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