Investigate adding rectifier transformations to improve stability in sampling

[SamuelBrand1]

Pertinent to above and @seabbs comment on using upjitter. Using the softplus transform instead of an upjitter (or clamp to do relu) stabilises the sampling. Also, using LogExpFunctions.xexpy to implement the effect of the AR process on observations in the stochastic model.
After these changes, an experiment to not use initial values to start NUTS still converges onto the correct posterior distributions, so at least in this context this is a big improvement in the stability of the model. I know @damonbayer has wrangled with this in Bayesian inference of ODEs so I wonder what your thoughts on using special functions to stabilise sampling are?

Originally posted by @SamuelBrand1 in #464 (comment)

This finding really makes me wonder if we should have some kind of (preferable smooth) rectifier transformation as default in a large chunk of the modelling. Maybe via TransformLatentModel etc?

• Check, and if possible apply, Transform... struct using softplus etc for chatzilena et al replication. #471
• New Transform struct as discussed Check, and if possible apply, Transform... struct using softplus etc for chatzilena et al replication. #471 (comment)
• Add stable form of exp transform to pipeline #479
• Change calculation method in EpiAwareBase.generate_latent_infs(epi_model::ExpGrowthRate, rt) #469

[seabbs]

step one is being able to do all the transforms we have in the new vignette using Transform right?

[SamuelBrand1]

Final sub task of this was closed by #496