## Monday, 14 August 2017

### The likelihood ratio and its use in the 'grooming gangs' news story

This blog has reported many times previously (see links below) about problems with using the likelihood ratio. Recall that the likelihood ratio is commonly used as a measure of the probative value of some evidence E for a hypothesis H; it is defined as the probability of E given H divided by the probability of E given not H.

There is especially great confusion in its use where we have data for the probability of H given E  rather than for the probability of E given H. Look at the somewhat confusing argument here in relation to the offence of 'child grooming' which is taken directly from the book McLoughlin, P. “Easy Meat: Inside Britain’s Grooming Gang Scandal.” (2016):

Given the sensitive nature of the grooming gangs story in the UK and the increasing number of convictions, it is important to get the maths right. The McLoughlin book is the most thoroughly researched work on the subject.  What the author of the book is attempting to determine is the likelihood ratio of the evidence E with respect to the hypothesis H where:

H: “Offence is committed by a Muslim” (so not H means “Offence is committed by a non-Muslim”)

E: “Offence is child grooming”

In this case, the population data cited by McLoughlin provides our priors P(H)=0.05 and, hence, P(not H)=0.95. But we also have the data on child grooming convictions that gives us P(H | E)=0.9 and, hence, P(not H | E)=0.1.

What we do NOT have here is direct data on either P(E|H) or P(E|not H). However, we can still use Bayes theorem to calculate the likelihood ratio since:

So, in the example we get:

Hence, while the method described in the book is confusing, the conclusion arrived at is (almost) correct (the slight error in the result, namely 170.94 instead of 171, is caused by the authors rounding  10 divided by 95% to 10.53).