#49: `expected_information_gain` enumerates discrete outcomes, so a
consumer recording continuous scores cannot ask which matchup to run
next. The issue flagged this as possibly a research question, since
"expected variance reduction under EP may not have a clean closed form
even for Gaussian likelihoods".
It does. Observing a scored event is a rank-one update to the precision
matrix, so Sherman-Morrison gives
reduction = (c^T L^-1 a)^2 / (v + a^T L^-1 a)
for target functional c and matchup contrast a. Verified against an
actual refit on four candidate matchups: agreement to 1e-9 relative.
Two consequences worth stating.
There is no expectation to take. The expression depends on which matchup
is played but not on how it turns out, because for a Gaussian likelihood
the posterior variance update is data-independent. Pinned by
`the_outcome_does_not_change_the_reduction`, which refits with scores of
(3, 1), (100, -50) and (0, 0) and gets the same answer. The name keeps
the term the active-learning literature uses; no averaging happens.
It is also far cheaper than its ranked counterpart — one linear solve
rather than a full inference pass per possible outcome — because
`c^T L^-1 a` and `a^T L^-1 a` share the same solve.
`target` is deliberately the same linear-functional shape as
`posterior_of`, as the issue proposed, so the two share a concept rather
than inventing two.
The load-bearing test is the refit comparison. An acquisition function
is the archetype of a surface that returns finite, plausible, monotone
numbers while being wrong, and then quietly selects worse matchups
forever; ranking behaviour alone would not catch that.
Closes#49
Co-Authored-By: Claude Opus 5 (1M context) <noreply@anthropic.com>
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