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Equation 5 · Part 3 · Frontier AI Model Comparisons: A First-Principles Introduction

Symbol B

P(A≻B)=11+e−(θA−θB).P(A \succ B) = \frac{1}{1 + e^{-(\theta_A - \theta_B)}}.
BB

What this part means

B occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.

Its job in the formula

B occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.

The passage around this formula

…rather than a raw win count [ 6 ] . The relevant object is a Bradley–Terry model: each system i is assigned a latent strength θi\theta_i , and the estimated probability that system A ’s output is preferred over system B ’s in a given comparison is P(A≻B)=11+e−(θA−θB)P(A \succ B) = \frac{1}{1 + e^{-(\theta_A - \theta_B)}}. What this equation exposes matters more than the arithmetic: θ\theta is fitted from the specific population of prompts and voters that generated the comparisons , not from a fixed task suite. A system’s rating is a statement about how it fares…

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