user_sample_sim.py}
# {xrst_spell
#     exp
#     mcmc
# }
# {xrst_comment_ch #}
#
# Sample Posterior Using Simulated Data
# #####################################
#
# Purpose
# *******
# The command
#
#     ``dismod_at`` *database* ``sample simulate`` *number_sample*
#
# samples form the posterior distribution of the :ref:`model_variables-name`
# using simulated data.
#
# Node Table
# **********
# For this example, the :ref:`node_table-name` is
# ::
#
#             north_america
#             /          \
#          mexico      canada
#
# Model Variables
# ***************
# All of the rates are constant w.r.t. age an time in this example.
#
# iota
# ====
# We use :math:`\iota_n` to denote the incidence rate for
# the parent region north_america which is a
# :ref:`fixed effect<model_variables@Fixed Effects, theta>` .
#
# u
# =
# We use :math:`u_m` (:math:`u_c`) to denote the
# :ref:`child rate effect<model_variables@Random Effects, u@Child Rate Effects>`
# for mexico (canada).
#
# x
# =
# The model variables for this example are combined into a single vector
# :math:`x \in \B{R}^3` where
# :math:`x_0 = \iota_n`,
# :math:`x_1 = u_m`, and
# :math:`x_2 = u_c`.
#
# Model Parameters
# ****************
#
# y
# =
# We use :math:`y_n, y_m, y_c` to denote the measured
# :ref:`avg_integrand@Integrand, I_i(a,t)@Sincidence`
# for north_america, mexico, and canada respectively.
#
# s
# =
# We use :math:`s_n, s_m, s_c` to denote the
# standard deviation of the measured noise
# for north_america, mexico, and canada respectively.
#
# delta
# =====
# We use :math:`\delta` to denote the standard deviation of
# the child rate effects.
# (The mean for the child rate effects is zero.)
#
# Values
# ======
# {xrst_spell_off}
# {xrst_code py}
import math
y_n    = 1e-2
y_m    = 2.0 * y_n
y_c    = y_n / 2.0
s_n    = y_n / 10.0
s_m    = y_m / 10.0
s_c    = y_c / 10.0
delta  = math.log( 2.0 )
# {xrst_code}
# {xrst_spell_on}
#
# 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 this example 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, \delta ) + h( u_c , 0 , \delta )
#
# number_sample
# *************
# This is the number of samples in the
# :ref:`sample_command@number_sample`
# specified by the sample command.
# {xrst_spell_off}
# {xrst_code py}
number_sample = 30
# {xrst_code}
# {xrst_spell_on}
#
# number_metropolis
# *****************
# This is the number of mcmc samples generated by the
# :ref:`metropolis-name` mcmc method.
# These samples are used to check the simulate method.
# {xrst_spell_off}
# {xrst_code py}
number_metropolis = 500 * number_sample
# {xrst_code}
# {xrst_spell_on}
#
# Truth Table
# ***********
# For the data simulation, we use the following model variable values:
# {xrst_spell_off}
# {xrst_code py}
iota_n_true = y_n
u_m_true    = math.log( y_m / y_n )
u_c_true    = math.log( y_c / y_n )
# {xrst_code}
# {xrst_spell_on}
# It follows that *y_m* = ``exp`` ( *u_m_true* ) * *iota_n_true* and
# *y_c* = ``exp`` ( *u_c_true* ) * *iota_n_true* .
#
# random_seed
# ***********
# Set one random seed for use by both the python code and dismod_at.
# In addition, if an error occurs, this random seed will be reported; see
# ``random_seed`` in the source code below.
# {xrst_spell_off}
# {xrst_code py}
import time
random_seed = int( time.time() )
# {xrst_code}
# {xrst_spell_on}
#
# Source Code
# ***********
# {xrst_literal
#     BEGIN PYTHON
#     END PYTHON
# }
#
# {xrst_end user_sample_sim.py}