morph.metrop: Morphometric Metropolis Algorithm

In mcmc: Markov Chain Monte Carlo

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

obj	see metrop.
initial	see metrop.
nbatch	see metrop.
blen	see metrop.
nspac	see metrop.
scale	see metrop.
outfun	unlike for metrop must be a function or missing; if missing the identity function, function(x) x, is used.
debug	see metrop.
morph	morph object used for transformations. See morph.
...	see metrop.

Details

morph.metrop implements morphometric methods for Markov chains. The caller specifies a log unnormalized probability density and a transformation. The transformation specified by the 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 morph, all parameters are passed to metrop, transformed when necessary. The scale parameter is not transformed.

If X is a real vector valued continuous random variable, and Y = f(X) where f is a diffeomorphism, then the pdf of Y is given by

f_Y(y) = f_X(f^{-1}(y)) | \nabla f^{-1}(y) |

where f_X is the pdf of X and \nabla f^{-1} is the Jacobian of f^{-1}. Because f is a diffeomorphism, a Markov chain for f_Y may be transformed into a Markov chain for f_X. Furthermore, these Markov chains are isomorphic (Johnson and Geyer, submitted) and have the same convergence rate. The morph variable provides a diffeomorphism, morph.metrop uses this diffeomorphism to induce the log unnormalized density, \log f_Y based on the user supplied log unnormalized density, \log f_X. morph.metrop runs a Metropolis algorithm for \log f_Y and transforms the resulting Markov chain into a Markov chain for f_X. The user accessible output components are the same as those that come from metrop, see the documentation for metrop for details.

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

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

The general idea is that a random-walk Metropolis sampler (what 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 morph) can produce geometrically ergodic chains when the tails of the log unnormalized density are too light for 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 morph with b = 0 (the default), which produces a transformation of the form g_1 in the notation used in the details section of the help for 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 g_1 behaves like a shift transformation on a ball of radius r centered at center, so these arguments to 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 r = 0 and center = 0 (the defaults) are used, but may not work as well as when r and 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 t distributions), one should use morph with r > 0 and p = 3, which produces a transformation of the form g_2 composed with g_1 in the notation used in the details section of the help for 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 t examples mentioned above. (See Johnson and Geyer, submitted, for the complete theory.)

Value

an object of class mcmc, subclass morph.metropolis. This object is a list containing all of the elements from an object returned by metrop, plus at least the following components:

morph	the morph object used for the transformations.
morph.final	the final state of the Markov chain on the transformed scale.

References

Johnson, L. T. and Geyer, C. J. (submitted) Variable Transformation to Obtain Geometric Ergodicity in the Random-walk Metropolis Algorithm.

See Also

metrop, morph.

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

mcmc documentation built on Nov. 17, 2023, 1:06 a.m.