constituents
- with \(\sigma\qty(z) = \frac{1}{1+e^{-z}}\) sigmoid model…
- \(i\) user
- \(x\) context (prompt, etc.)
- \(a,b\) candidate pairwise items
- \(r\qty(i,x,a)\) the representation of the choice
requirements
The Brady-Terry gives:
\begin{equation} P\qty(a \succ b | i, x) = \sigma \qty(r\qty(i,x,a) - r\qty(i,x,b)) \end{equation}
additional information
properties of Bradly-Terry
Suppose you are only observing binary cases, how do we induce Choice Data that are ordered preferences?
The following three statements are equivalent.
Brady Terry For \(r: X \to \qty(0, \infty)\) with
\begin{equation} P\qty( j \succ j’) = \sigma \qty(r_{j} - r_{j’}) \end{equation}
Triangle property for all \(a,b,c\)
\begin{equation} P\qty(j \succ j’) P\qty(j’ \succ j’’) P\qty(j ’’ \succ j) = P\qty(j \succ j’’) P \qty(j’’ \succ j’) P\qty(j’ \succ j) \end{equation}
Specific Objectivity Define odds \(o\qty(j,j’) = \log \frac{P\qty(j > j’)}{P\qty(j’ > j)}\), we have:
\begin{equation} o\qty(j, j’’) - o\qty(j’,j’’) = o\qty(j,j’) \end{equation}
assumptions of Bradly-Terry
- utility scores will still induce a transitive ordering
- adding some constant to all response scores changes no pairwise probability
- binary label means that you can’t represent no data or ties
- that the predictions are IID to the population you are studying