Sigmoid

Sigmoid Relationship to Bradley-Terry Model

Suppose we have scores $s_i,s_j$ for objects $i$ and $j$. Under the Bradley-Terry model, we can express the preference probability as follows:

$$\begin{array}{rl}P(i▹j)& =\frac{e^{s_i}}{e^{s_i}+e^{s_j}}\\ & =\frac{1}{(e^{s_i}+e^{s_j})/e^{s_i}}\\ & =\frac{1}{1+e^{s_j}/e^{s_i}}\\ & =\frac{1}{1+e^{s_j-s_i}}\\ & =\frac{1}{1+e^{-(s_i-s_j)}}\\ & =\sigma (s_i-s_j)\end{array}$$

Where $\sigma (x)=1/(1+e^{-x})$ is the sigmoid function.