Blurb:: Use derivatives to inform the MCMC proposal covariance.

Description::

This keyword selection results in definition of the MCMC proposal
covariance from the Hessian of the misfit function (negative log
likelihood), where this Hessian is defined from either a Gauss-Newton
approximation (using only first derivatives of the calibration terms)
or a full Hessian (using values, first derivatives, and second
derivatives of the calibration terms).  If this Hessian is
indeterminate, it will be corrected as described in  \cite TheoMan

<b> Default Behavior </b>
The default is \c prior based proposal covariance.  This is a more
advanced option that exploits structure in the form of the likelihood.

<b> Expected Output </b>

When derivatives are specified for defining the proposal covariance, the
misfit Hessian and its inverse (the MVN proposal covariance) will be
output to the standard output stream.

<b> Usage Tips </b>

The full Hessian of the misfit is used when either supported by the 
emulator in use (for PCE and surfpack GP, but not SC or dakota GP) or 
by the user's response specification (Hessian type is not "no_hessians"), 
in the case of no emulator.  When this full Hessian is indefinite and
cannot be inverted to form the proposal covariance, fallback to the
positive semi-definite Gauss-Newton Hessian is employed.

Since this proposal covariance is locally accurate, it should be
updated periodically using the \c update_period option.  While the
adaptive metropolis option can be used in combination with
derivative-based preconditioning, it is generally preferable to
instead decrease the proposal update period due to the improved local
accuracy of this approach.

Topics::	bayesian_calibration

Examples::
Generate a 2000 sample posterior chain, using derivatives to
initialize the proposal covariance at the start of the chain.
\verbatim
method,
        bayes_calibration queso
          samples = 2000 seed = 348
          delayed_rejection
          emulator pce sparse_grid_level = 2
          proposal_covariance derivatives
\endverbatim
Theory::
Faq::
See_Also::
