File: morph.metrop.Rd

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\name{morph.metrop}
\alias{morph.metrop}
\alias{morph.metrop.function}
\alias{morph.metrop.morph.metropolis}
\title{Morphometric Metropolis Algorithm}
\description{
  Markov chain Monte Carlo for continuous random vector using a
  Metropolis algorithm for an induced density.
}
\usage{
morph.metrop(obj, initial, nbatch, blen = 1, nspac = 1, scale = 1,
  outfun, debug = FALSE, morph, ...)
}
\arguments{
  \item{obj}{see \code{\link{metrop}}.}
  \item{initial}{see \code{\link{metrop}}.}
  \item{nbatch}{see \code{\link{metrop}}.}
  \item{blen}{see \code{\link{metrop}}.}
  \item{nspac}{see \code{\link{metrop}}.}
  \item{scale}{see \code{\link{metrop}}.}
  \item{outfun}{unlike for \code{\link{metrop}} must be a function or missing;
    if missing the identity function, \code{function(x) x}, is used.}
  \item{debug}{see \code{\link{metrop}}.}
  \item{morph}{morph object used for transformations.  See \code{\link{morph}}.}
  \item{...}{see \code{\link{metrop}}.}
}
\details{
  \code{morph.metrop} implements morphometric methods for Markov
  chains.  The caller specifies a log unnormalized probability density
  and a transformation.  The transformation specified by the
  \code{morph} parameter is used to induce a new log unnormalized
  probability   density, a Metropolis algorithm is
  run for the induced density.  The Markov chain is transformed back to
  the original scale.  Running the Metropolis algorithm for the induced
  density, instead of the original density, can result in a Markov chain
  with better convergence properties.  For more details see Johnson and Geyer
  (submitted).  Except for \code{morph}, all parameters are
  passed to \code{\link{metrop}}, transformed when necessary.  The
  \code{scale} parameter is \emph{not} transformed.

  If \eqn{X} is a real vector valued continuous random variable, and
  \eqn{Y = f(X)} where \eqn{f} is a diffeomorphism, then the pdf of
  \eqn{Y} is given by \deqn{f_Y(y) = f_X(f^{-1}(y)) | \nabla f^{-1}(y)
  |}{ fY(y) = fX(f^{-1}(y)) | del f^{-1}(y) |} where \eqn{f_X}{fX} is
  the pdf of \eqn{X} and \eqn{\nabla f^{-1}}{del f^{-1}} is the Jacobian
  of \eqn{f^{-1}}.  Because \eqn{f} is a diffeomorphism, a Markov chain
  for \eqn{f_Y}{fY} may be transformed into a Markov chain for
  \eqn{f_X}{fX}.  Furthermore, these Markov chains are isomorphic
  (Johnson and Geyer, submitted) and have the same convergence rate.
  The \code{\link{morph}} variable provides a diffeomorphism,
  \code{morph.metrop} uses this diffeomorphism to induce the log
  unnormalized density, \eqn{\log f_Y}{log fY} based on the user
  supplied log unnormalized density, \eqn{\log f_X}{log fX}.
  \code{morph.metrop} runs a Metropolis algorithm for \eqn{\log f_Y}{log
  fY} and transforms the resulting Markov chain into a Markov chain for
  \eqn{f_X}{fX}.  The user accessible output components are the same as
  those that come from \code{\link{metrop}}, see the documentation for
  \code{\link{metrop}} for details.

  Subsequent calls of \code{morph.metrop} may change to the
  transformation by specifying a new value for \code{morph}.

  Any of the other parameters to \code{morph.metrop} may also be
  modified in subsequent calls.  See \code{\link{metrop}} for more details.

  The general idea is that a random-walk Metropolis sampler
  (what \code{\link{metrop}} does) will not be geometrically
  ergodic unless the tails of the unnormalized density decrease
  superexponentially fast (so the tails of the log unnormalized density
  decrease faster than linearly).  It may not be geometrically ergodic
  even then (see Johnson and Geyer, submitted, for the complete theory).
  The transformations used by this function (provided by \code{\link{morph}})
  can produce geometrically ergodic chains when the tails of the log
  unnormalized density are too light for \code{\link{metrop}} to do so.

  When the tails of the unnormalized density are exponentially light but
  not superexponentially light (so the tails of the log unnormalized density
  are asymptotically linear, as in the case of exponential family models
  when conjugate priors are used, for example logistic regression, Poisson
  regression with log link, or log-linear models for categorical data), one
  should use \code{\link{morph}} with \code{b = 0} (the default), which
  produces a transformation of the form \eqn{g_1}{g1} in the notation
  used in the details section of the help for \code{\link{morph}}.
  This will produce a geometrically ergodic sampler if other features of the
  log unnormalized density are well behaved.  For example it will do so
  for the exponential family examples mentioned above.
  (See Johnson and Geyer, submitted, for the complete theory.)

  The transformation \eqn{g_1}{g1} behaves like a shift transformation
  on a ball of radius \code{r} centered at \code{center}, so these arguments
  to \code{\link{morph}} should be chosen so that a sizable proportion of
  the probability under the original (untransformed) unnormalized density
  is contained in this ball.  This function will work when \code{r = 0} and
  \code{center = 0} (the defaults) are used, but may not work as well as when
  \code{r} and \code{center} are well chosen.

  When the tails of the unnormalized density are not exponentially light
  (so the tails of the log unnormalized density decrease sublinearly, as
  in the case of univariate and multivariate \eqn{t} distributions), one
  should use \code{\link{morph}} with \code{r > 0} and \code{p = 3}, which
  produces a transformation of the form \eqn{g_2}{g2} composed
  with \eqn{g_1}{g1} in the notation
  used in the details section of the help for \code{\link{morph}}.
  This will produce a geometrically ergodic sampler if other features of the
  log unnormalized density are well behaved.  For example it will do so
  for the \eqn{t} examples mentioned above.
  (See Johnson and Geyer, submitted, for the complete theory.)
}
\value{
  an object of class \code{mcmc}, subclass \code{morph.metropolis}.
  This object is a list containing all of the elements from an object
  returned by \code{\link{metrop}}, plus at least the following
  components:
  \item{morph}{the morph object used for the transformations.}
  \item{morph.final}{the final state of the Markov chain on the
    transformed scale.}
}
\examples{
out <- morph.metrop(function(x) dt(x, df=3, log=TRUE), 0, blen=100,
  nbatch=100, morph=morph(b=1))
# change the transformation.
out <- morph.metrop(out, morph=morph(b=2))
out$accept
# accept rate is high, increase the scale.
out <- morph.metrop(out, scale=4)
# close to 0.20 is about right.
out$accept
}
\references{
Johnson, L. T. and Geyer, C. J. (submitted)
Variable Transformation to Obtain Geometric Ergodicity
    in the Random-walk Metropolis Algorithm.
}
\seealso{
  \code{\link{metrop}}, \code{\link{morph}}.
}
\keyword{misc}