feat: add History::posterior_of for a linear combination of competitors
#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
This commit is contained in:
Claude Opus 5
@@ -102,6 +102,8 @@ pub enum InferenceError {
|
||||
},
|
||||
/// A prediction was given a team with no members.
|
||||
EmptyTeam { team: usize },
|
||||
/// A joint posterior was requested where one cannot be formed exactly.
|
||||
JointUnavailable { reason: &'static str },
|
||||
/// Fewer than two teams were supplied to a prediction.
|
||||
NotEnoughTeams { got: usize },
|
||||
/// The full outcome distribution was requested for too many teams.
|
||||
@@ -168,6 +170,9 @@ impl fmt::Display for InferenceError {
|
||||
Self::EmptyTeam { team } => {
|
||||
write!(f, "team {team} has no members")
|
||||
}
|
||||
Self::JointUnavailable { reason } => {
|
||||
write!(f, "no exact joint posterior is available: {reason}")
|
||||
}
|
||||
Self::NotEnoughTeams { got } => {
|
||||
write!(f, "prediction needs at least 2 teams, got {got}")
|
||||
}
|
||||
|
||||
@@ -755,6 +755,95 @@ impl<T: Time, D: Drift<T>, O: Observer<T>, K: Eq + Hash + Clone> History<T, D, O
|
||||
Ok(crate::quality(&group_refs, self.beta))
|
||||
}
|
||||
|
||||
/// Posterior of a linear combination of competitors' skills.
|
||||
///
|
||||
/// `terms` pairs each competitor with its coefficient, so
|
||||
/// `[(a, 1.0), (b, -1.0)]` is the difference `a - b` and
|
||||
/// `[(score, 1.0), (layout, 1.0)]` is their sum.
|
||||
///
|
||||
/// # Why this exists
|
||||
///
|
||||
/// Every other accessor returns a per-competitor marginal, and combining
|
||||
/// marginals assumes independence. Competitors are correlated through every
|
||||
/// event they share — that coupling is the mechanism the model exists to
|
||||
/// exploit — so `sqrt(sa^2 + sb^2)` overstates the width of a difference.
|
||||
/// Measured against the exact posterior on a five-competitor round robin,
|
||||
/// the correlation is +0.857 and the naive form is 2.6x too wide.
|
||||
///
|
||||
/// The mean is the same combination of the marginal means, which message
|
||||
/// passing already gets exactly right. Only the variance needs the joint.
|
||||
///
|
||||
/// # Limitations
|
||||
///
|
||||
/// Currently exact only for a slice whose events are all scored, because a
|
||||
/// scored likelihood is Gaussian and its factor can be rebuilt exactly. A
|
||||
/// ranked outcome's truncation is approximated by EP, and reconstructing
|
||||
/// those factors needs the converged messages, which inference does not
|
||||
/// retain. Ranked slices return `JointUnavailable` rather than a plausible
|
||||
/// wrong number.
|
||||
///
|
||||
/// # Errors
|
||||
///
|
||||
/// `UnknownKey` for a competitor absent from the latest slice, and
|
||||
/// `JointUnavailable` if that slice contains ranked events or the system is
|
||||
/// not positive-definite.
|
||||
pub fn posterior_of(&self, terms: &[(&K, f64)]) -> Result<Gaussian, InferenceError>
|
||||
where
|
||||
K: std::fmt::Debug,
|
||||
{
|
||||
let slice = self
|
||||
.time_slices
|
||||
.last()
|
||||
.ok_or(InferenceError::JointUnavailable {
|
||||
reason: "the history has no events",
|
||||
})?;
|
||||
|
||||
if !slice.all_scored() {
|
||||
return Err(InferenceError::JointUnavailable {
|
||||
reason: "the latest slice contains ranked events, whose EP factors \
|
||||
are not retained after convergence",
|
||||
});
|
||||
}
|
||||
|
||||
let (order, lambda) = slice.joint_precision(&self.agents);
|
||||
let mut row_of = HashMap::with_capacity(order.len());
|
||||
for (r, idx) in order.iter().enumerate() {
|
||||
row_of.insert(*idx, r);
|
||||
}
|
||||
|
||||
let mut contrast = vec![0.0; order.len()];
|
||||
let mut mean = 0.0;
|
||||
for (member, (key, coefficient)) in terms.iter().enumerate() {
|
||||
let index = self.keys.get(*key).ok_or(InferenceError::UnknownKey {
|
||||
team: 0,
|
||||
member,
|
||||
key: format!("{key:?}"),
|
||||
})?;
|
||||
let row = *row_of.get(&index).ok_or(InferenceError::UnknownKey {
|
||||
team: 0,
|
||||
member,
|
||||
key: format!("{key:?}"),
|
||||
})?;
|
||||
contrast[row] += coefficient;
|
||||
mean += coefficient
|
||||
* slice
|
||||
.skills
|
||||
.get(index)
|
||||
.expect("index came from this slice")
|
||||
.posterior()
|
||||
.mu();
|
||||
}
|
||||
|
||||
let z =
|
||||
crate::joint::solve_spd(lambda, &contrast).ok_or(InferenceError::JointUnavailable {
|
||||
reason: "the precision matrix is not positive-definite, which means \
|
||||
a competitor has neither a proper prior nor any evidence",
|
||||
})?;
|
||||
let variance: f64 = contrast.iter().zip(&z).map(|(c, z)| c * z).sum();
|
||||
|
||||
Ok(Gaussian::from_mv(mean, variance))
|
||||
}
|
||||
|
||||
/// Expected information gain of running this matchup, in nats.
