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Bayesian inversion

If we had worked hard on a particular test corpus of documents to identify (always with respect to some particular query) which documents were and which were not, it would them be possible to carefully study which features $x_{i}$ were reliably found in relevant documents and which were not. Collecting such statistics for each feature would then allow us to estimate: \[ \Pr({\bf x} | \mathname{Rel}) \] the probabily of any particular set of features ${\bf x}$, given that we know it is \Rel. (Just which statistics we collect, and how, is discussed in more detail in Section §7.4 as part of a more general classification task.) The retrieval question requires that we ask the converse, the probability that for the document we are considering, it should be considered relevant. This inverstion is accomplished via the familiar Bayes Rule: \Pr(\mathname{Rel} | {\bf x}) = {\Pr({\bf x}| \mathname{Rel}) \Pr(\mathname{Rel}) \over \Pr({\bf x})}

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