1Blurb::
2Calculate the confidence intervals on estimates of first and second moments
3
4Description::
5During Bayesian calibration, a chain of samples is produced, which represents
6the underlying posterior distribution of model parameters. For each parameter
7sample, the corresponding model response is computed. The
8\c confidence_intervals keyword indicates the calculation of a 95% confidence
9interval on the estimated mean and variance of each parameter and each
10response.
11
12As of Dakota 6.10, these confidence intervals are calculated using the
13asymptotically valid interval estimator,
14
15\f[
16\bar{g}_{n} \pm t_{*}\frac{\hat{\sigma}_{n}}{\sqrt{n}},
17\f]
18
19where \f$\bar{g}_{n}\f$ is the moment (i.e. mean or variance), \f$t_{*}\f$ is
20the Student's \f$t\f$-value for the 95th quantile, \f$n\f$ is the sample size, and
21\f$\hat{\sigma}_{n}\f$ is an estimate of the
22standard error whose square is obtained using batch means estimation. The
23Markov chain produced during calibration is broken up into "batches," the
24sample moment is calculated for each batch, and \f$\hat{\sigma}_{n}\f$ is an
25unbiased estimate of the standard deviation in these batch moment calculations.
26
27<b> Expected Output </b>
28
29If \c confidence_intervals is specified, the 95% confidence interval for the
30mean and variance for each parameter and for each response will be output to
31the screen. If \c output is set to \c debug, the mean of the moment calculated
32for each batch will also be output to the screen.
33
34<b> Additional Discussion </b>
35
36Confidence intervals may be used to indicate to the user whether \c samples
37needs to be increased during the Bayesian calibration. For example, if the
38width of the intervals (one, many, or all) is below some threshold value, that
39may indicate that enough samples have been drawn.
40
41Examples::
42Below is a \c method block of a Dakota input file that indicates the calculation
43of confidence intervals
44
45\verbatim
46method,
47       bayes_calibration queso
48          dram
49          seed = 34785
50          chain_samples = 1000
51          chain_diagnostics
52             confidence_intervals
53\endverbatim
54
55The calculated confidence intervals are output to the screen under "Chain
56diagnostics":
57
58\verbatim
59Sample moment statistics for each posterior variable:
60                            Mean           Std Dev          Skewness          Kurtosis
61             E  2.8609959149e+07  1.4417714265e+05  8.0289072766e-01  7.8655956160e-02
62             w  2.5016445558e+00  3.8306697138e-03 -1.2217188066e-01  3.8866929786e-02
63
64Sample moment statistics for each response function:
65                            Mean           Std Dev          Skewness          Kurtosis
66        stress  2.6282814617e+03  8.9765361327e+01  1.3400226598e-01  4.9239052296e-02
67  displacement  2.9604502307e-01  1.0636886950e-02 -3.5080744509e-01 -1.2381836901e-01
68
69Chain diagnostics
70  95% Confidence Intervals of means
71                  E = [2.8570364780e+07, 2.8649553519e+07]
72                  w = [2.5009524557e+00, 2.5023366559e+00]
73             stress = [2.6120609285e+03, 2.6445019948e+03]
74       displacement = [2.9337418438e-01, 2.9871586175e-01]
75  95% Confidence Intervals of variances
76                  E = [1.5074828429e+10, 2.6499268497e+10]
77                  w = [1.1359880885e-05, 1.7988180028e-05]
78             stress = [6.2340446164e+03, 9.8815955721e+03]
79       displacement = [8.8187472572e-05, 1.3809925539e-04]
80\endverbatim
81