Statistics and the law: the prosecutor’s fallacy

This post arose from a recent meeting at the Royal Society. It was organised by Julie Maxton to discuss the application of statistical methods to legal problems. I found myself sitting next to an Appeal Court Judge who wanted more explanation of the ideas. Here it is.

Some preliminaries

The papers that I wrote recently were about the problems associated with the interpretation of screening tests and tests of significance. They don’t allude to legal problems explicitly, though the problems are the same in principle.  They are all open access. The first appeared in 2014:
http://rsos.royalsocietypublishing.org/content/1/3/140216

Since the first version of this post, March 2016, I’ve written two more papers and some popular pieces on the same topic. There’s a list of them at http://www.onemol.org.uk/?page_id=456.
I also made a video for YouTube of a recent talk.

In these papers I was interested in the false positive risk (also known as the false discovery rate) in tests of significance. It turned out to be alarmingly large. That has serious consequences for the credibility of the scientific literature. In legal terms, the false positive risk means the proportion of cases in which, on the basis of the evidence, a suspect is found guilty when in fact they are innocent. That has even more serious consequences.

Although most of what I want to say can be said without much algebra, it would perhaps be worth getting two things clear before we start.

The rules of probability.

(1) To get any understanding, it’s essential to understand the rules of probabilities, and, in particular, the idea of conditional probabilities. One source would be my old book, Lectures on Biostatistics (now free), The account on pages 19 to 24 give a pretty simple (I hope) description of what’s needed. Briefly, a vertical line is read as “given”, so Prob(evidence | not guilty) means the probability that the evidence would be observed given that the suspect was not guilty.

(2) Another potential confusion in this area is the relationship between odds and probability. The relationship between the probability of an event occurring, and the odds on the event can be illustrated by an example. If the probability of being right-handed is 0.9, then the probability of being not being right-handed is 0.1.  That means that 9 people out of 10 are right-handed, and one person in 10 is not. In other words for every person who is not right-handed there are 9 who are right-handed. Thus the odds that a randomly-selected person is right-handed are 9 to 1. In symbols this can be written

\[ \mathrm{probability=\frac{odds}{1 + odds}} \]

In the example, the odds on being right-handed are 9 to 1, so the probability of being right-handed is 9 / (1+9) = 0.9.

Conversely,

\[ \mathrm{odds =\frac{probability}{1 – probability}} \]

In the example, the probability of being right-handed is 0.9, so the odds of being right-handed are 0.9 / (1 – 0.9) = 0.9 / 0.1 = 9 (to 1).

With these preliminaries out of the way, we can proceed to the problem.

The legal problem

The first problem lies in the fact that the answer depends on Bayes’ theorem. Although that was published in 1763, statisticians are still arguing about how it should be used to this day.  In fact whenever it’s mentioned, statisticians tend to revert to internecine warfare, and forget about the user.

Bayes’ theorem can be stated in words as follows

\[ \mathrm{\text{posterior odds ratio} = \text{prior odds ratio} \times \text{likelihood ratio}} \]

“Posterior odds ratio” means the odds that the person is guilty, relative to the odds that they are innocent, in the light of the evidence, and that’s clearly what one wants to know.  The “prior odds” are the odds that the person was guilty before any evidence was produced, and that is the really contentious bit.

Sometimes the need to specify the prior odds has been circumvented by using the likelihood ratio alone, but, as shown below, that isn’t a good solution.

The analogy with the use of screening tests to detect disease is illuminating.

Screening tests

A particularly straightforward application of Bayes’ theorem is in screening people to see whether or not they have a disease.  It turns out, in many cases, that screening gives a lot more wrong results (false positives) than right ones.  That’s especially true when the condition is rare (the prior odds that an individual suffers from the condition is small).  The process of screening for disease has a lot in common with the screening of suspects for guilt. It matters because false positives in court are disastrous.

The screening problem is dealt with in sections 1 and 2 of my paper. or on this blog (and here). A bit of animation helps the slides, so you may prefer the Youtube version.

