fix(evidence): accumulate in log space and floor the per-link value

Per-link evidence was multiplied in linear space and logged only at the
end. Each link contributes a probability in (0, 1], so the product over an
n-team game decays geometrically: around a thousand links it flushes to
exactly 0.0 and `ln(0.0)` is `-inf`, which then propagates through the sum
in `History::log_evidence_internal` and takes the whole history with it.
`Game::free_for_all` builds one team per player, so this is reachable at
the competitor counts the T3 benchmarks target.

`Game`, `OwnedGame`, and `time_slice::Event` now carry `log_evidence`
directly, summed over links rather than multiplied then logged.

The cached per-link evidence is also floored at `f64::MIN_POSITIVE`. It
could legitimately reach zero or go negative: `1.0 - cdf(..)` rounds to
zero for a near-certain outcome, and the `erfc` approximation carries
~1e-7 error so `cdf` can exceed 1.0 and make the difference negative —
`ln` of which is NaN.

Existing log-evidence goldens are unchanged, confirming the accumulation
is numerically equivalent in the range where the old form worked.

Co-Authored-By: Claude Opus 5 (1M context) <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01DnsaJg74eNSva3PJjK2eej

src/factor/margin.rs

@@ -64,9 +64,13 @@ impl Factor for MarginFactor {
     }
 }
 
+/// Density of the observed margin under the cavity, clamped to a positive
+/// floor so a far-out observation cannot underflow to `0.0` and make
+/// `log_evidence` `-inf`.
 fn cavity_evidence(cavity: Gaussian, m_obs: f64, sigma: f64) -> f64 {
     let combined_sigma = (cavity.sigma().powi(2) + sigma.powi(2)).sqrt();
-    pdf(m_obs, cavity.mu(), combined_sigma)
+
+    pdf(m_obs, cavity.mu(), combined_sigma).max(f64::MIN_POSITIVE)
 }
 
 #[cfg(test)]

src/factor/trunc.rs

@@ -72,12 +72,20 @@ impl Factor for TruncFactor {
 }
 
 /// P(diff > margin) for non-tie, P(|diff| < margin) for tie.
+///
+/// Clamped to a positive floor: for a near-certain outcome the tail rounds to
+/// exactly 0.0, and the `erfc` approximation used by `cdf` carries ~1e-7 error
+/// so it can even return slightly more than 1.0, making the difference
+/// negative. Either would send `log_evidence` to `-inf` or NaN and poison the
+/// sum across the whole history.
 fn cavity_evidence(diff: Gaussian, margin: f64, tie: bool) -> f64 {
-    if tie {
+    let raw = if tie {
         cdf(margin, diff.mu(), diff.sigma()) - cdf(-margin, diff.mu(), diff.sigma())
     } else {
         1.0 - cdf(margin, diff.mu(), diff.sigma())
-    }
+    };
+
+    raw.clamp(f64::MIN_POSITIVE, 1.0)
 }
 
 #[cfg(test)]

src/game.rs

@@ -37,10 +37,17 @@ impl DiffFactor {
         }
     }
 
-    pub(crate) fn evidence(&self) -> f64 {
+    /// Log of this link's cached evidence.
+    ///
+    /// Accumulating in log space keeps a long diff chain from underflowing:
+    /// each link contributes a probability in `(0, 1]`, so the linear product
+    /// over an n-team game decays geometrically and flushes to zero — and
+    /// `ln(0.0)` is `-inf` — well within the team counts a large free-for-all
+    /// reaches.
+    pub(crate) fn log_evidence(&self) -> f64 {
         match self {
-            Self::Trunc(f) => f.evidence_cached.unwrap_or(1.0),
+            Self::Trunc(f) => f.evidence_cached.unwrap_or(1.0).ln(),
-            Self::Margin(f) => f.evidence_cached.unwrap_or(1.0),
+            Self::Margin(f) => f.evidence_cached.unwrap_or(1.0).ln(),
         }
     }
 
@@ -92,7 +99,7 @@ pub struct OwnedGame<T: Time, D: Drift<T>> {
     p_draw: f64,
     pub(crate) convergence: crate::ConvergenceOptions,
     pub(crate) likelihoods: Vec<Vec<Gaussian>>,
-    pub(crate) evidence: f64,
+    pub(crate) log_evidence: f64,
 }
 
 impl<T: Time, D: Drift<T>> OwnedGame<T, D> {
@@ -113,7 +120,7 @@ impl<T: Time, D: Drift<T>> OwnedGame<T, D> {
             &mut arena,
         );
         let likelihoods = g.likelihoods;
-        let evidence = g.evidence;
+        let log_evidence = g.log_evidence;
         Self {
             teams,
             result,
@@ -121,7 +128,7 @@ impl<T: Time, D: Drift<T>> OwnedGame<T, D> {
             p_draw,
             convergence,
             likelihoods,
-            evidence,
+            log_evidence,
         }
     }
 
