To extend the message-passing approach to handle non-Gaussian priors and non-linear observation models while maintaining scalability benefits, several strategies can be employed:
Non-Gaussian Priors: One approach is to approximate the non-Gaussian prior distribution as a Gaussian distribution using techniques like variational inference or moment matching. By transforming the non-Gaussian prior into a Gaussian form, the message-passing algorithm can still be applied efficiently. Another method is to use iterative linearization techniques to approximate the non-Gaussian prior as a series of linearized models, allowing for iterative updates within the message-passing framework.
Non-linear Observation Models: For non-linear observation models, the message-passing algorithm can be adapted to handle these by linearizing the models around the current estimates. This involves computing the Jacobian of the observation model and updating the messages based on the linearized form. By iteratively linearizing the non-linear observation models, the message-passing algorithm can still provide accurate estimates while accommodating the non-linearity.
Hybrid Approaches: A hybrid approach combining message passing with other techniques like particle filters or ensemble methods can be used to handle non-Gaussian priors and non-linear observation models. By integrating the strengths of different methods, the hybrid approach can leverage the scalability of message passing while accommodating the complexities of non-Gaussian and non-linear scenarios.
By incorporating these strategies, the message-passing approach can be extended to handle a wider range of probabilistic models with non-Gaussian priors and non-linear observation models, while still benefiting from its scalability and efficiency in large-scale data assimilation problems.