The rest of my paper applies similar ideas to tests of significance.  In that case the prior probability is the probability that there is in fact a real effect, or, in the legal case, the probability that the suspect is guilty before any evidence has been presented. This is the slippery bit of the problem both conceptually, and because it’s hard to put a number on it.

But the examples below show that to ignore it, and to use the likelihood ratio alone, could result in many miscarriages of justice.

In the discussion of tests of significance, I took the view that it is not legitimate (in the absence of good data to the contrary) to assume any prior probability greater than 0.5. To do so would presume you know the answer before any evidence was presented.  In the legal case a prior probability of 0.5 would mean assuming that there was a 50:50 chance that the suspect was guilty before any evidence was presented. A 50:50 probability of guilt before the evidence is known corresponds to a prior odds ratio of 1 (to 1)  If that were true, the likelihood ratio would be a good way to represent the evidence, because the posterior odds ratio would be equal to the likelihood ratio.

It could be argued that 50:50 represents some sort of equipoise, but in the example below it is clearly too high, and if it is less that 50:50, use of the likelihood ratio runs a real risk of convicting an innocent person.

The following example is modified slightly from section 3 of a book chapter by Mortera and Dawid (2008). Philip Dawid is an eminent statistician who has written a lot about probability and the law, and he’s a member of the legal group of the Royal Statistical Society.

My version of the example removes most of the algebra, and uses different numbers.

Example: The island problem

The “island problem” (Eggleston 1983, Appendix 3) is an imaginary example that provides a good illustration of the uses and misuses of statistical logic in forensic identification.

A murder has been committed on an island, cut off from the outside world, on which 1001 (= N + 1) inhabitants remain. The forensic evidence at the scene consists of a measurement, x, on a “crime trace” characteristic, which can be assumed to come from the criminal. It might, for example, be a bit of the DNA sequence from the crime scene.

Say, for the sake of example, that the probability of a random member of the population having characteristic x is P = 0.004 (i.e. 0.4% ), so the probability that a random member of the population does not have the characteristic is 1 – P = 0.996. The mainland police arrive and arrest a random islander, Jack. It is found that Jack matches the crime trace. There is no other relevant evidence.

How should this match evidence be used to assess the claim that Jack is the murderer? We shall consider three arguments that have been used to address this question. The first is wrong. The second and third are right. (For illustration, we have taken N = 1000, P = 0.004.)

(1) Prosecutor’s fallacy

Prosecuting counsel, arguing according to his favourite fallacy, asserts that the probability that Jack is guilty is 1 – P , or 0.996, and that this proves guilt “beyond a reasonable doubt”.

The probability that Jack would show characteristic x if he were not guilty would be 0.4% i.e. Prob(Jack has x | not guilty) = 0.004.  Therefore the probability of the evidence, given that Jack is guilty, Prob(Jack has x | Jack is guilty), is one 1 – 0.004 = 0.996.

But this is Prob(evidence | guilty) which is not what we want.  What we need is the probability that Jack is guilty, given the evidence, P(Jack is guilty | Jack has characteristic x).

To mistake the latter for the former is the prosecutor’s fallacy, or the error of the transposed conditional.

Dawid gives an example that makes the distinction clear.

“As an analogy to help clarify and escape this common and seductive confusion, consider the difference between “the probability of having spots, if you have measles” -which is close to 1  and “the probability of having measles, if you have spots” -which, in the light of the many alternative possible explanations for spots, is much smaller.”

(2) Defence counter-argument

Counsel for the defence points out that, while the guilty party must have characteristic x, he isn’t the only person on the island to have this characteristic. Among the remaining N = 1000 innocent islanders, 0.4% have characteristic x, so the number who have it will be NP = 1000 x 0.004 = 4 . Hence the total number of islanders that have this characteristic must be 1 + NP = 5 . The match evidence means that Jack must be one of these 5 people, but does not otherwise distinguish him from any of the other members of it.  Since just one of these is guilty, the probability that this is Jack is thus 1/5, or 0.2— very far from being “beyond all reasonable doubt”.

