Glue factors, likelihood computation, and filtering in state space models

Factor graphs of statistical models can be augmented by a glue factor that expresses some additional (initial, final, or otherwise “local”) condition. That applies, in particular, to (otherwise time-invariant) linear Gaussian state space models, which are thus generalized to pulse-like models that are localized anywhere in time. The model likelihood can then be computed by (forward-backward or forward-only) sum-product message passing, which leads to the concept of a likelihood filter. We propose to build (forward-only) likelihood filters from a bank of second-order linear systems. We also observe that such likelihood filters can be cascaded into a new sort of neural network that works naturally with multichannel time signals at multiple time scales.