Add Bayesian posterior sampling to Analysis and ParameterAnalysis

[henrikjacobsenfys]

Extends the Bayesian sampling from #237 to the remaining two Analysis classes.

Stacked on #237 — please merge that first; this PR targets the bayesian branch, and I will retarget it at develop once #237 lands.

No new sampling machinery. Each class supplies the three mixin hooks — build the fitter, bind the data, list the chain parameters — and everything else is shared, which was the point of building #237 mixin-first.

Analysis

sample_posterior(fit_method=...) mirrors fit():

# one chain per Q index, delegating to the Analysis1d objects
results = analysis.sample_posterior(fit_method='independent')      # list, one per Q
result  = analysis.sample_posterior(fit_method='independent', Q_index=5)

# one chain over every Q at once, through a MultiFitter
result  = analysis.sample_posterior(fit_method='simultaneous')

The simultaneous path refreshes each per-Q convolver against its masked energy grid before sampling, exactly as _fit_all_Q_simultaneously does, so the sampler sees the model evaluations the fit would have made. This was the open question at the end of #237, and it works: 9 parameters across 3 Q values sample in ~14s.

ParameterAnalysis

Samples the binding models. fit() built its MultiFitter inline, so the per-target data, functions, and models are now resolved by a shared _build_fit_inputs() used by both paths — which also guarantees fitting and sampling see the same targets, in the same order, with the same unit conversions.

Parameter labels

This needed rethinking for multi-Q. A multi-Q analysis holds one copy of each parameter per Q, all sharing a name — three parameters called Gaussian width. A summary showed three identical rows, and a name could not be used to select a parameter.

Labels now come from an overridable parameter_label(): Analysis qualifies by Q index, ParameterAnalysis by binding model, and both only when the bare name is actually ambiguous, so single-Q and single-binding cases keep their short names. The summary and bounds tables size themselves to the longest label instead of truncating at a fixed width.

Drive-by fix

Analysis.fit's docstring promised "a single FitResults object if fitting simultaneously". MultiFitter splits its combined result back up per dataset, so a list has always been returned. Docstring corrected to match the behaviour.

Tutorial

Tutorial 1 gains a Bayesian section on the two-step diffusion fit, and it surfaces something worth documenting: the posterior comes out about twelve times tighter than the reported least-squares uncertainties.

That gap is real, not a bug. The width fit has a reduced χ² of ~150, so lmfit inflates its reported uncertainties by √(reduced χ²) ≈ 12 on the assumption that a poor fit means understated input uncertainties. DREAM makes no such adjustment. Neither is simply right, and the gap is itself a signal that the two-step model is not capturing the data — which is exactly what the tutorial goes on to address by fitting everything simultaneously.

I originally intended to sample the full simultaneous diffusion model, but measured it at over ten minutes — too slow for a tutorial and for CI. The ParameterAnalysis step gives the same lesson in ~25s, and tutorial 1 as a whole runs in 41s.

Testing

• 41 new tests; 1630 pass in total.
• Unit tests mock the sampler. Integration tests run real chains (~55s) covering simultaneous multi-Q, independent-vs-simultaneous agreement, and ParameterAnalysis.
• One integration test doubles as a check that suggest_bounds refuses to invent a scale: a polynomial coefficient sitting at exactly zero with vanishing uncertainty is flagged rather than bounded, and absolute_floor is what resolves it.
• pixi run fix clean; pixi run check clean except a pre-existing CONTRIBUTING.md prettier difference that also fails on a clean tree at HEAD (local npx prettier is newer than CI's — CI passes).

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[codecov]

Codecov Report

❌ Patch coverage is 99.34641% with 2 lines in your changes missing coverage. Please review.
✅ Project coverage is 99.10%. Comparing base (a8946ce) to head (442a5bd).

Files with missing lines	Patch %	Lines
src/easydynamics/utils/posterior_plotting.py	96.15%	1 Missing and 1 partial ⚠️

Additional details and impacted files

@@             Coverage Diff              @@
##           bayesian     #238      +/-   ##
============================================
+ Coverage     98.47%   99.10%   +0.63%     
============================================
  Files            56       56              
  Lines          4533     4825     +292     
  Branches        774      831      +57     
============================================
+ Hits           4464     4782     +318     
+ Misses           36       18      -18     
+ Partials         33       25       -8     

Flag	Coverage Δ
unittests	99.10% <99.34%> (+0.63%)	⬆️

Flags with carried forward coverage won't be shown. Click here to find out more.

Files with missing lines	Coverage Δ
src/easydynamics/analysis/analysis.py	97.70% <100.00%> (+1.29%)	⬆️
src/easydynamics/analysis/bayesian_sampling.py	100.00% <100.00%> (+8.07%)	⬆️
src/easydynamics/analysis/parameter_analysis.py	99.59% <100.00%> (+0.08%)	⬆️
src/easydynamics/analysis/posterior.py	98.78% <100.00%> (+2.18%)	⬆️
src/easydynamics/utils/posterior_plotting.py	96.92% <96.15%> (+4.51%)	⬆️

... and 1 file with indirect coverage changes

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