Gaussian pdf divisions in expectation propagation

Expectation Propagation (EP) is a popular Message Passing algorithm. In contrast to Belief Propagation (BP), it projects beliefs to the exponential family at every update. The update for a posterior factor gets obtained by dividing the projected tilted pdf by the other approximate posterior factors. This division can easily pose problems such as negative variances. This may happen if the factor to be updated, which in many instances is a prior, is very spread out such as a super Gaussian pdf or a Gaussian mixture model. Upon closer inspection however, it turns out that the posterior and extrinsic variances are computed in EP using different pdfs. We propose two solutions to remedy the problem, both based on using the same pdf to compute extrinsic and posterior variances. A third solution is proposed based on revisiting the EP optimization criterion. Furthermore, in a Generalized Linear Model where the signal is composed of multiple sub-signals of constant magnitude, we propose employing the Von Mises distribution for messages in order to circumvent the issue of non-proper distributions. We had encountered this problem in semiblind channel estimation in which we exploit the finite alphabet for the unknown symbols.