@@ -142,7 +149,7 @@ impl<T: Time, D: Drift<T>> OwnedGame<T, D> {
             &mut arena,
         );
         let likelihoods = g.likelihoods;
-        let evidence = g.evidence;
+        let log_evidence = g.log_evidence;
         Self {
             teams,
             result: scores,
@@ -150,7 +157,7 @@ impl<T: Time, D: Drift<T>> OwnedGame<T, D> {
             p_draw: 0.0,
             convergence,
             likelihoods,
-            evidence,
+            log_evidence,
         }
     }
 
@@ -163,7 +170,7 @@ impl<T: Time, D: Drift<T>> OwnedGame<T, D> {
     }
 
     pub fn log_evidence(&self) -> f64 {
-        self.evidence.ln()
+        self.log_evidence
     }
 }
 
@@ -175,7 +182,7 @@ pub struct Game<'a, T: Time = i64, D: Drift<T> = crate::drift::ConstantDrift> {
     p_draw: f64,
     pub(crate) convergence: crate::ConvergenceOptions,
     pub(crate) likelihoods: Vec<Vec<Gaussian>>,
-    pub(crate) evidence: f64,
+    pub(crate) log_evidence: f64,
 }
 
 impl<'a, T: Time, D: Drift<T>> Game<'a, T, D> {
@@ -222,7 +229,7 @@ impl<'a, T: Time, D: Drift<T>> Game<'a, T, D> {
             p_draw,
             convergence,
             likelihoods: Vec::new(),
-            evidence: 0.0,
+            log_evidence: 0.0,
         };
 
         this.likelihoods(arena);
@@ -261,7 +268,7 @@ impl<'a, T: Time, D: Drift<T>> Game<'a, T, D> {
             p_draw: 0.0,
             convergence,
             likelihoods: Vec::new(),
-            evidence: 0.0,
+            log_evidence: 0.0,
         };
 
         this.likelihoods_scored(arena, score_sigma);
@@ -355,7 +362,7 @@ impl<'a, T: Time, D: Drift<T>> Game<'a, T, D> {
             arena.lhood_lose[n_teams - 1] = pw_last - links[n_diffs - 1].msg();
         }
 
-        let evidence: f64 = links.iter().map(|l| l.evidence()).product();
+        let log_evidence: f64 = links.iter().map(DiffFactor::log_evidence).sum();
 
         // Inverse permutation: inv_buf[orig_i] = sorted_i.
         arena.inv_buf.resize(n_teams, 0);
@@ -386,11 +393,11 @@ impl<'a, T: Time, D: Drift<T>> Game<'a, T, D> {
             })
             .collect::<Vec<_>>();
 
-        (evidence, likelihoods)
+        (log_evidence, likelihoods)
     }
 
     fn likelihoods(&mut self, arena: &mut ScratchArena) {
-        let (evidence, likelihoods) = self.run_chain(arena, |i, sort_buf, vars| {
+        let (log_evidence, likelihoods) = self.run_chain(arena, |i, sort_buf, vars| {
             let tie = self.result[sort_buf[i]] == self.result[sort_buf[i + 1]];
             let margin = if self.p_draw == 0.0 {
                 0.0
@@ -405,17 +412,17 @@ impl<'a, T: Time, D: Drift<T>> Game<'a, T, D> {
             let vid = vars.alloc(N_INF);
             DiffFactor::Trunc(TruncFactor::new(vid, margin, tie))
         });
-        self.evidence = evidence;
+        self.log_evidence = log_evidence;
         self.likelihoods = likelihoods;
     }
 
     fn likelihoods_scored(&mut self, arena: &mut ScratchArena, score_sigma: f64) {
-        let (evidence, likelihoods) = self.run_chain(arena, |i, sort_buf, vars| {
+        let (log_evidence, likelihoods) = self.run_chain(arena, |i, sort_buf, vars| {
             let m_obs = self.result[sort_buf[i]] - self.result[sort_buf[i + 1]];
             let vid = vars.alloc(N_INF);
             DiffFactor::Margin(MarginFactor::new(vid, m_obs, score_sigma))
         });
-        self.evidence = evidence;
+        self.log_evidence = log_evidence;
         self.likelihoods = likelihoods;
     }
 
@@ -433,7 +440,7 @@ impl<'a, T: Time, D: Drift<T>> Game<'a, T, D> {
     }
 
     pub fn log_evidence(&self) -> f64 {
-        self.evidence.ln()
+        self.log_evidence
     }
 }
 

src/time_slice.rs

@@ -87,7 +87,7 @@ struct Team {
 #[derive(Debug)]
 pub(crate) struct Event {
     teams: Vec<Team>,
-    evidence: f64,
+    log_evidence: f64,
     weights: Vec<Vec<f64>>,
     kind: EventKind,
 }
@@ -161,7 +161,7 @@ impl Event {
             }
         }
 
