R: Simulated Tempering and Umbrella Sampling

temper {mcmc}

R Documentation

Simulated Tempering and Umbrella Sampling

Description

Markov chain Monte Carlo for continuous random vectors using parallel or serial simulated tempering, also called umbrella sampling. For serial tempering the state of the Markov chain is a pair (i, x), where i is an integer between 1 and k and x is a vector of length p. This pair is represented as a single real vector c(i, x). For parallel tempering the state of the Markov chain is vector of vectors (x[1], ..., x[k]), where each x is of length p. This vector of vectors is represented as a k by p matrix.

Usage

temper(obj, initial, neighbors, nbatch, blen = 1, nspac = 1, scale = 1,
    outfun, debug = FALSE, parallel = FALSE, ...)

Arguments

`obj`	either an R function or an object of class `"tempering"` from a previous run. If a function, should evalutate the log unnormalized density log h(i, x) of the desired equilibrium distribution of the Markov chain for serial tempering (the same function is used for both serial and parallel tempering, see details below for further explanation). If an object, the log unnormalized density function is `obj$lud`, and missing arguments of `temper` are obtained from the corresponding elements of `obj`. The first argument of the log unnormalized density function is the state for simulated tempering (i, x) is supplied as an R vector `c(i, x)`; other arguments are arbitrary and taken from the `...` arguments of `temper`. The log unnormalized density functon should return `-Inf` for points of the state space having probability zero.
`initial`	for serial tempering, a real vector `c(i, x)` as described above. For parallel tempering, a real k by p matrix as described above. In either case, the initial state of the Markov chain.
`neighbors`	a logical symmetric matrix of dimension `k` by `k`. Elements that are `TRUE` indicate jumps or swaps attempted by the Markov chain.
`nbatch`	the number of batches.
`blen`	the length of batches.
`nspac`	the spacing of iterations that contribute to batches.
`scale`	controls the proposal step size for real elements of the state vector. For serial tempering, proposing a new value for the x part of the state (i, x). For parallel tempering, proposing a new value for the x[i] part of the state (x[1], ..., x[k]). In either case, the proposal is a real vector of length p. If scalar or vector, the proposal is `x + scale * z` where `x` is the part x or x[i] of the state the proposal may replace. If matrix, the proposal is `x + scale %% z`. If list, the length must be `k`, and each element must be scalar, vector, or matrix, and operate as described above. The i-th component of the list is used to update x when the state is (i, x)* or x[i] otherwise.
`outfun`	controls the output. If a function, then the batch means of `outfun(state, ...)` are returned. The argument `state` is like the argument `initial` of this function. If missing, the batch means of the real part of the state vector or matrix are returned, and for serial tempering the batch means of a multivariate Bernoulli indicating the current component are returned.
`debug`	if `TRUE` extra output useful for testing.
`parallel`	if `TRUE` does parallel tempering, if `FALSE` does serial tempering.
`...`	additional arguments for `obj` or `outfun`.

Details

Serial tempering simulates a mixture of distributions of a continuous random vector. The number of components of the mixture is k, and the dimension of the random vector is p. Denote the state (i, x), where i is a positive integer between 1 and k, and let h(i, x) denote the unnormalized joint density of their equilibrium distribution. The logarithm of this function is what obj or obj$lud calculates. The mixture distribution is the marginal for x derived from the equilibrium distribution h(i, x), that is,

h(x) = sum[i = 1 to k] h(i, x)

Parallel tempering simulates a product of distributions of a continuous random vector. Denote the state (x[1], ..., x[k]), then the unnormalized joint density of the equilibrium distribution is

h(x[1], ..., x[k]) = prod[i = 1 to k] h(i, x[i])

The update mechanism of the Markov chain combines two kinds of elementary updates: jump/swap updates (jump for serial tempering, swap for parallel tempering) and within-component updates. Each iteration of the Markov chain one of these elementary updates is done. With probability 1/2 a jump/swap update is done, and with probability 1/2 a with-component update is done.

Within-component updates are the same for both serial and parallel tempering. They are “random-walk” Metropolis updates with multivariate normal proposal, the proposal distribution being determined by the argument scale. In serial tempering, the x part of the current state (i, x) is updated preserving h(i, x). In parallel tempering, an index i is chosen at random and the part of the state x[i] representing that component is updated, again preserving h(i, x).

Jump updates choose uniformly at random a neighbor of the current component: if i indexes the current component, then it chooses uniformly at random a j such that neighbors[i, j] == TRUE. It then does does a Metropolis-Hastings update for changing the current state from (i, x) to (j, x).

Swap updates choose a component uniformly at random and a neighbor of that component uniformly at random: first an index i is chosen uniformly at random between 1 and k, then an index j is chosen uniformly at random such that neighbors[i, j] == TRUE. It then does does a Metropolis-Hastings update for swapping the states of the two components: interchanging x[i, ] and x[j, ] while perserving h(x[1], ..., x[k]).

