trueskill-tt/src/gaussian.rs

use std::ops;

use crate::{MU, N_INF, SIGMA};

/// A Gaussian distribution stored in natural parameters.
///
/// `pi  = 1 / sigma^2`  (precision)
/// `tau = mu * pi`      (precision-adjusted mean)
///
/// Multiplication and division in message passing become pure adds/subs of
/// the stored fields with no `sqrt` or reciprocal in the hot path. `mu()` and
/// `sigma()` are accessors computed on demand.
#[derive(Clone, Copy, PartialEq, Debug)]
pub struct Gaussian {
    pi: f64,
    tau: f64,
}

impl Gaussian {
    /// Construct from mean and standard deviation.
    pub const fn from_ms(mu: f64, sigma: f64) -> Self {
        if sigma == f64::INFINITY {
            Self { pi: 0.0, tau: 0.0 }
        } else if sigma == 0.0 {
            // Point mass at mu. tau = mu * pi = mu * inf.
            // For mu == 0 this is 0; for mu != 0 it is inf * mu = inf (IEEE).
            // Only N00 (mu=0, sigma=0) is used in practice.
            Self {
                pi: f64::INFINITY,
                tau: if mu == 0.0 { 0.0 } else { f64::INFINITY },
            }
        } else {
            let pi = 1.0 / (sigma * sigma);
            Self { pi, tau: mu * pi }
        }
    }

    /// Construct directly from natural parameters.
    #[inline]
    pub(crate) const fn from_natural(pi: f64, tau: f64) -> Self {
        Self { pi, tau }
    }

    #[inline]
    pub fn pi(&self) -> f64 {
        self.pi
    }

    #[inline]
    pub fn tau(&self) -> f64 {
        self.tau
    }

    #[inline]
    pub fn mu(&self) -> f64 {
        // A non-positive precision is an improper (uninformative) Gaussian — its mean is
        // undefined. Treat it like `pi == 0` and return 0. EP message cancellation can land
        // `pi` on a tiny negative value (round-off of exactly zero); without this guard
        // `tau / pi` would yield a spurious finite mean.
        if self.pi <= 0.0 {
            0.0
        } else {
            self.tau / self.pi
        }
    }

    #[inline]
    pub fn sigma(&self) -> f64 {
        // A non-positive precision is improper → infinite standard deviation. Guarding
        // `pi <= 0.0` (not just `== 0.0`) keeps `1.0 / pi.sqrt()` from returning NaN when EP
        // cancellation produces a tiny negative precision (round-off of exactly zero).
        if self.pi <= 0.0 {
            f64::INFINITY
        } else if self.pi.is_infinite() {
            0.0
        } else {
            1.0 / self.pi.sqrt()
        }
    }

    pub(crate) fn delta(&self, other: Gaussian) -> (f64, f64) {
        (
            (self.mu() - other.mu()).abs(),
            (self.sigma() - other.sigma()).abs(),
        )
    }

    pub(crate) fn exclude(&self, other: Gaussian) -> Self {
        let var = self.sigma().powi(2) - other.sigma().powi(2);
        if var <= 0.0 {
            // When sigma_self ≈ sigma_other (including ULP-level rounding differences
            // from the pi→sigma accessor round-trip), the excluded contribution is N00.
            // Computing from_ms(tiny_mu, 0.0) would give {pi:inf, tau:inf}, whose
            // mu() = inf/inf = NaN.  Returning N00 is correct: when both Gaussians
            // carry the same variance, the residual is a point mass at 0.
            return Gaussian::from_ms(0.0, 0.0);
        }
        let mu = self.mu() - other.mu();
        Self::from_ms(mu, var.sqrt())
    }

    pub(crate) fn forget(&self, variance_delta: f64) -> Self {
        let var = self.sigma().powi(2) + variance_delta;
        Self::from_ms(self.mu(), var.sqrt())
    }

