How Does CONDENSATION Behave with a Finite Number of Samples?
Condensationis a popular algorithm for sequential inference that resamples a sampled representation of the posterior. The algorithm is known to be asymptotically correct as the number of samples tends to infinity. However, the resampling phase involves a loss of information. The sequence of representations produced by the algorithm is a Markov chain, which is usually inhomogeneous. We show simple discrete examples where this chain is homogeneous and has absorbing states. In these examples, the representation moves to one of these states in time apparently linear in the number of samples and remains there. This phenomenon appears in the continuous case as well, where the algorithm tends to produce “clumpy” representations. In practice, this means that different runs of a tracker on the same data can give very different answers, while a particular run of the tracker will look stable. Furthermore, the state of the tracker can collapse to a single peak — which has non-zero probability of being the wrong peak — within time linear in the number of samples, and the tracker can appear to be following tight peaks in the posterior even in the absence of any meaningful measurement. This means that, if theoretical lower bounds on the number of samples are not available, experiments must be very carefully designed to avoid these effects.
KeywordsMarkov Chain State Transition Matrix Prior Density Stochastic Simulation Algorithm Homogeneous Markov Chain
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