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lines 6-98 of file: example/user/predict_sample.py
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{xrst_begin user_predict_sample.py}
{xrst_spell
}

Predict Using Posterior Samples
###############################

Purpose
*******
The command

| |tab|   ``dismod_at`` *database* ``predict sample fit_var`` *scale*

computes a prediction for each avgint table :ref:`avgint_table@avgint_id`
and each sample table :ref:`sample_table@sample_index` .

Note Table
**********
This example has only one node called ``world``

Model Variables
***************
There is only one model variable *iota* in this example and it is
constant w.r.t; age and time.


Measurement
***********
The data are measurements of
:ref:`avg_integrand@Integrand, I_i(a,t)@Sincidence` ; i.e.
a direct measurement of *iota* .
We use *n_data* for the number of measurements.
The data is simulated using a normal with mean equal to the true *iota*
and standard deviation equal to *std_data* .
This distribution is also used for the model of the measurement noise.
{xrst_code py}'''
iota_true = 0.01
n_data    = 100
std_data  = iota_true * 0.2
r'''{xrst_code}

Random Seed
***********
This is used to seed the random number generator used to simulate the noise
in the measurement values:
{xrst_code py}'''
import time
random_seed = int( time.time() )
r'''{xrst_code}

Number of Samples
*****************
Each sample has a complete set of the :ref:`model_variables-name` which
yields a different prediction :`
{xrst_code py}'''
number_samples = 100
r'''{xrst_code}

Sample Covariance
*****************
Let :math:`\iota` be *iota* ,
math:`y_i` be the i-th simulated measurement,
:math:`n` be the number of data points *n_data* ,
and :math:`\sigam` bi the measurement standard deviation *std_data* .
Dropping terms that do not depend on *iota* , the negative log-likelihood is

.. math::
   L( \iota ) = \frac{1}{2} \sum_{i=0}^{m-1}
      \left( \frac{y_i - \iota }{\sigma} \right)^2

Taking the second derivative w.r.t. *iota* is :math:`m / \sigma^2` .
Hence the samples for *iota* should have variance ;math:`\sigma^2 / m` .

Prediction Covariance
*********************
The samples are scaled by *scale* before making predictions:
{xrst_code py}'''
scale = 2.0
r'''{xrst_code}
This should make the variance for the predictions should be

   (scale * sigma)^2 / n_data


Source Code
***********
The source code below does not include the global settings defined above:
{xrst_literal
   # BEGIN SOURCE
   # END SOURCE
}

{xrst_end user_predict_sample.py}