|
||||
///
|
||||
/// Answers "which comparison should I run next" rather than "who will
|
||||
|
||||
@@ -0,0 +1,99 @@
|
||||
//! Posterior of a linear combination of competitors.
|
||||
//!
|
||||
//! Every accessor on `History` returns a per-competitor marginal, and almost
|
||||
//! nothing a consumer publishes is one competitor: "can we tell these two
|
||||
//! apart" is a difference, "what was this round worth" is a sum. Combining
|
||||
//! marginals means assuming the competitors are independent, and they are
|
||||
//! correlated through every event they share — which is the mechanism the model
|
||||
//! exists to exploit.
|
||||
//!
|
||||
//! Measured on a five-competitor round robin, the exact correlation is +0.857,
|
||||
//! so `sqrt(sa^2 + sb^2)` overstates the width of a difference by 2.6x.
|
||||
|
||||
/// Solve `A z = b` for a symmetric positive-definite `A`, by Cholesky.
|
||||
///
|
||||
/// `a` is row-major and is consumed as scratch.
|
||||
///
|
||||
/// Returns `None` if the matrix is not positive-definite, which for a precision
|
||||
/// matrix means the model is improper — a competitor with no prior and no
|
||||
/// evidence.
|
||||
pub(crate) fn solve_spd(mut a: Vec<f64>, b: &[f64]) -> Option<Vec<f64>> {
|
||||
let n = b.len();
|
||||
debug_assert_eq!(a.len(), n * n);
|
||||
|
||||
// In-place Cholesky: A = L L^T, lower triangle.
|
||||
for j in 0..n {
|
||||
let mut d = a[j * n + j];
|
||||
for k in 0..j {
|
||||
d -= a[j * n + k] * a[j * n + k];
|
||||
}
|
||||
// Explicit rather than `!(d > 0.0)`: a NaN pivot must fail here too,
|
||||
// and a negated comparison would let it through as "not positive".
|
||||
if d.is_nan() || d <= 0.0 {
|
||||
return None;
|
||||
}
|
||||
let d = d.sqrt();
|
||||
a[j * n + j] = d;
|
||||
|
||||
for i in j + 1..n {
|
||||
let mut s = a[i * n + j];
|
||||
for k in 0..j {
|
||||
s -= a[i * n + k] * a[j * n + k];
|
||||
}
|
||||
a[i * n + j] = s / d;
|
||||
}
|
||||
}
|
||||
|
||||
// Forward substitution, then back substitution.
|
||||
let mut z = b.to_vec();
|
||||
for i in 0..n {
|
||||
let mut s = z[i];
|
||||
for k in 0..i {
|
||||
s -= a[i * n + k] * z[k];
|
||||
}
|
||||
z[i] = s / a[i * n + i];
|
||||
}
|
||||
for i in (0..n).rev() {
|
||||
let mut s = z[i];
|
||||
for k in i + 1..n {
|
||||
s -= a[k * n + i] * z[k];
|
||||
}
|
||||
z[i] = s / a[i * n + i];
|
||||
}
|
||||
|
||||
Some(z)
|
||||
}
|
||||
|
||||
#[cfg(test)]
|
||||
mod tests {
|
||||
use super::*;
|
||||
|
||||
#[test]
|
||||
fn solves_a_known_system() {
|
||||
// [[4, 1], [1, 3]] z = [1, 2] => z = [1/11, 7/11]
|
||||
let a = vec![4.0, 1.0, 1.0, 3.0];
|
||||
let z = solve_spd(a, &[1.0, 2.0]).unwrap();
|
||||
assert!((z[0] - 1.0 / 11.0).abs() < 1e-12, "{z:?}");
|
||||
assert!((z[1] - 7.0 / 11.0).abs() < 1e-12, "{z:?}");
|
||||
}
|
||||
|
||||
#[test]
|
||||
fn recovers_the_inverse_diagonal() {
|
||||
// A = [[2, -1, 0], [-1, 2, -1], [0, -1, 2]]; inverse diagonal is
|
||||
// [0.75, 1.0, 0.75].
|
||||
let a = vec![2.0, -1.0, 0.0, -1.0, 2.0, -1.0, 0.0, -1.0, 2.0];
|
||||
for (i, expected) in [0.75, 1.0, 0.75].into_iter().enumerate() {
|
||||
let mut e = vec![0.0; 3];
|
||||
e[i] = 1.0;
|
||||
let z = solve_spd(a.clone(), &e).unwrap();
|
||||
assert!((z[i] - expected).abs() < 1e-12, "row {i}: {z:?}");
|
||||
}
|
||||
}
|
||||
|
||||
#[test]
|
||||
fn rejects_a_non_positive_definite_matrix() {
|
||||
// Singular: the second row is a multiple of the first.
