1Cademy - Log-Probability of a Ranked Sequence

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Log-Probability of a Ranked Sequence

The log-probability of an ordered list or ranked sequence of preferences, $\mathring{Y}$ , given an input $\mathbf{x}$ , can be defined as the sum of the conditional log-probabilities at each stage of selection. For an ordered list $\mathring{Y}$ given by $\mathbf{y}_{j_1} \succ \mathbf{y}_{j_2} \succ \cdots \succ \mathbf{y}_{j_N}$ , the overall log-probability is expressed as: $\log \Pr(\mathring{Y} | \mathbf{x}) = \log \Pr(\mathbf{y}_{j_1} \succ \mathbf{y}_{j_2} \succ \cdots \succ \mathbf{y}_{j_N} | \mathbf{x})$ $= \log \Pr(\mathbf{y}_{j_1} | \mathbf{x}, \{\mathbf{y}_{j_1},\mathbf{y}_{j_2},...,\mathbf{y}_{j_N}\}) + \log \Pr(\mathbf{y}_{j_2} | \mathbf{x}, \{\mathbf{y}_{j_2},...,\mathbf{y}_{j_N}\}) + \cdots + \log \Pr(\mathbf{y}_{j_N} | \mathbf{x}, \{\mathbf{y}_{j_N}\})$ $= \sum_{k = 1}^{N} \log \Pr(\mathbf{y}_{j_k} | \mathbf{x}, \mathring{Y}_{\ge k})$ where $\mathring{Y}_{\ge k}$ represents the subset of the list of outputs that remain unselected at the $k$ -th stage, i.e., $\mathring{Y}_{\ge k} = \{\mathbf{y}_{j_k},...,\mathbf{y}_{j_N}\}$ .

Updated 2026-05-02

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