Blurb::
Bayesian calibration

Description::

Bayesian calibration methods take prior information on parameter
values (in the form of prior distributions) and observational data
(e.g. from experiments) and infer posterior distributions on the
parameter values. When the computational simulation is then executed
with samples from the posterior parameter distributions, the results
that are produced are consistent with ("agree with") the experimental
data. Calibrating parameters from a computational simulation model
requires a likelihood function that specifies the likelihood of
observing a particular observation given the model and its associated
parameterization; %Dakota assumes a Gaussian likelihood function. The
algorithms that produce the posterior distributions on model
parameters are most commonly Monte Carlo Markov Chain (MCMC) sampling
algorithms. MCMC methods require many samples, often tens of
thousands, so in the case of model calibration, often emulators of the
computational simulation are used. For more details on the algorithms
underlying the methods, see the %Dakota User's manual.

%Dakota has four classes of Bayesian calibration methods: QUESO/DRAM,
GPMSA, DREAM, and WASABI.

<ol>

<li>The QUESO methods use components from the QUESO library
(Quantification of Uncertainty for Estimation, Simulation, and
Optimization) developed at The University of Texas at Austin.  %Dakota
uses its DRAM (Delayed Rejected Adaptive Metropolis) algorithm, and
variants, for the MCMC sampling.</li>

<li>GPMSA (Gaussian Process Models for Simulation Analysis) is an
approach developed at Los Alamos National Laboratory and %Dakota
currently uses the QUESO implementation.  It constructs Gaussian
process models to emulate the expensive computational simulation as
well as model discrepancy. GPMSA also has extensive features for
calibration, such as the capability to include a model discrepancy
term and the capability to model functional data such as time series
data.  This is an experimental capability and not all features are
available in %Dakota yet.</li>

<li> DREAM (DiffeRential Evolution Adaptive Metropolis) is a method
that runs multiple different chains simultaneously for global
exploration, and automatically tunes the proposal covariance during
the process by a self-adaptive randomized subspace sampling.
\cite Vrugt.</li>
 
<li> WASABI: Non-MCMC Bayesian inference via interval analysis </li>

</ol>

<b>Usage Tips</b>

The Bayesian capabilities are under active development.  At this
stage, the QUESO methods in %Dakota are the most advanced and robust,
followed by DREAM, followed by GPMSA and WASABI which are not yet
ready for production use.

The prior distribution is characterized by the properties of the
uncertain variables. Correlated priors are only supported for
unbounded normal, untruncated lognormal, uniform, exponential, gumbel,
frechet, and weibull distributions and require specification of \c
standardized_space, for example, for QUESO \ref
method-bayes_calibration-queso-standardized_space

Note that as of Dakota 6.2, the field responses and associated field
data may be used with QUESO and DREAM.  That is, the user can specify
field simulation data and field experiment data, and Dakota will
interpolate and provide the proper residuals to the Bayesian
calibration.

Topics::	bayesian_calibration, package_queso
Examples::
Theory::
Faq::
See_Also::	
