Exploration of the target distribution π₁ by different samplers AIMM, AMH, AGM-2 and AGM-3 throughout an experiment consisting of 100,000 iterations.

Each plot is in log-scale:

• the black dashed line is the target distribution π₁
• the blue line illustrates the adaptation of the proposal kernel: the incremental proposal for AIMM, the random walk proposal for AMH and the adapting Gaussian mixture for AGM-2 and AGM-3
• the histogram bars represent the Markov chain sampling distribution

AIMM increments 14 times throughout the simulation. The proposal progressively fits to the target while still allowing to increment if insufficiently supported regions of the state space are discovered. After 100,000 iterations the chain distribution matches π₁ up to a precision of 10^-5 (red landmark line).

AMH converges more slowly to the target distribution. In fact it takes more than 10,000 iterations for AMH to discover the three modes. After 100,000 iterations the left and right modes are still respectively over- and underestimated.

AGM-2's proposal kernel cannot match π₁ since it tries to adapt a mixture of two Gaussians to a distribution that has three modes. This results in a slower convergence as many proposed states are rejected. After 100,000 iterations the chain distribution is still missing part of the tail of π₁ at the precision level of 10^-5 (red landmark line).

AGM-3 appears to be the most efficient sampler in this scenario since the family of proposals is well chosen: in fact it contains π₁. This explains why AGM-3 nearly samples i.i.d. draws from π₁. In terms of convergence, the chain distribution of AGM-3 resembles that of AIMM after 100,000 iterations.

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