5.9 Doping posterior odds

Now let’s look at the posterior odds: what are the odds of the athlete doping versus not doping given a positive result of the test?

From Figure 5.2 we can extract the odds of a randomly selected athlete doping, versus not, prior to being tested. This is given by \(\frac{200}{9800}\), which can be simplified to \(\frac{1}{49}\) and also expressed as odds of \(1:49\), or 49 to 1 against doping. These prior odds represent a conversion of the base rate of 0.02 to odds format.

Using Bayes’ rule and the LR, the posterior odds are \[\begin{align} \text{posterior odds} &= \text{LR} \times \text{prior odds}, \\ \text{posterior odds} &= 19 \times \frac{1}{49}, \\ \text{posterior odds} &= \frac{19}{49}, \end{align}\] corresponding to posterior odds of \(19:49\), or 49 to 19 against doping given a positive test result. Converting this back to probability gives \(\frac{19}{19+49}=0.2794118\), about 0.28. This is the same as the answer in Section 3.19.

Notice that the likelihood ratio for a positive test gave support in favour of \(H_p\) (the athlete doping) but the posterior odds were still in favour of \(H_d\) (the athlete not doping). This was because the prior odds were more heavily in favour of \(H_d\) than the factor by which the LR supported \(H_p\). Committing the prosecutor’s fallacy here would not only give an inaccurate answer for the posterior odds of doping, but it would mistake the odds as being in favour of \(H_p\) when they are actually in favour of \(H_d\). This is an important point; the likelihood ratio alone does not tell us anything about the posterior odds in favour of a proposition, it only tells us the relative size of the posterior odds compared to the prior odds. For the fact finder, this confirms that LRs should be used to update beliefs about competing propositions (as well as using other evidence), and not as beliefs about competing propositions.