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See the chain run! The green dot shows the current position of the chain; the green line connects the previous position to the current position. The chain proceeds by alternately updating the x and y coordinates, uniformly from all choices which remain underneath the (blue) graph of the target function.

With a=1, the target is a standard Exp(1) distribution, so the mean should be (gradually!) converging to 1. For larger values of a, the convergence should be at least as good. But note that for very small values of a, the sampler is much less stable.

The applet accepts the following keyboard inputs. (You may need to "click" on the applet first.)

• Use 'f' and 's' to make the chain run faster or slower.
• Use 'r' to restart the simulation.
• Use '+' and '-' to modify the value of the exponent a (and restart).
• Use 't' to toggle on/off the displaying of the proposal line (in white) before each move.

For further discussion of slice samplers and auxiliary variable techniques, see the following recent papers:

• Spatial statistics and Bayesian computation, by J. Besag and P. Green (JRSSB 55, 1993).
• Auxiliary Variable Methods for Markov Chain Monte Carlo with Applications, by D. Higdon.
• Markov chain Monte Carlo methods based on `slicing' the density function, by R.M. Neal.
• Gibbs sampling for Bayesian nonconjugate and hierarchical models using auxiliary variables, by P. Damien, S. Walker, and J.C. Wakefield.
• On the use of auxiliary variables in Markov chain Monte Carlo, by A. Mira and L. Tierney.
• Convergence of slice sampler Markov chains and The polar slice sampler, by G.O. Roberts and J.S. Rosenthal.