The initial state must satisfy lud(initial, ...) > - Inf for serial tempering or must satisfy lud(initial[i, ], ...) > - Inf for each i for parallel tempering, where lud is either obj or obj$lud. That is, the initial state must have positive probability.

Value

an object of class "mcmc", subclass "tempering", which is a list containing at least the following components:

`batch`	the batch means of the continuous part of the state. If `outfun` is missing, an `nbatch` by `k` by `p` array. Otherwise, an `nbatch` by `m` matrix, where `m` is the length of the result of `outfun`.
`ibatch`	(returned for serial tempering only) an `nbatch` by `k` matrix giving batch means for the multivariate Bernoulli random vector that is all zeros except for a 1 in the `i`-th place when the current state is (i, x).
`acceptx`	fraction of Metropolis within-component proposals accepted. A vector of length `k` giving the acceptance rate for each component.
`accepti`	fraction of Metropolis jump/swap proposals accepted. A `k` by `k` matrix giving the acceptance rate for each allowed jump or swap component. `NA` for elements such that the corresponding elements of `neighbors` is `FALSE`.
`initial`	value of argument `initial`.
`final`	final state of Markov chain.
`initial.seed`	value of `.Random.seed` before the run.
`final.seed`	value of `.Random.seed` after the run.
`time`	running time of Markov chain from `system.time()`.
`lud`	the function used to calculate log unnormalized density, either `obj` or `obj$lud` from a previous run.
`nbatch`	the argument `nbatch` or `obj$nbatch`.
`blen`	the argument `blen` or `obj$blen`.
`nspac`	the argument `nspac` or `obj$nspac`.
`outfun`	the argument `outfun` or `obj$outfun`.

Description of additional output when debug = TRUE can be found in the vignette debug (../doc/debug.pdf).

Warning

If outfun is missing, then the log unnormalized density function can be defined without a ... argument and that works fine. One can define it starting ludfun <- function(state) and that works or ludfun <- function(state, foo, bar), where foo and bar are supplied as additional arguments to temper and that works too.

If outfun is a function, then both it and the log unnormalized density function can be defined without ... arguments if they have exactly the same arguments list and that works fine. Otherwise it doesn't work. Start the definitions ludfun <- function(state, foo) and outfun <- function(state, bar) and you get an error about unused arguments. Instead start the definitions ludfun <- function(state, foo, ...) and outfun <- function(state, bar, ...), supply foo and bar as additional arguments to temper, and that works fine.

In short, the log unnormalized density function and outfun need to have ... in their arguments list to be safe. Sometimes it works when ... is left out and sometimes it doesn't.

Of course, one can avoid this whole issue by always defining the log unnormalized density function and outfun to have only one argument state and use global variables (objects in the R global environment) to specify any other information these functions need to use. That too follows the R way. But some people consider that bad programming practice.

Examples

d <- 9
witch.which <- c(0.1, 0.3, 0.5, 0.7, 1.0)
ncomp <- length(witch.which)

neighbors <- matrix(FALSE, ncomp, ncomp)
neighbors[row(neighbors) == col(neighbors) + 1] <- TRUE
neighbors[row(neighbors) == col(neighbors) - 1] <- TRUE

ludfun <- function(state, log.pseudo.prior = rep(0, ncomp)) {
    stopifnot(is.numeric(state))
    stopifnot(length(state) == d + 1)
    icomp <- state[1]
    stopifnot(icomp == as.integer(icomp))
    stopifnot(1 <= icomp && icomp <= ncomp)
    stopifnot(is.numeric(log.pseudo.prior))
    stopifnot(length(log.pseudo.prior) == ncomp)
    theta <- state[-1]
    if (any(theta > 1.0)) return(-Inf)
    bnd <- witch.which[icomp]
    lpp <- log.pseudo.prior[icomp]
    if (any(theta > bnd)) return(lpp)
    return(- d * log(bnd) + lpp)
}

# parallel tempering
thetas <- matrix(0.5, ncomp, d)
out <- temper(ludfun, initial = thetas, neighbors = neighbors, nbatch = 20,
    blen = 10, nspac = 5, scale = 0.56789, parallel = TRUE, debug = TRUE)

# serial tempering
theta.initial <- c(1, rep(0.5, d))
# log pseudo prior found by trial and error
qux <- c(0, 9.179, 13.73, 16.71, 20.56)

out <- temper(ludfun, initial = theta.initial, neighbors = neighbors,
    nbatch = 50, blen = 30, nspac = 2, scale = 0.56789,
    parallel = FALSE, debug = FALSE, log.pseudo.prior = qux)

[Package mcmc version 0.7-3 Index]