A Markov chain Monte Carlo example
Written by Murali Haran,
Dept. of Statistics, Penn State University
This module works through an example of the use of Markov chain
Monte Carlo for drawing samples from a multidimensional distribution
and estimating expectations with respect to this distribution. The
algorithms used to draw the samples is generally refered to as the
Metropolis-Hastings algorithm of which the Gibbs sampler is a special
case. We describe a model that is easy to specify but requires samples
from a relatively complicated distribution for which classical Monte
Carlo sampling methods are impractical. We describe how to implement a
Markov chain Monte Carlo (MCMC) algorithm for this example.
The purpose of this is twofold: First to illustrate how MCMC
algorithms are easy to implement (at least in principle) in situations
where classical Monte Carlo methods do not work and second to provide
a glimpse of practical MCMC implementation issues. It is difficult to
work through a truly complex example of a Metropolis-Hastings
algorithm in a short tutorial. Our example is therefore necessarily
simple but working through it should provide a beginning MCMC user a
taste for how to implement an MCMC procedure for a problem where
classical Monte Carlo methods are unusable.
Datasets and other files used in this tutorial:
pdf files referred to in this tutorial that give technical details:
Introduction
Monte Carlo methods are a collection of techniques that use
pseudo-random (computer simulated) values to estimate solutions to
mathematical problems. In this tutorial, we will focus on using Monte
Carlo for Bayesian inference. In particular, we will use it for the
evaluation of expectations with respect to a probability
distribution. Monte Carlo methods can also be used for a variety of
other purposes, including estimating maxima or minima of functions (as
in likelihood-based inference) but we will not discuss these here.
Monte Carlo works as follows: Suppose we want to estimate an
expectation of a function g(x) with respect to the probability
distribution f. We denote this desired quantity m= E f
g(x). Often, m is analytically intractable (the integration or
summation required is too complicated). A Monte Carlo estimate of m is
obtained by simulating N pseudo-random values from the distribution f,
say X1,X2,..,XN and simply taking the
average of g(X1),g(X2),..,g(XN) to
estimate m. As N (number of samples) gets large, the estimate
converges to the true expectation m.
A toy example to calculate P(-1 < X
< 0) when X is a Normal(0,1) random variable:
xs <- rnorm(10000) # simulate 10,000 draws from N(0,1)
xcount <- sum((xs>-1) & (xs<0)) # count number of draws between -1 and 0
xcount/10000 # Monte Carlo estimate of probability
pnorm(0)-pnorm(-1) # Compare it to R's answer (cdf at 0) - (cdf at -1)
R has random number generators for most standard distributions and
there are many more general algorithms (such as rejection sampling)
for producing independent and identically distributed (i.i.d.) draws
from f. Another, very general approach for producing non i.i.d. draws
(approximately) from f is the Metropolis-Hastings algorithm.
Aside: A powerful technique for estimating expectations is
importance sampling, where we produce draws from a different
distribution, say q, and compute a specific weighted average of these
draws to obtain estimates of expectations with respect to f.
Markov chain Monte Carlo : For complicated distributions,
producing pseudo-random i.i.d. draws from f is often infeasible. In
such cases, the Metropolis-Hastings algorithm is used to produce a
Markov chain say X1,X2,..,XN where
the Xi's are
dependent draws that are approximately from the
desired distribution. As before, the average of
g(X1),g(X2),..,g(XN) is an estimate
that converges to m as N gets large. The
Metropolis-Hastings algorithm is very general and hence very
useful. In the following example we will see how it can be used for
inference for a model/problem where it would otherwise be impossible
to compute desired expectations.
Problem and model description
Our example uses a
dataset from the Chandra Orion Ultradeep Project (COUP). More
information on this is available at:
CASt Chandra Flares data set .
The raw data, which arrives approximately according to a Poisson
process, gives the individual photon arrival times (in seconds) and
their energies (in keV). The processed data we consider here is
obtained by grouping the events into evenly-spaced time bins (10,000
seconds width).
Our goal for this data analysis is to identify the change point and
estimate the intensities of the Poisson process before and after the
change point.
We describe a Bayesian model for this change point problem (Carlin
and Louis, 2000). Let Yt be the number of occurrences of some event at
time t. The process is observed for times 1 through n and we assume
that there is a change at time k, i.e., after time k, the event counts
are significantly different (higher or lower than before). The
mathematical description of the model is provided in change point model (pdf)
. While this is a simple model, it is adequate for illustrating
some basic principles for constructing an MCMC algorithm.
We first read in the data:
chptdat <- read.table("COUP551_rates.dat",skip=1, head=T)
We can begin with a simple time series plot as exploratory
analysis.