    /// EP damping in natural-parameter space: `α·new + (1−α)·self`.
    ///
    /// Used by within-game inference to stabilise oscillating fixed-point
    /// loops on hard graphs. `alpha = 1.0` returns `new` exactly;
    /// `alpha < 1.0` shrinks each per-step update.
    pub fn damp_natural(self, new: Gaussian, alpha: f64) -> Gaussian {
        Gaussian::from_natural(
            alpha * new.pi() + (1.0 - alpha) * self.pi(),
            alpha * new.tau() + (1.0 - alpha) * self.tau(),
        )
    }
}

impl Default for Gaussian {
    fn default() -> Self {
        Self::from_ms(MU, SIGMA)
    }
}

impl ops::Add<Gaussian> for Gaussian {
    type Output = Gaussian;
    /// Variance addition: (mu1 + mu2, sqrt(σ1² + σ2²)).
    /// Used for combining performance and noise; rare relative to mul/div.
    fn add(self, rhs: Gaussian) -> Self::Output {
        let mu = self.mu() + rhs.mu();
        let var = self.sigma().powi(2) + rhs.sigma().powi(2);
        Self::from_ms(mu, var.sqrt())
    }
}

impl ops::Sub<Gaussian> for Gaussian {
    type Output = Gaussian;
    /// (mu1 - mu2, sqrt(σ1² + σ2²)). Same sigma combination as Add.
    fn sub(self, rhs: Gaussian) -> Self::Output {
        let mu = self.mu() - rhs.mu();
        let var = self.sigma().powi(2) + rhs.sigma().powi(2);
        Self::from_ms(mu, var.sqrt())
    }
}

impl ops::Mul<Gaussian> for Gaussian {
    type Output = Gaussian;
    /// Factor product: nat-param add. Hot path — two f64 additions, no sqrt.
    fn mul(self, rhs: Gaussian) -> Self::Output {
        Self::from_natural(self.pi + rhs.pi, self.tau + rhs.tau)
    }
}

impl ops::Mul<f64> for Gaussian {
    type Output = Gaussian;
    fn mul(self, scalar: f64) -> Self::Output {
        if !scalar.is_finite() {
            return N_INF;
        }
        if scalar == 0.0 {
            // Scaling by 0 collapses to a point mass at 0 (sigma' = 0, mu' = 0).
            // This is N00, the additive identity, NOT N_INF.
            return Gaussian::from_ms(0.0, 0.0);
        }
        // sigma' = sigma * |scalar|  =>  pi' = pi / scalar²
        // mu'    = mu * scalar       =>  tau' = tau / scalar
        Self::from_natural(self.pi / (scalar * scalar), self.tau / scalar)
    }
}

impl ops::Div<Gaussian> for Gaussian {
    type Output = Gaussian;
    /// Cavity: nat-param sub. Hot path — two f64 subtractions, no sqrt.
    fn div(self, rhs: Gaussian) -> Self::Output {
        Self::from_natural(self.pi - rhs.pi, self.tau - rhs.tau)
    }
}

#[cfg(test)]
mod tests {
    use super::*;

    #[test]
    fn non_positive_precision_is_improper_not_nan() {
        // EP message cancellation can leave `pi` a tiny negative (round-off of exactly zero).
        // Such a Gaussian is improper/uninformative: mu() must be 0 and sigma() infinite, not
        // NaN. A NaN here propagates through the moment-space `Sub` in the game chain and
        // poisons every skill in the slice.
        let tiny_neg = Gaussian::from_natural(-5.55e-17, -8.88e-16);
        assert_eq!(tiny_neg.mu(), 0.0);
        assert!(tiny_neg.sigma().is_infinite());

        // A frankly-negative precision is treated the same way.
        let neg = Gaussian::from_natural(-1.0, 2.0);
        assert_eq!(neg.mu(), 0.0);
        assert!(neg.sigma().is_infinite());

        // Subtracting such a message must not produce NaN (the original failure path).
        let proper = Gaussian::from_ms(9.75, 1.256);
        let diff = proper - tiny_neg;
        assert!(diff.pi().is_finite() && !diff.pi().is_nan());
        assert!(diff.tau().is_finite() && !diff.tau().is_nan());
    }

