5.6 Updating odds

The LR has a very clear interpretation from Bayes’ rule, which was introduced in Chapter 3. The rule states that \[\text{posterior odds} = \text{LR} \times \text{prior odds}.\] The LR is the numerical factor by which we multiply prior odds in order to obtain posterior odds. The LR tells us how much to update odds in light of new information.

We saw in Table 5.1 that values of the LR equal to 1 meant that the evidence provided equal support for both propositions when compared to each other. Logically, if evidence provides equal support for two competing propositions, then it should not update any odds prior to collecting the evidence. If we look at Bayes’ theorem above and plug in an LR of 1, then this is exactly what happens. The prior odds for the propositions are equal to the posterior odds for them; observing the evidence did not change the odds.

We also saw that when the LR is greater than 1, then the evidence provides more support for \(H_p\) than proposition \(H_d\). The intuition is that the evidence should increase the odds to be more in favour of \(H_p\) (compared to \(H_d\)) than it previously was. This is confirmed with Bayes’ rule above; when the LR is greater than 1, then the prior odds in favour of \(H_p\) increases when becoming the posterior odds. Observing the evidence was found to be more probable assuming \(H_p\) than \(H_d\), and so the odds were increased in favour of \(H_p\) by a factor equal to the LR.

When the LR is less than 1, the evidence provides more support for \(H_d\) than \(H_p\). This tells us we should update the prior odds to be more in favour of proposition \(H_d\) (compared to \(H_p\)) than it previously was. According to Bayes’ rule, this is exactly what happens; an LR less than 1 means that the prior odds in favour of \(H_p\) are reduced when becoming the posterior odds. Reducing the odds of \(H_p\) means increasing the odds in favour of \(H_d\). Observing the evidence was found to be more probable assuming \(H_d\) than \(H_p\), and so the odds of \(H_p\) were reduced (increased in favour of \(H_d\)) by a factor equal to the LR.