Y <- chptdat[,2] # store data in Y
ts.plot(Y,main="Time series plot of change point data")
The plot suggests that the change point may be around 10.
Setting up the MCMC algorithm
Our goal is to simulate
multiple draws from the posterior distribution which is a
multidimensional distribution known only upto a (normalizing)
constant. From this multidimensional distribution, we can easily
derive the conditional distribution of each of the individual
parameters (one dimension at a time). This is described, along with
a description of the Metropolis-Hastings algorithm in full conditional distributions and M-H algorithm (pdf) .
Programming an MCMC algorithm in R
We will need an editor for our program. For instance, we can use
Wordpad (available under the Start button menu under
Accessories). Ideally, a more `intelligent' editor such as emacs (with
ESS or emacs speaks statistics installed) should be used to edit R
programs.
Please save code from the
MCMC template in R into a file and open this
file using the editor. Save this file as MCMCchpt.R .
Note that in this version of the code, all parameters are sampled except for k
(which is fixed at our guessed change point).
To load the program from the file MCMCchpt.R we use the "source" command.
(Reminder: It may be helpful to type: setwd("V:/") to set the
default directory to the place where you can save your files)
source("MCMCchpt.R") # with appropriate filepathname
We can now run the MCMC algorithm:
mchain <- mhsampler(NUMIT=1000,dat=Y) # call the function with appropriate arguments
MCMC output analysis
Now that we have output from our sampler, we can treat these
samples as data from which we can estimate quantities of
interest. For instance, to estimate the expectation of a marginal
distribution for a particular parameter, we would simply average all
draws for that parameter so to obtain an estimate of E(theta):
mean(mchain[1,]) # obtain mean of first row (thetas)
To get estimates for means for all parameters:
apply(mchain,1,mean) # compute means by row (for all parameters at once)
apply(mchain,1,median) # compute medians by row (for all parameters at once)
To obtain an estimate of the entire posterior distribution:
plot(density(mchain[1,]),main="smoothed density plot for theta posterior")
plot(density(mchain[2,]),main="smoothed density plot for lambda posterior")
hist(mchain[3,],main="histogram for k posterior")
To find the (posterior) probability that lambda is greater than 10:
sum(mchain[2,]>10)/length(mchain[2,])
Now comment the line that fixes k at our guess (add the # mark):
# currk <- KGUESS
Rerun the sampler with k also sampled.
mchain <- mhsampler(NUMIT=1000,dat=Y)
With the new output, you
can repeat the calculations above (finding means, plotting density
estimates etc.)
You can also study how your estimate for the expectation of the posterior
distribution for k changes with each iteration.
estvssamp(mchain[3,])
We would like to assess whether our Markov chain is moving around
quickly enough to produce good estimates (this property is often
called 'good mixing'). While this is in general difficult to do
rigorously,
estimates of the autocorrelation in the samples is an informal but
useful check. To obtain sample autocorrelations we use the acf plot function:
acf(mchain[1,],main="acf plot for theta")
acf(mchain[2,],main="acf plot for lambda")
acf(mchain[3,],main="acf plot for k")
acf(mchain[4,],main="acf plot for b1")
acf(mchain[5,],main="acf plot for b2")
If the samples are heavily autocorrelated we should rethink our
sampling scheme or, at the very least, run the chain for much
longer. Note that the autocorrelations are negligible for all
parameters except k which is heavily autocorrelated. This is easily
resolved for this example since the sampler is fast (we can run the
chain much longer very easily). In problems where producing additional
samples is more time consuming, such as complicated high dimensional
problems, improving the sampler `mixing' can be much more critical.
Why are there such strong autocorrelations for k? The acceptance rate
for k proposals (printed out with each MCMC run) are well below 10%
which suggests that k values are stagnant more than 90% of the time. A
better proposal for the Metropolis-Hastings update of a parameter can
help improve acceptance rates which often, in turn, reduces
autocorrelations. Try another proposal for k and see how it affects
autocorrelations. In complicated problems, carefully constructed
proposals can have a major impact on the efficiency of the MCMC
algorithm.
How do we choose starting values? In general, any value we
believe would be reasonable under the posterior distribution will
suffice. You can experiment with different starting values. For
instance: modify the starting value for k in the function (for
instance, try setting k=10), "source" the function in R and run the
sampler again as follows:
mchain2 <- mhsampler(NUMIT=1000,dat=Y)
You can study how your estimate for the expectation of the posterior
distribution for k changes with each iteration.
estvssamp(mchain2[3,])
Assessing accuracy and determining chain length
There are two
important issues to consider when we have draws from an MCMC
algorithm: (1) how do we assess the accuracy of our estimates based
on the sample (how do we compute Monte Carlo standard errors?) (2)
how long do we run the chain before we feel confident that our
results are reasonably accurate ?
