Possibly improve NUTS tuning

[amal-ghamdi]

General description of task

The demo https://github.com/CUQI-DTU/CUQIpy/blob/main/demos/demo37_experimental_mcmc_module_Gibbs.ipynb uses NUTS within Gibbs. For the NUTS example, before the fix in #621, the inferred signal is (exec. time ~20 min)

After this fix, the signal is (exec. time ~ 20 min)

Note that this fix not necessarily improve the inference (both inferences looks a bit similar). However, for small number of steps, the old implementation seems to perform better (10 steps).

num_sampling_steps = {
    "x" : 1,
    "s" : 1
}

sampler = HybridGibbs(target, sampling_strategy, num_sampling_steps)

sampler.warmup(10)
sampler.sample(10)

before fix:

After fix:

This is likely because tuning in NUTS suggests worse time step than the method _find_good_epsilon which is called at the beginning of sampling. Before fixing NUTS statefulness this method seems to be called every time the sampler is re-initialized (each Gibbs step)

Definition of Done (Feature/change)

• description1
• description2

Definition of Done (Mandatory)

• Documentation added (docstrings on all public methods/classes)
• Unit tests added/updated (and passing!)
• Code reviewed and approved by 2 CUQI-maintainers
• Online documentation is rendered correctly (esp. math)

Definition of Done (Optional)

• HowTo/tutorial added