ScienceClaw & Infinite Framework: FH-BT Model

• ScienceClaw + Infinite Framework is a probabilistic model that generalizes the classical Bradley-Terry framework using full historical data for paired competition outcomes.
• It employs logistic regression with gradient-based optimization and integrates Élő-style stochastic updates for real-time adaptation.
• The framework further extends to model draws, incorporate covariates, and apply low-rank matrix completion, achieving state-of-the-art predictive performance.

The Full-History Bradley-Terry (FH-BT) model is a statistical learning method for modelling and predicting paired competition outcomes, systematically integrating logistic regression principles with probabilistic strengths over the entire competition history. It generalizes the classical Bradley-Terry framework to yield principled supervised and online predictive capabilities for paired comparisons, featuring flexible batch and stochastic (Élő-style) learning, draw and feature modeling, and matrix-completion-based structural extensions. FH-BT finds application in sports outcome prediction and related domains demanding interpretable, computationally efficient probabilistic models of pairwise competition (Király et al., 2017).

1. Mathematical Foundations and Model Formulation

In the classical Bradley-Terry model, each competitor $i$ is associated with a latent strength parameter $\theta_i$. Competition outcomes $$Y_{ij} \in \{1= i \text{ wins}, 0 = i \text{ loses}\}$$ are modeled as $Y_{ij} \sim \operatorname{Bernoulli}(p_{ij})$ with

$p_{ij} = \sigma(\theta_i - \theta_j + h)$

where $h$ is an optional home-advantage term and $\sigma(x) = (1 + e^{-x})^{-1}$ denotes the logistic sigmoid.

The FH-BT model treats $\theta = (\theta_1,\ldots,\theta_Q)$ as fixed but unknown and fits these latent strengths by maximizing the likelihood over the full history of observed outcomes $D = \{(i_k, j_k, Y^{(k)})\}_{k=1}^N$. The batch log-likelihood for data $D$ is

$$\ell(\theta|D) = \sum_{k=1}^N \left[ Y^{(k)}_{i_k j_k}\log \sigma(\theta_{i_k} - \theta_{j_k} + h) + (1 - Y^{(k)}_{i_k j_k})\log(1 - \sigma(\theta_{i_k} - \theta_{j_k} + h)) \right].$$

This formulation admits efficient gradient-based optimization with explicit gradients:

$$\frac{\partial \ell}{\partial \theta_i} = \sum_{k: i_k = i}(Y^{(k)}_{i_k j_k} - p_{i_k j_k}) - \sum_{k: j_k = i}(Y^{(k)}_{i_k j_k} - p_{i_k j_k}).$$

The batch fit yields a global optimum $\hat{\theta}^\star$ (“full-history” fit), providing statistical consistency under standard assumptions (Király et al., 2017).

2. Online Updating and Élő Connection

FH-BT accommodates online adaptation through stochastic gradient updates. Upon observing a new result $(i, j, Y_{ij})$, latent strengths are updated as

$$\theta_i \gets \theta_i + K \cdot (Y_{ij} - p_{ij}), \qquad \theta_j \gets \theta_j - K \cdot (Y_{ij} - p_{ij}),$$

where $K$ (the “K-factor”) is a small step-size. This update is equivalent to stochastic gradient ascent on the FH-BT likelihood and generalizes the Élő system in a statistically principled manner, bridging classical heuristics and modern supervised learning (Király et al., 2017).

A two-stage online FH-BT strategy, utilizing an initial full-history batch fit followed by K-factor updates, outperforms pure online or naive batch re-fit strategies in predictive performance on real datasets (paired t-test $p \ll 10^{-4}$).

3. Extensions: Draws, Covariates, and Generalized Linear Modeling

The FH-BT model generalizes to accommodate draws and covariates via structured log-odds extensions. For ternary outcomes $$Y_{ij} \in \{\text{win}, \text{draw}, \text{lose}\}$$, a proportional-odds approach introduces a shared draw parameter $\varphi$:

• $\operatorname{logit} P(\text{win}) = L_{ij}$
• $$\operatorname{logit} P(\text{win or draw}) = L_{ij} + \varphi$$

with probabilities:

• $P(\text{win}) = \sigma(L_{ij})$
• $$P(\text{draw}) = \sigma(-L_{ij}) - \sigma(-L_{ij} - \varphi)$$
• $P(\text{lose}) = \sigma(-L_{ij} - \varphi)$

Covariate incorporation follows the GLM paradigm:

$$L_{ij} = \theta_i - \theta_j + h + \langle \lambda, X_{ij} \rangle$$

or more generally,

$$L_{ij} = \langle \beta, X_{ij} \rangle + \langle \gamma, X_i \rangle + \langle \delta, X_j \rangle + \alpha_{ij}$$

with structural constraints (e.g., low-rankness, anti-symmetry, shared intercepts) ensuring tractable parameterization. Allowing a single covariate, such as team promotion, yields significant log-loss improvements over the vanilla FH-BT model ($p \approx 0.002$) (Király et al., 2017).

4. Matrix-Completion and Structural Embeddings

FH-BT supports low-rank extensions inspired by matrix-completion formulations. The log-odds matrix $L \in \mathbb{R}^{Q\times Q}$ is anti-symmetric ($L = -L^T$) and, in the classical (one-factor) case, has rank two. Structural extensions include:

• Two-factor FH-BT: $L = uv^T - vu^T$
• Rank-4 extension: $$L = uv^T - vu^T + \theta\mathbf{1}^T - \mathbf{1}\theta^T$$

Alternatively, nuclear norm regularization may be applied:

$$\min_L \| \hat{H} - L \|_F^2 + \lambda \| L \|_{\ast} \quad \text{subject to} \quad L + L^T = 0$$

where $\hat{H} = \log(W) - \log(W)^T$ is the empirical log-odds and $\|\cdot\|_\ast$ denotes the nuclear norm (Király et al., 2017).

These structural variants enable controlled expressive power for capturing latent factors beyond simple strength hierarchies, improving predictive accuracy when match outcomes are governed by more complex latent structures.

5. Empirical Validation and Performance

Experimental validation proceeds on both synthetic and real-world data. On synthetically generated outcomes using rank-2 or rank-4 log-odds, FH-BT rank-matched extensions significantly outperform the classical BT model in both accuracy and log-likelihood (one-sided Wilcoxon $p \ll 10^{-4}$).

For English Premier League data, key results include:

• The best FH-BT variant achieves approximately 52.8% win/draw/lose classification accuracy (95% CI: [50.6%, 55.0%]) and mean log-loss near $-0.980$
• Contemporary betting odds (Bet365) yield 54.1%/$-0.967$

These results demonstrate that FH-BT's enhanced expressivity yields state-of-the-art predictions close to those of betting markets (Király et al., 2017).

6. Significance and General Properties

FH-BT provides a statistically principled, GLM-style probabilistic model for paired competition. It unifies:

• Supervised probabilistic prediction for binary and ternary outcomes
• Batch (full-history) and online (stochastic, Élő-style) learning
• Covariate incorporation without loss of parsimony or computational efficiency
• Structured, low-rank latent factor embeddings

The model’s ability to leverage full historical data for latent strength inference, together with online updates for real-time adaptation, offers a robust framework applicable not only to sports prediction but to any paired comparison setting requiring interpretable, adaptive probabilistic modeling (Király et al., 2017).