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Dakota Reference Manual
Version 6.15
Explore and Predict with Confidence
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Calculate the confidence intervals on estimates of first and second moments
Alias: none
Argument(s): 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,
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.
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]