docs: clarify reff is computed from inverse of unsmoothed importance ratios

[devmotion]

Summary

• Fix reff incorrectly documented #76: reff was documented as the relative efficiency of the unsmoothed importance ratios, but it is the relative efficiency of their inverse (matching Stan's loo::psis definition r_eff = ess(1/r) / S). Updated the PSISResult, psis/psis!, and ess_is docstrings accordingly.
• Fix an analogue of Calculation of relative efficiency in loo docstring PosteriorStats.jl#54 in the psis jldoctest: the example computed reff directly from log_ratios, which gives a different ESS than the correct computation on the inverse ratios. Switched to ess(softmax(-log_ratios; dims=(1, 2)); ...) — equivalent to ess(1 ./ exp.(log_ratios); ...) but numerically stable. Added LogExpFunctions to the docs environment so softmax is available, and regenerated the expected output table.
• Bumped version to 0.9.9.

Test plan

• Documenter.doctest(PSIS; doctestfilters=[r"└.*"]) passes
• Full docs/make.jl build succeeds
• Pkg.test() passes (511/511)

[codecov]

Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 97.44%. Comparing base (ca17ad6) to head (158ae3f).
⚠️ Report is 1 commits behind head on main.

Additional details and impacted files

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  Coverage   97.44%   97.44%           
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  Files           6        6           
  Lines         274      274           
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  Hits          267      267           
  Misses          7        7           

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[sethaxen]

Thanks for the fix! Indeed, this is what the loo::psis docstring says.

For completeness, this is due to (Section 3.2.8 of the PSIS paper):

FYI in the next breaking release we will switch to using the relative tail-ESS to determine relative efficiency. This is preferable both because the Pareto diagnostic is entirely determined by the tail behavior and because the tail-ESS is invariant to monotone transformations (so whether we use weights, log-weights, or inverse-(log-)weights, we get the same value). posterior now does this by default.

Downgrade docs failure seems unrelated to this PR.