Blurb::
Calculate the confidence intervals on estimates of first and second moments
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,


.. math:: 

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

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