Regarding (1): Computing standard errors for a Monte Carlo estimate for
an i.i.d. (classical Monte Carlo) sampler is easy, as shown for the
toy example on estimating P(-1 < X < 0) when X is a Normal(0,1) random
variable. Simply obtain the sample standard deviation of the g(xi)
values and divide by square root of n (the number of samples). Since
Markov chains produce dependent draws, computing precise Monte Carlo
standard errors for such samplers is a very difficult problem in
general. For (2): Draws produced by classical Monte Carlo methods
(such as rejection sampling) produced draws from the correct
distribution. The MCMC algorithm produces draws that are
asymptotically from the correct distribution. All the draws we see
after a finite number of iterations are therefore only approximately
from the correct distribution. Determining how long we have to run
the chain before we feel sufficiently confident that the MCMC
algorithm has produced reasonably accurate draws from the
distribution is therefore a very difficult problem. Most rigorous
solutions are too specific or tailored towards relatively simple
situations while more general approaches tend to be heuristic.
There are many ways to compute Monte Carlo standard errors. See, for instance, Practical Markov chain Monte Carlo and the
references therein. We describe a simple but reasonable way of calculating it: the consistent batch means method in R and a brief description (pdf) .
To compute MC s.error via batch means, download the bm
function from the batchmeans.R file above and source the file into
R. We can now calculate standard error estimates for each of the five
parameter estimates:
bm(mchain[1,])
bm(mchain[2,])
bm(mchain[3,])
bm(mchain[4,])
bm(mchain[5,])
Are these standard errors acceptable ?
There is a vast literature on different proposals for dealing with the
latter issue (how long to run the chain) but they are all heuristics
at best. The links at the bottom of this page (see section titled
"Some resources") provide references to learn more about suggested
solutions. One method that is fairly simple, theoretically justified
in some cases and seems to work reasonably well in practice is as
follows: run the MCMC algorithm and periodically
compute Monte Carlo standard errors. Once the Monte Carlo standard
errors are below some (user-defined) threshold, stop the
simulation.
Often MCMC users do not run their simulations long enough. For
complicated problems run lengths in the millions (or more) are
typically suggested (although this may not always be feasible). For
our example run the MCMC algorithm again, this time for 100000
iterations (set NUMIT=100000).
mchain2 <- mhsampler(NUMIT=100000,dat=Y)
You can now obtain estimates of the posterior
distribution of the parameters as before and compute the new Monte
Carlo standard error. Note whether the estimates and corresponding MC
standard error have changed with respect to the previous sampler.
Making changes to the model
If we were to change the prior distributions on some of the individual
parameters, only relatively minor changes may be needed in the
program. For instance if the Inverse Gamma prior on b1 and b2 were
replaced by Gamma(0.01,100) priors on them, we would only have to
change the lines in the code corresponding to the updates of b1 and b2
(we would need to perform a Metropolis-Hastings update of each
parameter). The rest of the code would remain unchanged. Modifying the
program to make it sample from the posterior for the modified model is a useful exercise. For
the modified full conditionals see modified
full conditional distributions.
An obvious modification to this model would be to allow for more than
one change point. A very sophisticated model that may be useful in
many change point problems is one where the number of change points is
also treated as unknown. In this case the number of Poisson
parameters (only two of them in our example: theta and lambda) is also
unknown. The posterior distribution is then a mixture over
distributions of varying dimensions (the dimensions change with the
number of change points in the model). This requires an advanced
version of the Metropolis-Hastings algorithm known as
reversible-jump Metropolis Hastings due to Peter Green. Some
related information is available at the HSSS
variable dimension MCMC workshop
Some resources
The
"CODA" and BOA packages in R implement many well known output analysis
techniques. Charlie Geyer's MCMC package
in R is another free resource. There is also MCMC software from
the popular
WINBugs project .
In
addition to deciding how long to run the sampler and how to compute
Monte Carlo standard error, there are many possibilities for choosing
how to update the parameters and more sophisticated methods used to
make the Markov chain move around the posterior distribution
efficiently. The literature on such methods is vast. The following references are a useful starting point.
Acknowledgment: The model is borrowed from Chapter 5 of "Bayes and
Empirical Bayes Methods for Data Analysis" by Carlin and Louis (2000).
The data example was provided by Konstantin Getman (Penn State University).