Exploration of the target distribution π2 in dimension 2 by different samplers AIMM, AGM (two implementations for both) and AMH throughout a simulation of 200,000 iterations.

Each sampler's efficiency is illustrated by two panels: we report the sample path of the Markov chain on the left panel and the proposal distribution on the right panel, for a subset of the 200,000 iterations. The ellipses location corresponds to the mean of Gaussian kernels and their size gives a 0.75 confidence region. The color of the ellipses corresponds to the weight of the component in the mixture. Kernels with weight less than 10-4 were not represented. Finally, in both panels five level lines (at levels 10-4, 10-3, 10-2, 10-1, 1) of π2 illustrate the high density regions of π2.

  1. AIMM implemented with a defensive kernel (in thicker and dashed line) located close to the mode: AIMM increments and converges quickly since the early states of the Markov chain are already in high density regions of π2. After 200,000 iterations, it can be visually checked that the chain has visited the five level sets of π2 in their respective proportion.
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  2. AIMM implemented with a defensive kernel (in thicker and dashed line) located in a region where π2 has essentially no probability mass. In this case AIMM's convergence is slower and designs intermediate kernels (the loose ellipses) that help the chain to jump to higher density regions. Nevertheless, AIMM manages to design a relevant proposal that resembles that obtained in the first implementation. Note that the weight of those intermediate components vanishes quickly: they therefore only serve transitory.

  3. AGM implemented with a mixture of 100 Gaussian kernels, initially centered at random locations. Those kernels located in low density regions are unable to jump closer to higher density regions because of the inertia resulting from AGM's adaptation scheme. Their weight thus vanishes and only a handful of components survive. This explains the sample path of the chain that misses a large part of the support of π2 . In fact, the chain hardly visits areas where the density is lower than 10-2.

  4. AGM implemented with a mixture of 100 Gaussian kernels, initially centered in the high density regions of π2. Even in this favourable setup, AGM cannot maintain a balanced mixture.

  5. AMH implemented with an adaptive random walk proposal using a defensive covariance matrix (in red and then blue) with probability 0.05 and the empirical covariance matrix based on the past history of the chain (in light blue) with probability 0.95. Even though AMH converges more slowly than AIMM and yields a large rejection rate, it is more robust on this example than AGM. After 200,000 iterations, the chain seems to have explored the state space accordingly to π2's topology.

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