trueskill-tt/README.md

# TrueSkill - Through Time

Rust port of [TrueSkillThroughTime.py](https://github.com/glandfried/TrueSkillThroughTime.py).

## Other implementations

- [ttt-scala](https://github.com/ankurdave/ttt-scala)
- [ChessAnalysis #F](https://github.com/lucasmaystre/ChessAnalysis)
- [TrueSkillThroughTime.jl](https://github.com/glandfried/TrueSkillThroughTime.jl)
- [TrueSkillThroughTime.R](https://github.com/glandfried/TrueSkillThroughTime.R)
- [TrueSkill Through Time: Revisiting the History of Chess](https://www.microsoft.com/en-us/research/wp-content/uploads/2008/01/NIPS2007_0931.pdf)
- [TrueSkill Through Time. The full scientific documentation](https://glandfried.github.io/publication/landfried2021-learning/)

## Drift

Skill drift models how a competitor's true skill can change between appearances.
Each time they reappear after a gap, their skill uncertainty is widened by the
drift model before the new evidence is incorporated.

Drift is represented by the `Drift` trait (`src/drift.rs`), generic over the
history's time type:

```text
pub trait Drift<T: Time>: Copy + Debug + Send + Sync {
    fn variance_delta(&self, from: &T, to: &T) -> f64;
    fn variance_for_elapsed(&self, elapsed: i64) -> f64;
}
```

Both methods return the amount to add to `σ²`, not to `σ`. `variance_delta`
works from two timestamps; `variance_for_elapsed` takes an already-computed
elapsed count, and is used on the paths that cache it. `Gaussian::forget`
applies the result entirely in variance space — `from_mv(mu, variance() +
variance_delta)` — taking no square root.

That block is a quotation rather than a doctest. The custom-drift example below
is compiled by CI, so it is what actually pins the signature.

### ConstantDrift

The built-in `ConstantDrift` implements a linear random walk — skill uncertainty
grows proportionally to time:

```text
variance_delta = elapsed * γ²
```

This is the standard TrueSkill Through Time model. Pass a `ConstantDrift(gamma)`
when constructing a `Rating`:

```rust
use trueskill_tt::{ConstantDrift, Gaussian, Rating};

// gamma = 0.1 means skill can shift ~0.1 per time unit.
let rating: Rating<i64, ConstantDrift> =
    Rating::new(Gaussian::from_ms(0.0, 6.0), 1.0, ConstantDrift(0.1));

assert_eq!(rating.drift().0, 0.1);
```

The type annotation is load-bearing: `ConstantDrift` implements `Drift<T>` for
every `T: Time`, so without it `T` is ambiguous.

### Custom drift

Implement `Drift<T>` to express any other model. For example, a drift that
saturates after a long absence, with uncertainty growing as the square root of
elapsed time instead of linearly:

```rust
use trueskill_tt::{Drift, Gaussian, History, Rating, Time};

#[derive(Clone, Copy, Debug)]
struct SqrtDrift {
    gamma: f64,
}

impl<T: Time> Drift<T> for SqrtDrift {
    fn variance_delta(&self, from: &T, to: &T) -> f64 {
        let elapsed = from.elapsed_to(to).max(0) as f64;
        elapsed.sqrt() * self.gamma * self.gamma
    }

    fn variance_for_elapsed(&self, elapsed: i64) -> f64 {
        (elapsed.max(0) as f64).sqrt() * self.gamma * self.gamma
    }
}

// On a single Rating:
let rating: Rating<i64, SqrtDrift> =
    Rating::new(Gaussian::from_ms(0.0, 6.0), 1.0, SqrtDrift { gamma: 0.5 });

// Or for a whole History, via the builder:
let history = History::builder().drift(SqrtDrift { gamma: 0.5 }).build();

assert_eq!(rating.beta(), 1.0);
assert_eq!(history.log_evidence(), 0.0);
```

`HistoryBuilder::drift` is the only way to set a history's drift model; there is
no `gamma()` shorthand. The default is `ConstantDrift(GAMMA)`.

### Per-competitor drift

A `History` has one drift model, but individual competitors can scale it.
`Member::with_drift_scale(s)` multiplies the drift *variance* that competitor
accumulates, so `s` is in the same units as `gamma`: `ConstantDrift(g)` at
scale `s` behaves exactly as `ConstantDrift(g * s)` would, for that competitor
alone.

`0.0` pins a competitor still. That is what makes a **fixed reference point**
expressible in the same graph as moving competitors — a bot at a known
strength, a rating floor, a course difficulty:

```rust
use trueskill_tt::{ConstantDrift, Event, History, Member, Outcome, Team};

let mut h = History::builder().drift(ConstantDrift(0.1)).build();

h.add_events(vec![Event {
    time: 0,
    teams: [
        Team::with_members([Member::new("player")]),
        // A course does not improve. Pin it, and the round's evidence
        // lands on the player instead of being split between the two.
        Team::with_members([Member::new("layout_7").with_drift_scale(0.0)]),
    ]
    .into_iter()
    .collect(),
    outcome: Outcome::winner(0, 2),
}])
.unwrap();

h.converge().unwrap();
```

Like `with_prior`, the scale is **competitor configuration captured at first
appearance** — setting it on a key the history already knows has no effect. It
must be finite and non-negative; ingestion otherwise fails with
`InferenceError::InvalidParameter`.

Note that the fluent `EventBuilder` (`h.event(t).team([...])`) sets weights but
not `drift_scale` or `prior`; those need the typed `Event` / `Team` / `Member`
shape shown above.

