`posterior_of`, `posterior_of_at` and `expected_variance_reduction` each
built the joint precision matrix, factorised it, asked one question and
threw it away. The factorisation is O(n^3) in the history's appearances
and depends only on the fit, so a caller asking about every pair in a
standings table, every cell in a grid, or every candidate in an
active-learning sweep paid for the same factorisation once per question.
`History::joint()` returns a `Joint` handle that pays it once. Measured
on 1976 appearances, 90 queries: 68.4s one-shot against 745ms factorise
plus 93ms of queries — 81.6x, with bit-identical answers. Per query,
Criterion at 480 appearances: 9.0ms one-shot against 48us cached, 187x.
The handle borrows the history, which is what makes it correct with no
invalidation logic: the borrow checker forbids adding events or refitting
while it is alive, so there is no window in which the factorisation could
describe a fit that no longer exists. It also makes the lifetime of the
n^2 factor explicit rather than parking it in the history forever — at
4000 appearances that is 128MB, which is not something to cache silently.
Every question the joint answers turns out to be a bilinear form,
c^T A^-1 a = (L^-1 c) . (L^-1 a)
so no caller ever needs L^-1 c itself. Replacing the general solve with a
forward substitution drops the back substitution as wasted work, halving
a query, and removes a failure mode: a variance as `c . (A^-1 c)` is a
difference of products that can round negative, where `|L^-1 c|^2` is a
sum of squares and cannot.
The one-shot calls are unchanged in cost and now delegate to the handle,
so the two paths cannot drift apart. tests/joint_handle.rs asserts they
agree bit for bit, including at pinned times, under UnknownKeys::Prior,
and across candidate matchups.
Refs #51
Co-Authored-By: Claude Opus 5 (1M context) <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_011hcFjNDmHXZF8URGLku5zZ
#46: every accessor returns a per-competitor marginal, and almost
nothing a consumer publishes is one competitor. Combining marginals
assumes independence, and competitors are correlated through every event
they share.
`posterior_of(&[(a, 1.0), (b, -1.0)])` returns the posterior of that
combination with the correlation intact. Validated against the exact
linear-Gaussian posterior on both a tree and a loopy fixture, for
differences and for single competitors: agreement to 1e-9 relative in
every case.
The investigation that preceded this is why it is not a covariance
accessor. Marginals from loopy message passing are about half the true
width, and ignoring correlation overstates a difference — the two errors
partially cancel, leaving 1.327x rather than 2.646x. Bolting true
correlations onto the existing marginals would have given 0.765 against
a true 1.524, which is overconfident: the direction the reporter
specifically called unsafe. Rebuilding the joint from the factor
structure fixes both at once, and a single-competitor query now returns
the exact marginal rather than the narrow one.
The precision matrix depends only on structure — who played whom, with
what weights and what noise — not on the observed outcomes, and the
means were already exact. So only the second moment is reconstructed.
Known limits, all deliberate and documented on the method:
- Latest slice only. A functional spanning times, such as "current
versus career", needs the time-expanded joint and is not covered.
- Scored events only. A ranked outcome's truncation is EP-approximated
and its converged factors are not retained after inference, so ranked
slices return `JointUnavailable` rather than a plausible wrong number.
- Dense Cholesky, O(n^3) per query in the slice's competitor count:
38.8us at 50, 5.66ms at 400, 49.1ms at 800. Fine for the sizes this
serves today; caching the factorization per slice would make repeat
queries O(n^2), and sparsity is the next step after that.
Refs #46, #47, #48
Co-Authored-By: Claude Opus 5 (1M context) <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_011hcFjNDmHXZF8URGLku5zZ
logaritmiskandClaude Opus 5