Dakota Reference Manual: confidence_intervals

| Dakota Reference Manual Version 6.15 Explore and Predict with Confidence

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confidence_intervals

Calculate the confidence intervals on estimates of first and second moments

## Specification

Alias: none

Argument(s): none

## Description

During Bayesian calibration, a chain of samples is produced, which represents the underlying posterior distribution of model parameters. For each parameter sample, the corresponding model response is computed. The `confidence_intervals` keyword indicates the calculation of a 95% confidence interval on the estimated mean and variance of each parameter and each response.

As of Dakota 6.10, these confidence intervals are calculated using the asymptotically valid interval estimator,

where is the moment (i.e. mean or variance), is the Student's -value for the 95th quantile, is the sample size, and is an estimate of the standard error whose square is obtained using batch means estimation. The Markov chain produced during calibration is broken up into "batches," the sample moment is calculated for each batch, and is an unbiased estimate of the standard deviation in these batch moment calculations.

Expected Output

If `confidence_intervals` is specified, the 95% confidence interval for the mean and variance for each parameter and for each response will be output to the screen. If `output` is set to `debug`, the mean of the moment calculated for each batch will also be output to the screen.

Additional Discussion

Confidence intervals may be used to indicate to the user whether `samples` needs to be increased during the Bayesian calibration. For example, if the width of the intervals (one, many, or all) is below some threshold value, that may indicate that enough samples have been drawn.

## Examples

Below is a `method` block of a Dakota input file that indicates the calculation of confidence intervals

    method,
           bayes_calibration queso
              dram
              seed = 34785
              chain_samples = 1000
              chain_diagnostics
                 confidence_intervals

The calculated confidence intervals are output to the screen under "Chain diagnostics":

    Sample moment statistics for each posterior variable:
                                Mean           Std Dev          Skewness          Kurtosis
                 E  2.8609959149e+07  1.4417714265e+05  8.0289072766e-01  7.8655956160e-02
                 w  2.5016445558e+00  3.8306697138e-03 -1.2217188066e-01  3.8866929786e-02

    Sample moment statistics for each response function:
                                Mean           Std Dev          Skewness          Kurtosis
            stress  2.6282814617e+03  8.9765361327e+01  1.3400226598e-01  4.9239052296e-02
      displacement  2.9604502307e-01  1.0636886950e-02 -3.5080744509e-01 -1.2381836901e-01

    Chain diagnostics
      95% Confidence Intervals of means
                      E = [2.8570364780e+07, 2.8649553519e+07]
                      w = [2.5009524557e+00, 2.5023366559e+00]
                 stress = [2.6120609285e+03, 2.6445019948e+03]
           displacement = [2.9337418438e-01, 2.9871586175e-01]
      95% Confidence Intervals of variances
                      E = [1.5074828429e+10, 2.6499268497e+10]
                      w = [1.1359880885e-05, 1.7988180028e-05]
                 stress = [6.2340446164e+03, 9.8815955721e+03]
           displacement = [8.8187472572e-05, 1.3809925539e-04]