Metropolis and Metropolis-Hastings Algorithms

The Metropolis algorithm is named after its inventor, the American physicist and computer scientist Nicholas C. Metropolis. The algorithm is simple but practical, and it can be used to obtain random samples from any arbitrarily complicated target distribution of any dimension that is known up to a normalizing constant. The Bayesian procedures use a special case of the Metropolis algorithm called the Gibbs sampler to obtain posterior samplers.

Suppose you want to obtain samples from a univariate distribution with probability density function . Suppose is the th sample from . To use the Metropolis algorithm, you need to have an initial value and a symmetric proposal density . For the th iteration, the algorithm generates a sample from based on the current sample , and it makes a decision to either accept or reject the new sample. If the new sample is accepted, the algorithm repeats itself by starting at the new sample. If the new sample is rejected, the algorithm starts at the current point and repeats. The algorithm is self-repeating, so it can be carried out as long as required. In practice, you have to decide the total number of samples needed in advance and stop the sampler after that many iterations have been completed.

Suppose is a symmetric distribution. The proposal distribution should be an easy distribution from which to sample, and it must be such that , meaning that the likelihood of jumping to from is the same as the likelihood of jumping back to from . The most common choice of the proposal distribution is the normal distribution with a fixed . The Metropolis algorithm can be summarized as follows:

1. Set . Choose a starting point . This can be an arbitrary point as long as .

2. Generate a new sample, , by using the proposal distribution .

3. Calculate the following quantity:

4. Sample from the uniform distribution .

5. Set if ; otherwise set .

6. Set . If , the number of desired samples, return to step 2. Otherwise, stop.

Note that the number of iteration keeps increasing regardless of whether a proposed sample is accepted.

This algorithm defines a chain of random variates whose distribution will converge to the desired distribution , and so from some point forward, the chain of samples is a sample from the distribution of interest. In Markov chain terminology, this distribution is called the stationary distribution of the chain, and in Bayesian statistics, it is the posterior distribution of the model parameters. The reason that the Metropolis algorithm works is beyond the scope of this documentation, but you can find more detailed descriptions and proofs in many standard textbooks, including Roberts (1996) and Liu (2001). The random-walk Metropolis algorithm is used in PROC MCMC.

You are not limited to a symmetric random-walk proposal distribution in establishing a valid sampling algorithm. A more general form, the Metropolis-Hastings (MH) algorithm, was proposed by Hastings (1970). The MH algorithm uses an asymmetric proposal distribution: . The difference in its implementation comes in calculating the ratio of densities:

Other steps remain the same.

The extension of the Metropolis algorithm to a higher-dimensional is straightforward. Suppose is the parameter vector. To start the Metropolis algorithm, select an initial value for each and use a multivariate version of proposal distribution , such as a multivariate normal distribution, to select a -dimensional new parameter. Other steps remain the same as those previously described, and this Markov chain eventually converges to the target distribution of . Chib and Greenberg (1995) provide a useful tutorial on the algorithm.

Independence Sampler

Another type of Metropolis algorithm is the "independence" sampler. It is called the independence sampler because the proposal distribution in the algorithm does not depend on the current point as it does with the random-walk Metropolis algorithm. For this sampler to work well, you want to have a proposal distribution that mimics the target distribution and have the acceptance rate be as high as possible.

1. Set . Choose a starting point . This can be an arbitrary point as long as .

2. Generate a new sample, , by using the proposal distribution . The proposal distribution does not depend on the current value of .

3. Calculate the following quantity:

4. Sample from the uniform distribution .

5. Set if ; otherwise set .

6. Set . If , the number of desired samples, return to step 2. Otherwise, stop.

A good proposal density should have thicker tails than those of the target distribution. This requirement sometimes can be difficult to satisfy especially in cases where you do not know what the target posterior distributions are like. In addition, this sampler does not produce independent samples as the name seems to imply, and sample chains from independence samplers can get stuck in the tails of the posterior distribution if the proposal distribution is not chosen carefully. The independence sampler is used in PROC MCMC.

Gibbs Sampler

The Gibbs sampler, named by Geman and Geman (1984) after the American physicist Josiah W. Gibbs, is a special case of the Metropolis sampler in which the proposal distributions exactly match the posterior conditional distributions and proposals are accepted 100% of the time. Gibbs sampling requires you to decompose the joint posterior distribution into full conditional distributions for each parameter in the model and then sample from them. The sampler can be efficient when the parameters are not highly dependent on each other and the full conditional distributions are easy to sample from. Some researchers favor this algorithm because it does not require an instrumental proposal distribution as Metropolis methods do. However, while deriving the conditional distributions can be relatively easy, it is not always possible to find an efficient way to sample from these conditional distributions.