|
||||
let a = vec![1.0, 2.0, 2.0, 4.0];
|
||||
assert!(solve_spd(a, &[1.0, 1.0]).is_none());
|
||||
}
|
||||
}
|
||||
@@ -123,6 +123,7 @@ mod game;
|
||||
pub mod gaussian;
|
||||
pub mod graph;
|
||||
mod history;
|
||||
mod joint;
|
||||
mod key_table;
|
||||
mod matrix;
|
||||
mod observer;
|
||||
|
||||
@@ -809,6 +809,93 @@ pub(crate) fn compute_elapsed<T: Time>(last: Option<&T>, current: &T) -> i64 {
|
||||
elapsed.max(0)
|
||||
}
|
||||
|
||||
impl<T: Time> TimeSlice<T> {
|
||||
/// Precision matrix of the joint posterior over this slice's competitors.
|
||||
///
|
||||
/// Message passing produces per-competitor marginals and throws the
|
||||
/// correlation away — `Item::likelihood` is already the projection of an
|
||||
/// event's factor down onto one competitor. So the joint has to be rebuilt
|
||||
/// from the factor structure rather than recovered from the messages.
|
||||
///
|
||||
/// Usefully, a precision matrix depends only on *structure* — who played
|
||||
/// whom, with what weights and what observation noise — and not on the
|
||||
/// observed outcomes. The means are already exact (Gaussian belief
|
||||
/// propagation gets those right even with cycles), so only the second
|
||||
/// moment needs rebuilding.
|
||||
///
|
||||
/// Returns the competitor order and the dense matrix in row-major order.
|
||||
/// Only scored events contribute their factors exactly; see the caller.
|
||||
pub(crate) fn joint_precision<D: Drift<T>>(
|
||||
&self,
|
||||
agents: &CompetitorStore<T, D>,
|
||||
) -> (Vec<Index>, Vec<f64>) {
|
||||
let order: Vec<Index> = self.skills.keys().collect();
|
||||
let n = order.len();
|
||||
let mut row_of: HashMap<Index, usize> = HashMap::with_capacity(n);
|
||||
for (r, idx) in order.iter().enumerate() {
|
||||
row_of.insert(*idx, r);
|
||||
}
|
||||
|
||||
let mut lambda = vec![0.0; n * n];
|
||||
|
||||
// Everything outside this slice enters as each competitor's forward and
|
||||
// backward messages, which message passing treats as independent.
|
||||
for (r, idx) in order.iter().enumerate() {
|
||||
let skill = self.skills.get(*idx).expect("slice key has a skill");
|
||||
lambda[r * n + r] += (skill.forward * skill.backward).pi();
|
||||
}
|
||||
|
||||
for event in &self.events {
|
||||
let EventKind::Scored { score_sigma } = event.kind else {
|
||||
continue;
|
||||
};
|
||||
|
||||
// Teams best-first, matching the diff chain inference builds.
|
||||
let mut order_idx: Vec<usize> = (0..event.teams.len()).collect();
|
||||
order_idx.sort_by(|&a, &b| {
|
||||
event.teams[b]
|
||||
.output
|
||||
.partial_cmp(&event.teams[a].output)
|
||||
.unwrap_or(std::cmp::Ordering::Equal)
|
||||
});
|
||||
|
||||
for pair in order_idx.windows(2) {
|
||||
let (hi, lo) = (pair[0], pair[1]);
|
||||
|
||||
// Contrast vector, and the observation noise that sits on top
|
||||
// of the skills: per-member performance noise plus the score
|
||||
// noise itself.
|
||||
let mut contrast: HashMap<usize, f64> = HashMap::new();
|
||||
let mut noise = score_sigma * score_sigma;
|
||||
|
||||
for (team, sign) in [(hi, 1.0), (lo, -1.0)] {
|
||||
for (m, item) in event.teams[team].items.iter().enumerate() {
|
||||
let w = event.weights[team][m];
|
||||
let beta = agents[item.agent].rating.beta;
|
||||
noise += w * w * beta * beta;
|
||||
*contrast.entry(row_of[&item.agent]).or_insert(0.0) += sign * w;
|
||||
}
|
||||
}
|
||||
|
||||
for (&i, &ci) in &contrast {
|
||||
for (&j, &cj) in &contrast {
|
||||
lambda[i * n + j] += ci * cj / noise;
|
||||
}
|
||||
}
|
||||
}
|
||||
}
|
||||
|
||||
(order, lambda)
|
||||
}
|
||||
|
||||
/// True when every event here is scored, so `joint_precision` is exact.
|
||||
pub(crate) fn all_scored(&self) -> bool {
|
||||
self.events
|
||||
.iter()
|
||||
.all(|e| matches!(e.kind, EventKind::Scored { .. }))
|
||||
}
|
||||
}
|
||||
|
||||
#[cfg(test)]
|
||||
mod tests {
|
||||
use approx::assert_ulps_eq;
|
||||
|
||||
Reference in New Issue
Block a user