(3) Bayesian argument

The probability of the having characteristic x (the evidence) would be Prob(evidence | guilty) = 1 if Jack were guilty, but if Jack were not guilty it would be 0.4%, i.e. Prob(evidence | not guilty) = P. Hence the likelihood ratio in favour of guilt, on the basis of the evidence, is

\[ LR=\frac{\text{Prob(evidence } | \text{ guilty})}{\text{Prob(evidence }|\text{ not guilty})} = \frac{1}{P}=250 \]

In words, the evidence would be 250 times more probable if Jack were guilty than if he were innocent.  While this seems strong evidence in favour of guilt, it still does not tell us what we want to know, namely the probability that Jack is guilty in the light of the evidence: Prob(guilty | evidence), or, equivalently, the odds ratio -the odds of guilt relative to odds of innocence, given the evidence,

To get that we must multiply the likelihood ratio by the prior odds on guilt, i.e. the odds on guilt before any evidence is presented. It’s often hard to get a numerical value for this. But in our artificial example, it is possible. We can argue that, in the absence of any other evidence, Jack is no more nor less likely to be the culprit than any other islander, so that the prior probability of guilt is 1/(N + 1), corresponding to prior odds on guilt of 1/N.

We can now apply Bayes’s theorem to obtain the posterior odds on guilt:

\[ \text {posterior odds} = \text{prior odds} \times LR = \left ( \frac{1}{N}\right ) \times \left ( \frac{1}{P} \right )= 0.25 \]

Thus the odds of guilt in the light of the evidence are 4 to 1 against. The corresponding posterior probability of guilt is

\[ Prob( \text{guilty } | \text{ evidence})= \frac{1}{1+NP}= \frac{1}{1+4}=0.2 \]

This is quite small –certainly no basis for a conviction.

This result is exactly the same as that given by the Defence Counter-argument’, (see above). That argument was simpler than the Bayesian argument. It didn’t explicitly use Bayes’ theorem, though it was implicit in the argument. The advantage of using the former is that it looks simpler. The advantage of the explicitly Bayesian argument is that it makes the assumptions more clear.

In summary The prosecutor’s fallacy suggested, quite wrongly, that the probability that Jack was guilty was 0.996. The likelihood ratio was 250, which also seems to suggest guilt, but it doesn’t give us the probability that we need. In stark contrast, the defence counsel’s argument, and equivalently, the Bayesian argument, suggested that the probability of Jack’s guilt as 0.2. or odds of 4 to 1 against guilt. The potential for wrong conviction is obvious.

Conclusions.

Although this argument uses an artificial example that is simpler than most real cases, it illustrates some important principles.

(1) The likelihood ratio is not a good way to evaluate evidence, unless there is good reason to believe that there is a 50:50 chance that the suspect is guilty before any evidence is presented.

(2) In order to calculate what we need, Prob(guilty | evidence), you need to give numerical values of how common the possession of characteristic x (the evidence) is the whole population of possible suspects (a reasonable value might be estimated in the case of DNA evidence),  We also need to know the size of the population.  In the case of the island example, this was 1000, but in general, that would be hard to answer and any answer might well be contested by an advocate who understood the problem.

These arguments lead to four conclusions.

(1) If a lawyer uses the prosecutor’s fallacy, (s)he should be told that it’s nonsense.

(2) If a lawyer advocates conviction on the basis of likelihood ratio alone, s(he) should be asked to justify the implicit assumption that there was a 50:50 chance that the suspect was guilty before any evidence was presented.

(3) If a lawyer uses Defence counter-argument, or, equivalently, the version of Bayesian argument given here, (s)he should be asked to justify the estimates of the numerical value given to the prevalence of x in the population (P) and the numerical value of the size of this population (N).  A range of values of P and N could be used, to provide a range of possible values of the final result, the probability that the suspect is guilty in the light of the evidence.

(4) The example that was used is the simplest possible case.  For more complex cases it would be advisable to ask a professional statistician. Some reliable people can be found at the Royal Statistical Society’s section on Statistics and the Law.

