The benefits of down-sampling in the particle filter

The choice of proposal distribution in the particle filter is one of the most important design choices, and also one of the trickiest one to implement. There are basically three main options: the prior, the likelihood and the optimal proposal that combines the prior and the likelihood. The optimal proposal however, can not be obtained in most cases. The prior proposal is although easy to implement, it does not incorporate the information available otherwise from the recent observation. The prior may thus work fine for low signal to noise ratio (SNR), where the recent observation does not carry much information. However, defining the critical value of the SNR is not that obvious. On the other hand, the likelihood as a proposal always includes the information from the recent observation, but it requires that the measurement dimension is at least equal to the state dimension. We here formalize the problem, and point out an approach based on down-sampling the model. One main advantage of down-sampling is that it can decrease the problem of particle degeneracy.