Suppose is the parameter vector, is the likelihood, and is the prior distribution. The full posterior conditional distribution of is proportional to the joint posterior density; that is,

For instance, the one-dimensional conditional distribution of given , is computed as the following:

The Gibbs sampler works as follows:

1. Set , and choose an arbitrary initial value of .

2. Generate each component of as follows:

   • draw from

   • draw from

   • ...

   • draw from

3. Set . If , the number of desired samples, return to step 2. Otherwise, stop.

The name "Gibbs" was introduced by Geman and Geman (1984). Gelfand et al. (1990) first used Gibbs sampling to solve problems in Bayesian inference. See Casella and George (1992) for a tutorial on the sampler. The GENMOD, LIFEREG, and PHREG procedures update parameters using the Gibbs sampler.

Adaptive Rejection Sampling Algorithm

The GENMOD, LIFEREG, and PHREG procedures use the adaptive rejection sampling (ARS) algorithm to sample parameters sequentially from their univariate full conditional distributions. The ARS algorithm is a rejection algorithm that was originally proposed by Gilks and Wild (1992). Given a log-concave density (the log of the density is concave), you can construct an envelope to the density by using linear segments. You then use the linear segment envelope as a proposal density (it becomes a piecewise exponential density on the original scale and is easy to generate samplers from) in the rejection sampling. The log-concavity condition is met in some of the models fit by the procedures. For example, the posterior densities for the regression parameters in the generalized linear models are log-concave under flat priors. When this condition fails, the ARS algorithm calls for an additional Metropolis-Hasting step (Gilks, Best, and Tan; 1995), and the modified algorithm becomes the adaptive rejection metropolis sampling (ARMS) algorithm. The GENMOD, LIFEREG, and PHREG procedures can recognize whether a model is log-concave and select the appropriate sampler for the problem at hand.

The GENMOD, LIFEREG, and PHREG procedures implement the ARMS algorithm based on code kindly provided by Walter R. Gilks, University of Leeds (Gilks; 2003), to obtain posterior samples. For a detailed description and explanation of the algorithm, see Gilks and Wild (1992) and Gilks, Best, and Tan (1995).

Burn-in, Thinning, and Markov Chain Samples

Burn-in refers to the practice of discarding an initial portion of a Markov chain sample so that the effect of initial values on the posterior inference is minimized. For example, suppose the target distribution is and the Markov chain was started at the value . The chain might quickly travel to regions around 0 in a few iterations. However, including samples around the value in the posterior mean calculation can produce substantial bias in the mean estimate. In theory, if the Markov chain is run for an infinite amount of time, the effect of the initial values decreases to zero. In practice, you do not have the luxury of infinite samples. In practice, you assume that after iterations, the chain has reached its target distribution and you can throw away the early portion and use the good samples for posterior inference. The value of is the burn-in number.

With some models you might experience poor mixing (or slow convergence) of the Markov chain. This can happen, for example, when parameters are highly correlated with each other. Poor mixing means that the Markov chain slowly traverses the parameter space (see the section Visual Analysis via Trace Plots for examples of poorly mixed chains) and the chain has high dependence. High sample autocorrelation can result in biased Monte Carlo standard errors. A common strategy is to thin the Markov chain in order to reduce sample autocorrelations. You thin a chain by keeping every th simulated draw from each sequence. You can safely use a thinned Markov chain for posterior inference as long as the chain converges. It is important to note that thinning a Markov chain can be wasteful because you are throwing away a fraction of all the posterior samples generated. MacEachern and Berliner (1994) show that you always get more precise posterior estimates if the entire Markov chain is used. However, other factors, such as computer storage or plotting time, might prevent you from keeping all samples.

To use the GENMOD, LIFEREG, MCMC, and PHREG procedures, you need to determine the total number of samples to keep ahead of time. This number is not obvious and often depends on the type of inference you want to make. Mean estimates do not require nearly as many samples as small-tail percentile estimates. In most applications, you might find that keeping a few thousand iterations is sufficient for reasonably accurate posterior inference. In all four procedures, the relationship between the number of iterations requested, the number of iterations kept, and the amount of thinning is as follows:

where is the rounding operator.