# Adaptive Metropolis within Gibbs algorithm Markov chain Monte Carlo for continuous random vector using an adaptive Metropolis within Gibbs algorithm. ```r adaptMetropGibbs(ltd, starting, tuning=1, accept.rate=0.44, batch = 1, batch.length=25, report=100, verbose=TRUE, ...) ``` ## Arguments - `ltd`: an function that evaluates the log target density of the desired equilibrium distribution of the Markov chain. First argument is the starting value vector of the Markov chain. Pass variables used in the `ltd` via the argument of `aMetropGibbs`. - `starting`: a real vector of parameter starting values. - `tuning`: a scalar or vector of initial Metropolis tuning values. The vector must be of `length(starting)`. If a scalar is passed then it is expanded to `length(starting)`. - `accept.rate`: a scalar or vector of target Metropolis acceptance rates. The vector must be of `length(starting)`. If a scalar is passed then it is expanded to `length(starting)`. - `batch`: the number of batches. - `batch.length`: the number of sampler iterations in each batch. - `report`: the number of batches between acceptance rate reports. - `verbose`: if `TRUE`, progress of the sampler is printed to the screen. Otherwise, nothing is printed to the screen. - `...`: currently no additional arguments. ## Returns A list with the following tags: - **p.theta.samples**: a `coda` object of posterior samples for the parameters. - **acceptance**: the Metropolis acceptance rate at the end of each batch. - **ltd**: `ltd` - **accept.rate**: `accept.rate` - **batch**: `batch` - **batch.length**: `batch.length` - **proc.time**: the elapsed CPU and wall time (in seconds). ## References Roberts G.O. and Rosenthal J.S. (2006). Examples of Adaptive MCMC. [http://probability.ca/jeff/ftpdir/adaptex.pdf](http://probability.ca/jeff/ftpdir/adaptex.pdf) Preprint. Rosenthal J.S. (2007). AMCMC: An R interface for adaptive MCMC. **Computational Statistics and Data Analysis**. 51:5467-5470. ## Note This function is a rework of Rosenthal (2007) with some added niceties. ## Author(s) Andrew O. Finley finleya@msu.edu , Sudipto Banerjee sudiptob@biostat.umn.edu ## Examples ```r ## Not run: rmvn <- function(n, mu=0, V = matrix(1)){ p <- length(mu) if(any(is.na(match(dim(V),p)))) stop("Dimension problem!") D <- chol(V) t(matrix(rnorm(n*p), ncol=p)%*%D + rep(mu,rep(n,p))) } ########################### ##Fit a spatial regression ########################### set.seed(1) n <- 50 x <- runif(n, 0, 100) y <- runif(n, 0, 100) D <- as.matrix(dist(cbind(x, y))) phi <- 3/50 sigmasq <- 50 tausq <- 20 mu <- 150 s <- (sigmasq*exp(-phi*D)) w <- rmvn(1, rep(0, n), s) Y <- rmvn(1, rep(mu, n) + w, tausq*diag(n)) X <- as.matrix(rep(1, length(Y))) ##Priors ##IG sigma^2 and tau^2 a.sig <- 2 b.sig <- 100 a.tau <- 2 b.tau <- 100 ##Unif phi a.phi <- 3/100 b.phi <- 3/1 ##Functions used to transform phi to continuous support. logit <- function(theta, a, b){log((theta-a)/(b-theta))} logit.inv <- function(z, a, b){b-(b-a)/(1+exp(z))} ##Metrop. target target <- function(theta){ mu.cand <- theta[1] sigmasq.cand <- exp(theta[2]) tausq.cand <- exp(theta[3]) phi.cand <- logit.inv(theta[4], a.phi, b.phi) Sigma <- sigmasq.cand*exp(-phi.cand*D)+tausq.cand*diag(n) SigmaInv <- chol2inv(chol(Sigma)) logDetSigma <- determinant(Sigma, log=TRUE)$modulus[1] out <- ( ##Priors -(a.sig+1)*log(sigmasq.cand) - b.sig/sigmasq.cand -(a.tau+1)*log(tausq.cand) - b.tau/tausq.cand ##Jacobians +log(sigmasq.cand) + log(tausq.cand) +log(phi.cand - a.phi) + log(b.phi -phi.cand) ##Likelihood -0.5*logDetSigma-0.5*(t(Y-X%*%mu.cand)%*%SigmaInv%*%(Y-X%*%mu.cand)) ) return(out) } ##Run a couple chains n.batch <- 500 batch.length <- 25 inits <- c(0, log(1), log(1), logit(3/10, a.phi, b.phi)) chain.1 <- adaptMetropGibbs(ltd=target, starting=inits, batch=n.batch, batch.length=batch.length, report=100) inits <- c(500, log(100), log(100), logit(3/90, a.phi, b.phi)) chain.2 <- adaptMetropGibbs(ltd=target, starting=inits, batch=n.batch, batch.length=batch.length, report=100) ##Check out acceptance rate just for fun plot(mcmc.list(mcmc(chain.1$acceptance), mcmc(chain.2$acceptance))) ##Back transform chain.1$p.theta.samples[,2] <- exp(chain.1$p.theta.samples[,2]) chain.1$p.theta.samples[,3] <- exp(chain.1$p.theta.samples[,3]) chain.1$p.theta.samples[,4] <- 3/logit.inv(chain.1$p.theta.samples[,4], a.phi, b.phi) chain.2$p.theta.samples[,2] <- exp(chain.2$p.theta.samples[,2]) chain.2$p.theta.samples[,3] <- exp(chain.2$p.theta.samples[,3]) chain.2$p.theta.samples[,4] <- 3/logit.inv(chain.2$p.theta.samples[,4], a.phi, b.phi) par.names <- c("mu", "sigma.sq", "tau.sq", "effective range (-log(0.05)/phi)") colnames(chain.1$p.theta.samples) <- par.names colnames(chain.2$p.theta.samples) <- par.names ##Discard burn.in and plot and do some convergence diagnostics chains <- mcmc.list(mcmc(chain.1$p.theta.samples), mcmc(chain.2$p.theta.samples)) plot(window(chains, start=as.integer(0.5*n.batch*batch.length))) gelman.diag(chains) ########################## ##Example of fitting a ##a non-linear model ########################## ##Example of fitting a non-linear model set.seed(1) ######################################################## ##Simulate some data. ######################################################## a <- 0.1 #-Inf < a < Inf b <- 0.1 #b > 0 c <- 0.2 #c > 0 tau.sq <- 0.1 #tau.sq > 0 fn <- function(a,b,c,x){ a+b*exp(x/c) } n <- 200 x <- seq(0,1,0.01) y <- rnorm(length(x), fn(a,b,c,x), sqrt(tau.sq)) ##check out your data plot(x, y) ######################################################## ##The log target density ######################################################## ##Define the log target density used in the Metrop. ltd <- function(theta){ ##extract and transform as needed a <- theta[1] b <- exp(theta[2]) c <- exp(theta[3]) tau.sq <- exp(theta[4]) y.hat <- fn(a, b, c, x) ##likelihood logl <- sum(dnorm(y, y.hat, sqrt(tau.sq), log=TRUE)) ##priors IG on tau.sq and normal on a and transformed b, c, d logl <- (logl -(IG.a+1)*log(tau.sq)-IG.b/tau.sq +sum(dnorm(theta[1:3], N.mu, N.v, log=TRUE)) ) ##Jacobian adjustment for tau.sq logl <- logl+log(tau.sq) return(logl) } ######################################################## ##The rest ######################################################## ##Priors IG.a <- 2 IG.b <- 0.01 N.mu <- 0 N.v <- 10 theta.tuning <- c(0.01, 0.01, 0.005, 0.01) ##Run three chains with different starting values n.batch <- 1000 batch.length <- 25 theta.starting <- c(0, log(0.01), log(0.6), log(0.01)) run.1 <- adaptMetropGibbs(ltd=ltd, starting=theta.starting, tuning=theta.tuning, batch=n.batch, batch.length=batch.length, report=100) theta.starting <- c(1.5, log(0.05), log(0.5), log(0.05)) run.2 <- adaptMetropGibbs(ltd=ltd, starting=theta.starting, tuning=theta.tuning, batch=n.batch, batch.length=batch.length, report=100) theta.starting <- c(-1.5, log(0.1), log(0.4), log(0.1)) run.3 <- adaptMetropGibbs(ltd=ltd, starting=theta.starting, tuning=theta.tuning, batch=n.batch, batch.length=batch.length, report=100) ##Back transform samples.1 <- cbind(run.1$p.theta.samples[,1], exp(run.1$p.theta.samples[,2:4])) samples.2 <- cbind(run.2$p.theta.samples[,1], exp(run.2$p.theta.samples[,2:4])) samples.3 <- cbind(run.3$p.theta.samples[,1], exp(run.3$p.theta.samples[,2:4])) samples <- mcmc.list(mcmc(samples.1), mcmc(samples.2), mcmc(samples.3)) ##Summary plot(samples, density=FALSE) gelman.plot(samples) burn.in <- 5000 fn.pred <- function(theta,x){ a <- theta[1] b <- theta[2] c <- theta[3] tau.sq <- theta[4] rnorm(length(x), fn(a,b,c,x), sqrt(tau.sq)) } post.curves <- t(apply(samples.1[burn.in:nrow(samples.1),], 1, fn.pred, x)) post.curves.quants <- summary(mcmc(post.curves))$quantiles plot(x, y, pch=19, xlab="x", ylab="f(x)") lines(x, post.curves.quants[,1], lty="dashed", col="blue") lines(x, post.curves.quants[,3]) lines(x, post.curves.quants[,5], lty="dashed", col="blue") ## End(Not run) ```

adaptMetropGibbs function