Blurb::
Specify probability levels at which to compute credible and prediction intervals
Description::
Credible and prediction intervals of model responses are computed for specified
probabilities. Credible intervals are calculated from the response function
values corresponding to the final MCMC chain. Calculation of prediction
intervals consider these response values as well as the experimental
uncertainty, which is specified by the user via the ``experiment_variance_type``
command.

*Expected Output*

If ``probability_levels`` is specified, Dakota will create a table containing
the credibile intervals for each response function. The corresponding table
containing the prediction intervals will also be created if a
``experiment_variance_type`` has been specified. This information is output to
the screen and to a file. In addition, the output file contains the means and
standard deviations of each response function and Gaussian approximations of
the 5/95 credible and prediction intervals, in which the lower bound is two
standard deviations below the mean and the upper bound is two standard
deviations above the mean.

*Usage Tips*

Only one probability level needs to be specified for each desired interval. Both
corresponding end points of the intervals are automatically calculated. For
example, if 0.05 is specified, both the 0.05 and 0.95 probability levels are
output to the screen and output file.

*Additional Discussion*

Credible intervals propagate uncertainties in parameter density information to
the quantity of interest and quantify how well the model fits the provided data.
Prediction intervals propagate both parameter and experimental measurement
uncertainties and contain the next experimental or simulated observation with
the specified probability.
Topics::

Examples::
Below is a Dakota input file specifying the calculation of credible and
prediction intervals


.. code-block::

    method,
     bayes_calibration queso
       chain_samples = 1000 seed = 348
       dram
     proposal_covariance
       diagonal values 1.0e6 1.0e-1
     probability_levels 0.05  0.1
          0.075 0.1
    
    variables,
     uniform_uncertain 2
              upper_bounds  1.e8 10.0
              lower_bounds 1.e6 0.1
              initial_point 2.85e7 2.5
              descriptors 'E' 'w'
            continuous_state 4
              initial_state 3 40000 500 1000
              descriptors 't' 'R' 'X' 'Y'
    
    interface,
            direct
              analysis_driver = 'mod_cantilever'
    
    responses,
            calibration_terms = 2
            calibration_data_file = 'dakota_cantilever_queso.withsigma.dat'
              freeform
              num_experiments = 10
              experiment_variance_type = 'scalar'
            descriptors = 'stress' 'displacement'
            no_gradients
            no_hessians


The resulting screen output below shows the table of credible and prediction
intervals.


.. code-block::

    Credibility Intervals for stress
                      Response Level    Probability Level
                      ----------------- -----------------
                      2.4764049695e+03  5.0000000000e-02
                      2.8242874802e+03  9.5000000000e-01
                      2.4990608791e+03  1.0000000000e-01
                      2.7952985803e+03  9.0000000000e-01
    Credibility Intervals for displacement
                      Response Level    Probability Level
                      ----------------- -----------------
                      2.7409870925e-01  7.5000000000e-02
                      3.0991296255e-01  9.2500000000e-01
                      2.7538816802e-01  1.0000000000e-01
                      3.0889319332e-01  9.0000000000e-01
    Prediction Intervals for stress
                      Response Level    Probability Level
                      ----------------- -----------------
                      2.0964882850e+03  5.0000000000e-02
                      3.1993026765e+03  9.5000000000e-01
                      2.1822183238e+03  1.0000000000e-01
                      3.1099058450e+03  9.0000000000e-01
    Prediction Intervals for displacement
                      Response Level    Probability Level
                      ----------------- -----------------
                      2.3559036055e-01  7.5000000000e-02
                      3.5097481218e-01  9.2500000000e-01
                      2.4016170870e-01  1.0000000000e-01
                      3.4701712866e-01  9.0000000000e-01


Theory::

Faq::

See_Also::
