On a possible division by zero in the Interacting Multiple Model (IMM) filter

The Interacting Multiple Model (IMM) is arguably the most successful approximation algorithm to the conditional expectation in jump linear systems. One of the key steps of the algorithm is the computation of the mixing probabilities which are used to define refined initial conditions for the different Kalman filters comprising the IMM. This calculation is based on a straightforward application of Bayes rule and involves a division by the probability of the current mode conditioned on the measurement history. While in typical applications this probability never vanishes, there are situations in which this is not necessarily the case. Consequently, direct, blind application of the IMM is impossible due to the potential division by zero. We investigate the reasons for the potential division by zero, analyze the causes and propose a simple solution to avoid such situation without modifying the original algorithm flow.