Correct Jacobian for the initial Rt prior

[sbfnk-bot]

Description

This PR closes #1475.

Since #1396 the user's prior on the initial reproduction number is applied to the derived initial Rt, R[1], in init_priors_lp(). R_mean is declared real<lower = 0>, so Stan already adds log(R_mean) for the lower-bound transform, and the hand-written target += log(R[1]) then double-counts it, leaving a spurious +log(R_mean) in the target. The prior is shifted upwards by exp(sdlog^2) — a LogNormal(mean = 2, sd = 1) prior behaves as if its mean were 2.5.

The correction relative to the sampled parameter is log(R[1]) - log(R_mean). This PR makes that change, corrects the derivation in the GP implementation vignette (which assumed the internal parameter was sampled on the log scale), and adds a prior-recovery regression test.

A prior-only fit recovers the requested prior after the change:

	median R[1]
requested LogNormal(mean = 2, sd = 1)	1.789
before	2.234
after	1.799

Initial submission checklist

• My PR is based on a package issue and I have explicitly linked it.
• I have tested my changes locally.
• I have added or updated unit tests where necessary.
• I have updated the documentation if required and rebuilt docs if yes.
• I have followed the established coding standards (and checked using lintr::lint_package()).
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[github-actions]

Thank you for your contribution sbfnk 🚀! Your synthetic_recovery markdown is ready for download 👉 here 👈!
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[github-actions]

This is how benchmark results would change (along with a 95% confidence interval in relative change) if 7e9c085 is merged into main:

• ✔️estimate_dist: 628ms -> 635ms [-1%, +3.21%]
• ✔️estimate_infections_gp: 23.8s -> 13.9s [-131.03%, +48.03%]
• ✔️estimate_infections_rw: 12.1s -> 6.41s [-153.81%, +59.49%]
• ✔️estimate_secondary: 1.42s -> 1.4s [-9.28%, +6.98%]
• ✔️estimate_truncation: 1.31s -> 1.36s [-3.79%, +10.47%]
  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 1be44b4 is merged into main:

• ✔️estimate_dist: 750ms -> 757ms [-0.69%, +2.51%]
• ✔️estimate_infections_gp: 17.9s -> 20.5s [-35.56%, +65.12%]
• ✔️estimate_infections_rw: 28.5s -> 11.2s [-223.34%, +102.02%]
• ✔️estimate_secondary: 1.63s -> 1.7s [-1.22%, +10.15%]
• ✔️estimate_truncation: 1.65s -> 1.64s [-12.98%, +12.08%]
  Further explanation regarding interpretation and methodology can be found in the documentation.

[seabbs]

Nice

[github-actions]

This is how benchmark results would change (along with a 95% confidence interval in relative change) if 629739c is merged into main:

• ✔️estimate_dist: 539ms -> 534ms [-1.91%, +0.13%]
• ✔️estimate_infections_gp: 11.3s -> 11s [-11.19%, +5.35%]
• ✔️estimate_infections_rw: 5.53s -> 5.2s [-14.53%, +2.92%]
• ❗🐌estimate_secondary: 1.09s -> 1.2s [+4.03%, +16.98%]
• ✔️estimate_truncation: 1.08s -> 1.11s [-5.43%, +12.22%]
  Further explanation regarding interpretation and methodology can be found in the documentation.