morph.metrop {mcmc} R Documentation

## 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

fY(y) = fX(f^{-1}(y)) | del f^{-1}(y) |

where fX is the pdf of X and del f^{-1} is the Jacobian of f^{-1}. Because f is a diffeomorphism, a Markov chain for fY may be transformed into a Markov chain for fX. 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 fY based on the user supplied log unnormalized density, log fX. `morph.metrop` runs a Metropolis algorithm for log fY and transforms the resulting Markov chain into a Markov chain for fX. 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 g1 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 g1 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 g2 composed with g1 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.

`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
```

[Package mcmc version 0.9-6 Index]