    #[test]
    fn test_add() {
        let n = Gaussian::from_ms(25.0, 25.0 / 3.0);
        let m = Gaussian::from_ms(0.0, 1.0);
        let r = n + m;
        assert!((r.mu() - 25.0).abs() < 1e-12);
        assert!((r.sigma() - 8.393118874676116).abs() < 1e-10);
    }

    #[test]
    fn test_sub() {
        let n = Gaussian::from_ms(25.0, 25.0 / 3.0);
        let m = Gaussian::from_ms(1.0, 1.0);
        let r = n - m;
        assert!((r.mu() - 24.0).abs() < 1e-12);
        assert!((r.sigma() - 8.393118874676116).abs() < 1e-10);
    }

    #[test]
    fn test_mul() {
        let n = Gaussian::from_ms(25.0, 25.0 / 3.0);
        let m = Gaussian::from_ms(0.0, 1.0);
        let r = n * m;
        assert!((r.mu() - 0.35488958990536273).abs() < 1e-10);
        assert!((r.sigma() - 0.992876838486922).abs() < 1e-10);
    }

    #[test]
    fn test_div() {
        let n = Gaussian::from_ms(25.0, 25.0 / 3.0);
        let m = Gaussian::from_ms(0.0, 1.0);
        let r = m / n;
        assert!((r.mu() - (-0.3652597402597402)).abs() < 1e-10);
        assert!((r.sigma() - 1.0072787050317253).abs() < 1e-10);
    }

    #[test]
    fn test_n00_is_add_identity() {
        // N00 (sigma=0) is the additive identity for the variance-convolution Add op.
        // N_INF (sigma=inf) is the identity for the EP-product Mul op.
        let g = Gaussian::from_ms(3.0, 2.0);
        let n00 = Gaussian::from_ms(0.0, 0.0);
        let r = n00 + g;
        assert!((r.mu() - g.mu()).abs() < 1e-12);
        assert!((r.sigma() - g.sigma()).abs() < 1e-12);
    }

    #[test]
    fn test_mul_is_factor_product() {
        // n * m in nat-params should be pi_n + pi_m, tau_n + tau_m
        let n = Gaussian::from_ms(2.0, 3.0);
        let m = Gaussian::from_ms(1.0, 2.0);
        let r = n * m;
        let expected_pi = n.pi() + m.pi();
        let expected_tau = n.tau() + m.tau();
        assert!((r.pi() - expected_pi).abs() < 1e-15);
        assert!((r.tau() - expected_tau).abs() < 1e-15);
    }

    #[test]
    fn test_div_is_cavity() {
        let n = Gaussian::from_ms(2.0, 1.0);
        let m = Gaussian::from_ms(1.0, 2.0);
        let r = n / m;
        let expected_pi = n.pi() - m.pi();
        let expected_tau = n.tau() - m.tau();
        assert!((r.pi() - expected_pi).abs() < 1e-15);
        assert!((r.tau() - expected_tau).abs() < 1e-15);
    }

    #[test]
    fn damp_natural_alpha_one_returns_new() {
        let old = Gaussian::from_ms(1.0, 2.0);
        let new = Gaussian::from_ms(5.0, 0.5);
        let damped = old.damp_natural(new, 1.0);
        assert_eq!(damped.pi(), new.pi());
        assert_eq!(damped.tau(), new.tau());
    }

    #[test]
    fn damp_natural_alpha_zero_returns_self() {
        let old = Gaussian::from_ms(1.0, 2.0);
        let new = Gaussian::from_ms(5.0, 0.5);
        let damped = old.damp_natural(new, 0.0);
        assert_eq!(damped.pi(), old.pi());
        assert_eq!(damped.tau(), old.tau());
    }

    #[test]
    fn damp_natural_alpha_half_is_midpoint_in_natural_params() {
        let old = Gaussian::from_ms(1.0, 2.0);
        let new = Gaussian::from_ms(5.0, 0.5);
        let damped = old.damp_natural(new, 0.5);
        let expected_pi = 0.5 * new.pi() + 0.5 * old.pi();
        let expected_tau = 0.5 * new.tau() + 0.5 * old.tau();
        assert!((damped.pi() - expected_pi).abs() < 1e-12);
        assert!((damped.tau() - expected_tau).abs() < 1e-12);
    }
}