Lab 3: Markov chain Monte Carlo
Nial Friel
1 November 2019
Aim of this lab
In the lab we will focus on implementing MCMC for an example which we have explored in class. The
objective will be understand the role that the proposal variance plays in determining the efficiency of the
Markov chain.
A brief re-cap of MCMC
Recall that MCMC is a framework for simulating from a target distribution π(θ) by constructing a Markov
chain with whose stationary distribution is π(θ). Then, if we run the chain for long enough, simulated values
from the chain can be treated as a (dependent) sample from the target distribution and used as a basis for
summarising important features of π.
Under certain regularity conditions, the Markov chain sample path mimics a random sample from π. Given
realisations {θt : t = 0, 1, . . . } from such a chain, typical asymptotic results include the distributional
convergence of the realisations, ie
θt → π(θ)
both in distribution and as t → ∞. Also the consistency of the ergodic average, for any scalar functional φ,
MCMC留学生作业代做、代写R程序设计作业
So this leads to the question of how one can generate such a Markov chain, in general. As we’ve seen in
lectures, the Metropolis-Hastings algorithm provides a general means to do this.
Metropolis-Hastings algorithm
Recall that at iteration t of the Metropolis-Hastings algorithm one simulates a proposed state, φ from some
proposal distribution which depends on the current state, θt. We denote this proposal density by q(θt, φ).
This proposed state is then accepted with probability denoted by α(θt, φ) (as described below) and the next
state of the Markov chain becomes θt+1 = φ, otherwise, θt+1 = θt.
We algorithmically describe this algorithm as follows:
Step 1. Given the current state, θt, generate a proposed new state, φ, from the distribution q(θt, φ).
Step 2. Calculate.
Step 3. With probability α(θt, φ), set θt+1 = φ, else set θt+1 = θt.
Step 4. Return to step 1.
1
Here we write a generic function to implement the Metroplis-Hastings algorithm.
metropolis.hastings <- function(f, # the target distribution
g, # the proposal distribution
random.g, # a sample from the proposal distribution
x0, # initial value for chain, in R it is x[1]
sigma, # proposal st. dev.
chain.size=1e5){ # chain size
x <- c(x0, rep(NA,chain.size-1)) # initialize chain
for(i in 2:chain.size) {
y <- random.g(x[i-1],sigma) # generate Y from g(.|xt) using random.g
#alpha <- min(1, f(y)*g(x[i-1],y,sigma)/(f(x[i-1])*g(y,x[i-1],sigma)))
alpha <- min(1, f(y)*g(y,x[i-1],sigma)/(f(x[i-1])*g(x[i-1],y,sigma)))
if( runif(1)<alpha){x[i] <- y}
else{x[i] <- x[i-1]}}x}
A first (simple) example
Here we will re-visit a simple example which we explored in lectures where we wish to sample from a N(0, 1)
distribution using another normal distribution as a proposal distribution. Recall that the objective was
explore the effect of the variance of the proposal distribution on the efficiency of the resulting Markov chain.
To implement this using our generic code we do the following, where we first use a poor choice of proposal
variance (σ2 = 202):sigma=20
f <- function(x) dnorm(x,0,1)
random.g <- function(x,sigma) rnorm(1,x,sigma) # q(x,y)
g <- function(x,y,sigma) dnorm(y,x,sigma)
x0 <- 4
chain.size <- 1e4
x <- metropolis.hastings(f,g,random.g,x0,sigma,chain.size)
We can then explore the output of this chain as follows:
par(mfrow=c(1,2))
plot(x, xlab="x", ylab="f(x)", main=paste("Trace plot of x[t], sigma=", sigma),col="red", type="l")
xmin = min(x[(0.2*chain.size) : chain.size])
xmax = max(x[(0.2*chain.size) : chain.size])
xs.lower = min(-4,xmin)
2
xs.upper = max(4,xmax)
xs <- seq(xs.lower,xs.upper,len=1e3)
hist(x[(0.2*chain.size) : chain.size],50, xlim=c(xs.lower,xs.upper),col="blue",xlab="x", main="Metropolis-Hastings",freq=FALSE)
lines(xs,f(xs),col="red", lwd=1)
0 2000 6000 10000
Trace plot of x[t], sigma= 20
Metropolis−Hastings
We see that the chain apears not to have mixed very well (as evidenced from the trace plot). Also, the
histrogram does not yield an excellent summary of the target density (plotted in red on the right hand plot).
