See the chain run! An explanation is below.

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

- Use the numbers '0' through '9' to set the animation speed level higher or lower.
- Use 'r' to restart the simulation, or 'z' to just zero the empirical count, or 's' to toggle whether or not to show the (black) empirical distribution.
- Use 'g' to cycle the target distribution between certain specific values, and randomly-generated values, and a special "counter-example" target.
- Use '+' and '-' to increase/decrease the number of states (and restart the simulation).
- Use 'y' to always adapt (default), or 'n' to never adapt, or 'd' to adapt with probability 1/iteration, or 'o' to fix gamma=1, or 't' to fix gamma=2, or 'F' to fix gamma=50.
- Use 'p' and 'm' to increase/decrease the current value of gamma.
- Use 'A' to jump to the left-most state, or 'B' to jump to the right-most state.
- Use '>' and '<' to increase/decrease the target probability of state 2 (and restart the simulation) for the counter-example target.
- At fast animation speed levels, you can press any other key (e.g. 'space') at any time to get an instantaneous snapshot of the iteration in progress.

Proposal: The proposal distribution is uniform on the white disks, from x-gamma to x+gamma (but excluding x itself). The yellow disk then shows the actual proposal state.

Accept/Reject: The yellow disk turns green if the proposal is accepted, or red if it is rejected. The (black) current state is updated accordingly.

Adaption: *If* adaption is turned on (with 'y'), the algorithm adapts by increasing gamma by 1 if the previous proposal was accepted, or decreasing gamma by 1 (to a minimum of 1) if the previous proposal was rejected.

Empirical distribution: The empirically estimated distribution is graphed with black bars. If the simulation correctly preserved stationarity of the target distribution, then the black and blue bars should converge in height.

Comparison of means: The small vertical blue line at the top shows the target mean, while the small vertical black line shows the current empirical mean. If the simulation correctly preserved stationarity, then the two lines should converge.

Conclusion:
With the adapt option turned on (with 'y'),
once the chain reaches state 1 with gamma=1,
it tends to get *stuck* there for a very long time, causing the
empirical distribution to significantly *overweight* state 1.
This shows that, counter-intuitively, this adaptive algorithm does
*not* preserve stationarity of the target distribution.
However, if we instead select diminishing adaption probabilities
(with 'd'), or no adaptions (with 'n'), then convergence is preserved.

Final remark: The example presented here is on a *discrete*
state space, but this is not essential. Indeed, if the above target and
proposal distributions are each convolved with a Normal(0, 0.000001)
distribution, this produces an example on a continuous state space
(with continuous, everywhere-positive densities) which has virtually
identical behaviour, and similarly fails to converge.

For further discussion of adaptive MCMC algorithms and related examples, see e.g.:

- G.O. Roberts and J.S. Rosenthal, Examples of Adaptive MCMC. J. Comp. Graph. Stat. 18(2) (2009), 349-367.
- G.O. Roberts and J.S. Rosenthal, Coupling and Ergodicity of Adaptive MCMC. J. Appl. Prob. 44 (2007), 458-475.
- J.S. Rosenthal, AMCMC: An R/C package for running Adaptive MCMC. Comp. Stat. Data Anal. 51 (2007), 5467-5470.
- K. Latuszynski, G.O. Roberts, and J.S. Rosenthal, Adaptive Gibbs samplers and related MCMC methods. Ann. Appl. Prob., to appear.
- Y.F. Atchadé and J.S. Rosenthal, On Adaptive Markov Chain Monte Carlo Algorithms. Bernoulli 11 (2005), 815-828.
- H. Haario, E. Saksman, and J. Tamminen, An adaptive Metropolis algorithm. Bernoulli 7 (2001), 223-242.
- C. Andrieu and E. Moulines, On the Ergodicity Properties of some Adaptive MCMC Algorithms. Ann. Appl. Prob. 16 (2006), 1462-1505.
- P. Giordani and R. Kohn, Adaptive Independent Metropolis-Hastings by Fast Estimation of Mixtures of Normals. Preprint, 2006.
- E. Turro, N. Bochkina, A.M.K. Hein, and S. Richardson (2007), BGX: a Bioconductor package for the Bayesian integrated analysis of Affymetrix GeneChips. BMC Bioinformatics 8 (2007), 439-448.
- S. Richardson, L. Bottolo, and J.S. Rosenthal, Bayesian models for sparse regression analysis of high dimensional data. Valencia IX Bayesian Meeting conference proceedings, 2010.

Applet by Jeffrey S. Rosenthal (contact me).

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