slicesample

Slice sampler

Syntax

rnd = slicesample(initial,nsamples,'pdf',pdf)
rnd = slicesample(initial,nsamples,'logpdf',logpdf)
[rnd,neval] = slicesample(initial,...)
[rnd,neval] = slicesample(initial,...,Name,Value)


Description

rnd = slicesample(initial,nsamples,'pdf',pdf) generates nsamples random samples using the slice sampling method (see Algorithms). pdf gives the target probability density function (pdf). initial is a row vector or scalar containing the initial value of the random sample sequences.

rnd = slicesample(initial,nsamples,'logpdf',logpdf) generates samples using the logarithm of the pdf.

[rnd,neval] = slicesample(initial,...) returns the average number of function evaluations that occurred in the slice sampling.

[rnd,neval] = slicesample(initial,...,Name,Value) generates random samples with additional options specified by one or more Name,Value pair arguments.

Input Arguments

initial

Initial point, a scalar or row vector. Set initial so pdf(initial) is a strictly positive scalar. length(initial) is the number of dimensions of each sample.

nsamples

Positive integer, the number of samples that slicesample generates.

pdf	Handle to a function that generates the probability density function, specified with @. pdf can be unnormalized, meaning it need not integrate to 1.

logpdf

Handle to a function that generates the logarithm of the probability density function, specified with @. logpdf can be the logarithm of an unnormalized pdf.

Name-Value Pair Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside quotes. You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

'burnin'

Nonnegative integer, the number of samples to generate and discard before generating the samples to return. The slice sampling algorithm is a Markov chain whose stationary distribution is proportional to that of the pdf argument. Set burnin to a high enough value that you believe the Markov chain approximately reaches stationarity after burnin samples. Default: 0

'thin'

Positive integer, where slicesample discards every thin - 1 samples and returns the next. The slice sampling algorithm is a Markov chain, so the samples are serially correlated. To reduce the serial correlation, choose a larger value of thin. Default: 1

'width'

Width of the interval around the current sample, a scalar or vector of positive values. slicesample begins with this interval and searches for an appropriate region containing the points of pdf that evaluate to a large enough value. If width is a scalar and the samples have multiple dimensions, slicesample uses width for each dimension. If width is a vector, it should have the same length as initial. Default: 10

Output Arguments

rnd	nsamples-by-length(initial) matrix, where each row is one sample.

neval

Scalar, the mean number of function evaluations per sample. neval includes the burnin and thin evaluations, not just the evaluations of samples returned in rnd. Therefore the total number of function evaluations is neval*(nsamples*thin + burnin).

Examples

Generate Random Samples From a Multimodal Density

This example shows how to generate random samples from a multimodal density using slicesample.

Define a function proportional to a multimodal density.

rng default  % For reproducibility
f = @(x) exp(-x.^2/2).*(1 + (sin(3*x)).^2).*...
    (1 + (cos(5*x).^2));
area = integral(f,-5,5);

Generate 2000 samples from the density, using a burn-in period of 1000, and keeping one in five samples.

N = 2000;
x = slicesample(1,N,'pdf',f,'thin',5,'burnin',1000);

Plot a histogram of the sample.

[binheight,bincenter] = hist(x,50);
h = bar(bincenter,binheight,'hist');
h.FaceColor = [.8 .8 1];

Scale the density to have the same area as the histogram, and superimpose it on the histogram.

hold on
h = gca;
xd = h.XLim;
xgrid = linspace(xd(1),xd(2),1000);
binwidth = (bincenter(2)-bincenter(1));
y = (N*binwidth/area) * f(xgrid);
plot(xgrid,y,'r','LineWidth',2)
hold off

The samples seem to fit the theoretical distribution well, so the burnin value seems adequate.

Tips

• There are no definitive suggestions for choosing appropriate values for burnin, thin, or width. Choose starting values of burnin and thin, and increase them, if necessary, to give the requisite independence and marginal distributions. See Neal [1] for details of the effect of adjusting width.

Algorithms

At each point in the sequence of random samples, slicesample selects the next point by “slicing” the density to form a neighborhood around the previous point where the density is above some value. Consequently, the sample points are not independent. Nearby points in the sequence tend to be closer together than they would be from a sample of independent values. For many purposes, the entire set of points can be used as a sample from the target distribution. However, when this type of serial correlation is a problem, the burnin and thin parameters can help reduce that correlation.

slicesample uses the slice sampling algorithm of Neal [1]. For numerical stability, it converts a pdf function into a logpdf function. The algorithm to resize the support region for each level, called “stepping-out” and “stepping-in,” was suggested by Neal.

References

[1] Neal, Radford M. "Slice Sampling." Ann. Stat. Vol. 31, No. 3, pp. 705–767, 2003. Available at Project Euclid.