If you do ask a professional statistician, and they present you with a lot of mathematics, you should still ask these questions about precisely what assumptions were made, and ask for an estimate of the range of uncertainty in the value of Prob(guilty | evidence) which they produce.

Postscript: real cases

Another paper by Philip Dawid, Statistics and the Law, is interesting because it discusses some recent real cases: for example the wrongful conviction of Sally Clark because of the wrong calculation of the statistics for Sudden Infant Death Syndrome.

On Monday 21 March, 2016, Dr Waney Squier was struck off the medical register by the General Medical Council because they claimed that she misrepresented the evidence in cases of Shaken Baby Syndrome (SBS).

This verdict was questioned by many lawyers, including Michael Mansfield QC and Clive Stafford Smith, in a letter. “General Medical Council behaving like a modern inquisition”

The latter has already written “This shaken baby syndrome case is a dark day for science – and for justice“..

The evidence for SBS is based on the existence of a triad of signs (retinal bleeding, subdural bleeding and encephalopathy). It seems likely that these signs will be present if a baby has been shake, i.e Prob(triad | shaken) is high. But this is irrelevant to the question of guilt. For that we need Prob(shaken | triad). As far as I know, the data to calculate what matters are just not available.

It seem that the GMC may have fallen for the prosecutor’s fallacy. Or perhaps the establishment won’t tolerate arguments. One is reminded, once again, of the definition of clinical experience: “Making the same mistakes with increasing confidence over an impressive number of years.” (from A Sceptic’s Medical Dictionary by Michael O’Donnell. A Sceptic’s Medical Dictionary BMJ publishing, 1997).

Appendix (for nerds). Two forms of Bayes’ theorem

The form of Bayes’ theorem given at the start is expressed in terms of odds ratios. The same rule can be written in terms of probabilities. (This was the form used in the appendix of my paper.) For those interested in the details, it may help to define explicitly these two forms.

In terms of probabilities, the probability of guilt in the light of the evidence (what we want) is

\[ \text{Prob(guilty } | \text{ evidence}) = \text{Prob(evidence } | \text{ guilty}) \frac{\text{Prob(guilty })}{\text{Prob(evidence })} \]

In terms of odds ratios, the odds ratio on guilt, given the evidence (which is what we want) is

\[ \frac{ \text{Prob(guilty } | \text{ evidence})} {\text{Prob(not guilty } | \text{ evidence}} =
\left ( \frac{ \text{Prob(guilty)}} {\text {Prob((not guilty)}} \right )
\left ( \frac{ \text{Prob(evidence } | \text{ guilty})} {\text{Prob(evidence } | \text{ not guilty}} \right ) \]

or, in words,

\[ \text{posterior odds of guilt } =\text{prior odds of guilt} \times \text{likelihood ratio} \]

This is the precise form of the equation that was given in words at the beginning.

A derivation of the equivalence of these two forms is sketched in a document which you can download.

Follow-up

23 March 2016

It’s worth pointing out the following connection between the legal argument (above) and tests of significance.

(1) The likelihood ratio works only when there is a 50:50 chance that the suspect is guilty before any evidence is presented (so the prior probability of guilt is 0.5, or, equivalently, the prior odds ratio is 1).

(2) The false positive rate in signiifcance testing is close to the P value only when the prior probability of a real effect is 0.5, as shown in section 6 of the P value paper.

However there is another twist in the significance testing argument. The statement above is right if we take as a positive result any P < 0.05. If we want to interpret a value of P = 0.047 in a single test, then, as explained in section 10 of the P value paper, we should restrict attention to only those tests that give P close to 0.047. When that is done the false positive rate is 26% even when the prior is 0.5 (and much bigger than 30% if the prior is smaller –see extra Figure), That justifies the assertion that if you claim to have discovered something because you have observed P = 0.047 in a single test then there is a chance of at least 30% that you’ll be wrong. Is there, I wonder, any legal equivalent of this argument?