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.
either an R function or an object of class "tempering" from
a previous run. If a function, should evaluate 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
function 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 preserving 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.