Change all priors to use <dist_spec>

[sbfnk]

Description

This PR closes #525 by changing all parameters to use the new interface.

In doing so and in order to keep this manageable the internals of handling parameters had to be updated a fair bit. This is now done similarly to delays in parameter arrays rather than named parameters. In principle this should make it easier to extend the model later with more parameters / probability distributions / time-varying parameters etc.

Things that could be added later fairly easy if deemed useful:

1. switch delay distribution parameters to use the new stan setup for parameters (this might simplify code so probably a good idea)
2. allow uncertain parameters of prior distributions other than delays

From tests it seems functionality has been preserved but it would be good for people to check if this is actually an improvement.

One thing to note is that now all parameters are initialised from (truncated) standard normal distributions so will have to monitor if this leads to any issues.

Initial submission checklist

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After the initial Pull Request

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[seabbs]

Wow - this is a lot. On the face of it looks good but going to need some time to digest

[github-actions]

This is how benchmark results would change (along with a 95% confidence interval in relative change) if deedb9e is merged into main:

• ✔️default: 24.1s -> 20.8s [-28.62%, +0.99%]
• ✔️no_delays: 26.2s -> 26.2s [-10.42%, +10.39%]
• ✔️random_walk: 9.66s -> 9.62s [-7.59%, +6.74%]
• 🚀stationary: 14.3s -> 11.8s [-30.72%, -3.89%]
• ✔️uncertain: 37.4s -> 36.8s [-13.63%, +10.13%]
  Further explanation regarding interpretation and methodology can be found in the documentation.

[github-actions]

This is how benchmark results would change (along with a 95% confidence interval in relative change) if 333de46 is merged into main:

• ✔️default: 22.4s -> 21.7s [-16.65%, +10.78%]
• ✔️no_delays: 27s -> 27.8s [-11.93%, +18.38%]
• ✔️random_walk: 9.69s -> 9.99s [-2.32%, +8.41%]
• 🚀stationary: 16.1s -> 12.5s [-43.91%, -0.72%]
• ✔️uncertain: 39.8s -> 38.4s [-14.72%, +8.04%]
  Further explanation regarding interpretation and methodology can be found in the documentation.

[sbfnk]

One thing to note is that the lengthscale prior is currently not included in this, because of the extra logic in

EpiNow2/inst/stan/functions/gaussian_process.stan

Line 201 in 2f27a3f

if (ls_sdlog > 0) {

It might be worth deciding on a single shape of the prior and then this could also be specified as a <dist_spec> (though this currently doesn't support specifying a minimum, which would either have to be dropped or functionality added).

[sbfnk]

Related to that merging this would enable a more comprehensive exploration of parameter choices and identifiability than what's done in #855 (not requiring editing the stan model) as we can fix parameters and/or try out different distributions with different parameters.

[seabbs]

which would either have to be dropped or functionality added

I think we would need to add this as from memory it really isn't identifiable without setting some bounds