Dakota Reference Manual: probability_levels

| Dakota Reference Manual Version 6.15 Explore and Predict with Confidence

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probability_levels

Specify probability levels at which to compute credible and prediction intervals

## Specification

Alias: none

Argument(s): REALLIST

Default: No CDF/CCDF response levels to compute

Child Keywords:

| Required/Optional | Description of Group | Dakota Keyword | Dakota Keyword Description
--- | --- | --- | --- | ---
| Optional | num_probability_levels | This capability is currently inactive

## 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.

## Examples

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

    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.

    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