Now we use a much smaller proposal variance (σ2 = 0.22) and explore how this has effected the Markov chain.
sigma=0.2
f <- function(x) dnorm(x,0,1)
random.g <- function(x,sigma) rnorm(1,x,sigma) # q(x,y)
g <- function(x,y,sigma) dnorm(y,x,sigma)
x0 <- 4
chain.size <- 1e4
x <- metropolis.hastings(f,g,random.g,x0,sigma,chain.size)
We can then explore the output of this chain as follows:
par(mfrow=c(1,2))
plot(x, xlab="x", ylab="f(x)", main=paste("Trace plot of x[t], sigma=", sigma),col="red", type="l")
xmin = min(x[(0.2*chain.size) : chain.size])
xmax = max(x[(0.2*chain.size) : chain.size])
xs.lower = min(-4,xmin)
xs.upper = max(4,xmax)
xs <- seq(xs.lower,xs.upper,len=1e3)
3
hist(x[(0.2*chain.size) : chain.size],50, xlim=c(xs.lower,xs.upper),col="blue",xlab="x", main="Metropolis-Hastings",freq=FALSE)
lines(xs,f(xs),col="red", lwd=1)
0 2000 6000 10000
Trace plot of x[t], sigma= 0.2
Metropolis−Hastings
Again, the trace plot indicates that the chain has not mixed very well, although the histogram of the Markov
chain trace is in much better agreement with the target density.
Finally, we consider the case where the proposal variance is set to σ2 = 32.
sigma=3
f <- function(x) dnorm(x,0,1)
random.g <- function(x,sigma) rnorm(1,x,sigma) # q(x,y)
g <- function(x,y,sigma) dnorm(y,x,sigma)
x0 <- 4
chain.size <- 1e4
x <- metropolis.hastings(f,g,random.g,x0,sigma,chain.size)
We can then explore the output of this chain as follows:
par(mfrow=c(1,2))
plot(x, xlab="x", ylab="f(x)", main=paste("Trace plot of x[t], sigma=", sigma),col="red", type="l")
xmin = min(x[(0.2*chain.size) : chain.size])
xmax = max(x[(0.2*chain.size) : chain.size])
xs.lower = min(-4,xmin)
xs.upper = max(4,xmax)
xs <- seq(xs.lower,xs.upper,len=1e3)
hist(x[(0.2*chain.size) : chain.size],50, xlim=c(xs.lower,xs.upper),col="blue",xlab="x", main="Metropolis-Hastings",freq=FALSE)
lines(xs,f(xs),col="red", lwd=1)
Trace plot of x[t], sigma= 3
Now the output, in terms of the trace plot and histogram suggests that the algorithm is mixing well and that
the output from the Metropolis-Hastings algorithm provides a good summary of the target distribution.
5
Exercises
1. Consider again the problem explored in the lab.
• Modify the code to monitor the rate at which the proposed values are accepted, for each of the three
scenarios above (σ = 20, 0.2 or 3). Explain how this analysis can help to decide how to tune the proposal
variance. [15 marks]
• For scenario, estimate the probability that X > 2, where X ∼ N(0, 1) using the output from the
Metropolis-Hastings algorithm. Provide an approximate standard error in each case. (Note that you
can compare your estimates to the true value 0.02275) [10 marks]
2. Consider sampling from Beta(2.5, 4.5) distribution using the an independence sampler with a uniform
proposal distribution. Provide R code to do this. Produce output from your code to illustrate the
performance of your algorithm.
[25 marks]
Hand-in date: November 13th at 12pm
因为专业，所以值得信赖。如有需要，请加QQ：99515681 或邮箱：[email protected]
微信：codehelp