Adapting the Variance-Covariance Matrix
As has been noted before, there have been instances of chains simply failing to mix. While this seems rather silly in principle (sooner or later, after all, some mixing should happen and it should eventually be good enough to get convergence by various ergodicity theorems), it is rather annoying in practice.
In thinking about this problem, I have concluded that, despite the assurances of BCIMA that the chain is actually from the target distribution, it provides no assurance that the resultant chain (when the length is picked a priori) will be useful at all. This might be called the curse of slow mixing. Part of the point of targeting a 20% acceptance ratio is that the probability that, in continuous state spaces, the chain has failed to move at least once is a 32%, thus, if one were to generate five times the number of desired points and then take every fifth (a reasonable if time expensive procedure), then one can reasonably expect good mixing of the final subsample. But dropping down to a 10% acceptance ratio, the probability of failing to move after five steps is 59%.
While there is no particular reason to believe that any particular target is appropriate, I have made the decision to include a target mixing rate in the ABCIMA code because of exactly the above fact: if the mixing rate is not reasonably high (between 10% and 30%), then it is hard to generate chains of sufficient length as to be useful without taking a very long time to compute. Since points burned in the process of Backward Coupling are dead computations weight by every measure, it seems intelligent to attempt to make good use of them by using them to attain chains which use a proposal density which is expected to yield a productive chain. This, by almost any measure, is certainly one which mixes enough to explore the space.
A word of note: if the proposal is reasonably close to the target and the state space is continuous, then almost any mixing rate above a certain floor is a good one. The proposal being close to the target means that the candidate samples are coming proportionally from the right parts of the state space and the high mixing means that the chain will shuffle around between points in the high-density regions of the chain.