-        self.evidence = g.evidence;
+        self.log_evidence = g.log_evidence;
     }
 }
 
@@ -296,7 +296,7 @@ impl<T: Time> TimeSlice<T> {
 
             Event {
                 teams,
-                evidence: 0.0,
+                log_evidence: 0.0,
                 weights,
                 kind: kinds[e],
             }
@@ -353,7 +353,7 @@ impl<T: Time> TimeSlice<T> {
                     }
                 }
 
-                event.evidence = g.evidence;
+                event.log_evidence = g.log_evidence;
             }
         } else {
             self.sweep_color_groups(agents);
@@ -530,26 +530,28 @@ impl<T: Time> TimeSlice<T> {
             let teams = event.within_priors(online, forward, &self.skills, agents);
             let result = event.outputs();
             match event.kind {
-                EventKind::Ranked => Game::ranked_with_arena(
+                EventKind::Ranked => {
-                    teams,
+                    Game::ranked_with_arena(
-                    &result,
+                        teams,
-                    &event.weights,
+                        &result,
-                    self.p_draw,
+                        &event.weights,
-                    self.convergence,
+                        self.p_draw,
-                    arena,
+                        self.convergence,
-                )
+                        arena,
-                .evidence
+                    )
-                .ln(),
+                    .log_evidence
-                EventKind::Scored { score_sigma } => Game::scored_with_arena(
+                }
-                    teams,
+                EventKind::Scored { score_sigma } => {
-                    &result,
+                    Game::scored_with_arena(
-                    &event.weights,
+                        teams,
-                    score_sigma,
+                        &result,
-                    self.convergence,
+                        &event.weights,
-                    arena,
+                        score_sigma,
-                )
+                        self.convergence,
-                .evidence
+                        arena,
-                .ln(),
+                    )
+                    .log_evidence
+                }
             }
         };
 
@@ -560,7 +562,7 @@ impl<T: Time> TimeSlice<T> {
                     .map(|event| run_event(event, &mut arena))
                     .sum()
             } else {
-                self.events.iter().map(|event| event.evidence.ln()).sum()
+                self.events.iter().map(|event| event.log_evidence).sum()
             }
         } else if online || forward {
             self.events
@@ -584,7 +586,7 @@ impl<T: Time> TimeSlice<T> {
                         .flat_map(|team| &team.items)
                         .any(|item| targets.contains(&item.agent))
                 })
-                .map(|event| event.evidence.ln())
+                .map(|event| event.log_evidence)
                 .sum()
         }
     }

tests/degenerate_inputs.rs

@@ -193,3 +193,55 @@ fn convergence_reports_are_finite_across_many_teams() {
         }
     }
 }
+
+/// A long diff chain underflows a linear evidence product: each link
+/// contributes a probability in (0, 1], so ~1000 links flush the product to
+/// exactly 0.0 and `ln(0.0)` is `-inf`. Accumulating in log space keeps it
+/// finite.
+#[test]
+fn log_evidence_survives_a_long_diff_chain() {
+    let holders: Vec<[R; 1]> = (0..1200).map(|_| [rating()]).collect();
+    let teams: Vec<&[R]> = holders.iter().map(|t| t.as_slice()).collect();
+    let game = Game::ranked(
+        &teams,
+        Outcome::ranking(0..holders.len() as u32),
+        &GameOptions::default(),
+    )
+    .unwrap();
+
+    let log_evidence = game.log_evidence();
+    assert!(
+        log_evidence.is_finite(),
+        "1200-team log-evidence must be finite, got {log_evidence}"
+    );
+    assert!(
+        log_evidence < 0.0,
+        "log-evidence of a probability must be negative, got {log_evidence}"
+    );
+}
+
+/// A near-certain outcome rounds the losing tail to exactly zero in the
+/// `erfc` approximation; the evidence floor keeps `ln` finite.
+#[test]
+fn log_evidence_finite_for_near_certain_outcome() {
+    let overwhelming = R::new(Gaussian::from_ms(5_000.0, 0.5), 1.0, ConstantDrift(0.0));
+    let hopeless = R::new(Gaussian::from_ms(-5_000.0, 0.5), 1.0, ConstantDrift(0.0));
+    let a = [overwhelming];
+    let b = [hopeless];
+    let teams: Vec<&[R]> = vec![&a, &b];
+
+    let game = Game::ranked(&teams, Outcome::winner(0, 2), &GameOptions::default()).unwrap();
+    assert!(
+        game.log_evidence().is_finite(),
+        "got {}",
+        game.log_evidence()
+    );
+
+    // And the reverse — a colossal upset — must also stay finite.
+    let upset = Game::ranked(&teams, Outcome::winner(1, 2), &GameOptions::default()).unwrap();
+    assert!(
+        upset.log_evidence().is_finite(),
+        "upset log-evidence must be finite, got {}",
+        upset.log_evidence()
+    );
+}