The heuristic I learned in a Bayesian filtering class is to use a diagonal matrix and inflate the prior predictive variance by a small amount each iteration (1-

The heuristic I learned in a Bayesian filtering class is to use a diagonal matrix and inflate the prior predictive variance by a small amount each iteration (1-5%) so that the data can easily tamp it down if there’s any kind of signal. The problem then becomes model selection over those inflation factors.