## Scored outcomes

Use `Outcome::scores([...])` when you have continuous per-team scores rather
than just ranks. Adjacent score margins flow into a `MarginFactor` that adds
soft Gaussian evidence about the latent performance diff. Configure
`HistoryBuilder::score_sigma(σ)` to control how much you trust the margins
(smaller σ = more trust).

```rust
use trueskill_tt::History;

let mut h = History::builder().score_sigma(2.0).build();
h.event(1)
    .team(["alice"])
    .team(["bob"])
    .scores([21.0, 9.0])
    .commit()
    .unwrap();
h.converge().unwrap();
```

## Prediction

`predict_outcome` gives the full distribution over finishing orders. Each entry
is a rank vector in the same shape `Outcome::ranking` takes — equal ranks mean a
tie — so an outcome feeds straight back into inference.

```rust
use trueskill_tt::History;

let mut h = History::builder().p_draw(0.1).build();
h.record_winner(&"alice", &"bob", 1).unwrap();
h.converge().unwrap();

let p = h.predict_outcome(&[&[&"alice"], &[&"bob"]]).unwrap();

// Probabilities are exhaustive and disjoint, so they sum to one.
assert!((p.total() - 1.0).abs() < 1e-6);

let (best, likelihood) = p.most_likely().unwrap();
println!("most likely: {best:?} at {likelihood:.3}");
println!("draw:        {:.3}", p.probability_of(&[0, 0]));
```

Supports any number of teams. Because the outcome space grows factorially, the
full distribution is capped at `MAX_PREDICTED_TEAMS`; two cheaper entry points
stay available at any size:

- `predict_win_probabilities(teams)` — `P(team i finishes strictly first)`,
  quadratic in team count.
- `predict_ranking(teams, ranks)` — one specific finishing order.

Unknown keys are an error by default, not a silent omission: a team the history
has never seen cannot produce a confident-looking probability. The error names
the key, and every key must already be known — pre-filter with `lookup` or
`current_skill` if your caller cannot guarantee that.

If predicting for competitors you have never seen is the point rather than a
mistake, say so once:

```rust
use trueskill_tt::{History, UnknownKeys};

let h = History::builder().unknown_keys(UnknownKeys::Prior).build();
```

An unknown competitor is then answered from the configured prior, which is the
honest reading — you have no evidence about them — and correctly *widens* a team
that contains one. There is deliberately no "skip the member" mode: a team's
performance is the sum of its members, so dropping one would make the model more
certain because it knows less.

### Asking about one competitor

`Gaussian` answers tail questions directly, which is what a stopping rule needs:

```rust
use trueskill_tt::History;

let mut h = History::builder().build();
h.record_winner(&"alice", &"bob", 1).unwrap();
let _ = h.converge().unwrap();

let skill = h.current_skill(&"alice").unwrap();

// "How sure am I that this is below the cutoff?" — a probability, not a
// `mu + z * sigma` band whose confidence drifts as sigma changes.
let _ = skill.probability_below(20.0);

// Use this rather than `1.0 - probability_below(x)`: the complement cancels
// away every digit in the upper tail, which is where a stopping rule lives.
let _ = skill.probability_above(30.0);
```

## Which match to play next

`quality()` measures whether a matchup is *fair*. That is not the same as
whether it is *informative*, and the two only coincide for two evenly matched
competitors. When each observation costs something, ask
`expected_information_gain` instead — the outcome-weighted divergence between
what you believe now and what you would believe afterwards.

```rust
use trueskill_tt::History;

let mut h = History::builder().build();
for t in 1..=10 {
    h.record_winner(&"veteran", &"regular", t).unwrap();
    h.record_winner(&"regular", &"veteran", t + 100).unwrap();
}
h.record_winner(&"veteran", &"newcomer", 500).unwrap();
h.converge().unwrap();

let settled = h.expected_information_gain(&[&[&"veteran"], &[&"regular"]]).unwrap();
let unknown = h.expected_information_gain(&[&[&"veteran"], &[&"newcomer"]]).unwrap();

// Playing the newcomer teaches you more than replaying a settled rivalry.
assert!(unknown > settled);
```

The result is in nats, and is bounded by the entropy of the outcome: at most
`ln 2 ≈ 0.693` for a two-way result, `ln 3` once draws are possible, `ln k` for
`k` outcomes. A value near zero means you already know how it ends.

This costs one full inference pass **per possible outcome**, so it is far more
expensive than `quality()`. Scoring every pairing among `n` competitors is
`O(n² × outcomes)` passes — shortlist with `quality()` or
`predict_win_probabilities` first, then score only the shortlist.

## Todo

- [x] Implement approx for Gaussian
- [x] Add more tests from `TrueSkillThroughTime.jl`
- [x] Generalise a time axis — `Time` is now a trait (`Untimed`, `i64`), not an enum
- [x] Add examples (`examples/atp.rs`, `examples/scored.rs`)
- [x] Add Observer (`Observer` / `NullObserver`)
- [x] Benchmark the inference loop (`benches/batch.rs`, `benches/history_converge.rs`, `benches/ingest.rs`)
- [x] N-team `predict_outcome` with draw mass, and `expected_information_gain`
- [x] Cross-check `quality()` against [sublee/trueskill](https://github.com/sublee/trueskill/tree/master) — N identical teams follow the closed form `(1/5)^((n-1)/2)` for the conventional parameters, asserted for n = 2..10, and the n=3/n=5 values (0.200, 0.040) match the reference package

## License

Licensed under either of

- Apache License, Version 2.0 ([LICENSE-APACHE](LICENSE-APACHE) or
  <http://www.apache.org/licenses/LICENSE-2.0>)
- MIT license ([LICENSE-MIT](LICENSE-MIT) or
  <http://opensource.org/licenses/MIT>)

at your option.

### Contribution

Unless you explicitly state otherwise, any contribution intentionally submitted
for inclusion in the work by you, as defined in the Apache-2.0 license, shall be
dual licensed as above, without any additional terms or conditions.