Efficient approximate inference in Bayesian networks with continuous variables

Inference is one key objective in a Bayesian network (BN), and it aims to estimate the posterior distributions of state variables based on evidence (observations). While efficient analytical inference algorithms (either approximate or exact) for BN with discrete variables have been well-established in the literature, the inference in BN with continuous variables is still challenging if the BN is non-linear and/or non-Gaussian. In this case we can either discretize the continuous variable and utilize the inference approaches for discrete BN; or we have to use sampling-based methods such as MCMC for static BN and particle filter for dynamic BN. This paper proposes a network collapsing technique based on the concept of probability integral transform to convert a multi-layer BN to an equivalent simple two-layer BN, so that the unscented Kalman filter can be applied to the collapsed BN and the posterior distributions of state variables can be obtained analytically. For dynamic BN, the proposed method is also able to propagate the state variables to the next time step analytically using the unscented transform, based on the assumption that the posterior distributions of state variables are Gaussian. Thus the proposed method achieves a very fast approximate solution, making it particularly suitable for dynamic BN where inference and uncertainty propagation are required over many time steps.