More user friendly demos
Yield descriptive error messages if invalid input is passed to the PoissonianEnergy likelihood. Furthermore, never start the minimization from exactly zero as this is potentially unstable in general and users tend to copy-paste code from the demos for new project.
Starting close to or even worse exactly at zero with MGVI is error prone for simple models with a linear response and a simple correlated field. The main problem are the zeros that appear in the Jacobians for everything but the excitations as a result of the linearity of the model in exactly these. While for the minimization itself zero-curvature for the Fisher metric is not necessarily problematic because of the prior curvature, it results in problems for the sampling. Namely sampling along axis with zero Fisher curvature results in prior samples being drawn for these axis.
On the one hand this is desirable because in the absence of information from the likelihood, the prior should be the only quantity that matters. On the other hand, this results in the model being sometimes impossible to optimize appropriately. This is because often for a flexible model the prior samples for the non-linear parameters are too diverse to jointly optimize their mean in a sensible way. As an example, for prior samples for the fluctuations parameter, since samples with a too high fluctuation parameter compared to the data are compensated by samples with a too low fluctuation parameter compared to the data. However, since it is critical to fix the fluctuations parameter before anything else can be optimized in a sensible way, in essence no optimization can nor does happens.