Calculate the confidence intervals on estimates of first and second moments


  • Alias: None

  • Arguments: None


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,

\[\bar{g}_{n} \pm t_{*}\frac{\hat{\sigma}_{n}}{\sqrt{n}},\]

where \(\bar{g}_{n}\) is the moment (i.e. mean or variance), \(t_{*}\) is the Student’s \(t\) -value for the 95th quantile, \(n\) is the sample size, and \(\hat{\sigma}_{n}\) 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 \(\hat{\sigma}_{n}\) 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.