= Bradley-Terry maximum-likelihood estimate on a path
{c}
{title2=$\widehat\theta_i=\prod_{k=i}^{n-1}p_k/(1-p_k)$}
For observed neighbor win fractions $p_0,\ldots,p_{n-1}\in(0,1)$ on the <path graph> $0,1,\ldots,n$, the normalized <Bradley-Terry model> estimate with $\widehat\theta_n=1$ is $\widehat\theta_i=\prod_{k=i}^{n-1}p_k/(1-p_k)$. This is backward propagation of empirical odds along the path. Every edge proportion is fitted exactly, and strict concavity in edge log-ratios proves that this is the unique global maximum.
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