Prosecutor Fallacy in Stephen Lawrence case?

If you believe the 'respectable' media reporting of the Stephen Lawrence case then there were at least two blatant instances of the prosecutor fallacy committed by expert forensic witnesses.

For example the BBC reported forensic scientist Edward Jarman as saying that

"..the blood stain on the accused's jacket was caused by fresh blood with a one-in-a-billion chance of not being victim Stephen Lawrence's."

Numerous other reports contained headlines with similar claims about this blood match. The Daily Mail (above) states explicitly:

".. the chances of that blood belonging to anyone but Stephen were rated at a billion to one, the Old Bailey heard."

Similar claims were made about the fibres found on the defendants' clothing that matched those from Stephen Lawrence's clothes.

If the expert witnesses had indeed made the assertions as claimed in the media reports then they would have committed a well known probability fallacy which, in this case, would have grossly exaggerated the strength of the prosecution evidence. The fallacy - called the prosecutor fallacy - has a long history (see, for example our reports here and here as well as summary explanation here).  Judges and lawyers are expected to ensure the fallacy is avoided because of its potential to mislead the jury. In practice the fallacy continues to be made in courts. However, despite the media reports the fallacy was not made in the Stephen Lawrence trial. The experts did not make the assertions that the media claimed they made. Rather, it was the media who misunderstood what the experts said and it was they who made the prosecutor fallacy. What is still disturbing is that, if media reporters with considerable legal knowledge made the fallacy then it is almost certain that the jury misinterpreted the evidence in the same way. In other words, although the prosecutor fallacy was not stated in court, it may still have been made in the jury's decision-making.

To understand the fallacy and its impact formally, suppose:

• H is the hypothesis "the blood on Dobson's jacket did not come from Stephen Lawrence"
• E is the evidence "blood found matches Lawrence's DNA".

What the media stated is that the forensic evidence led to the probability of H (given E) being a billion to one.

But, in fact, the forensic experts did not (and could not) conclude anything about the probability of H (given E). What the media have done is confuse this probability with the (very different) probability of E given H.

What the experts were stating was that (providing there was no cross contamination or errors made) the probability of E given H is one in a billion. In other words what the experts were asserting was

"The blood found on the jacket matches that of Lawrence and such a match is found in only one in a billion people. Hence the chances of seeing this blood match evidence is one in a billion if the blood did not come from Lawrence".

In theory (since there are about 7 billion people on the planet) there should be about 7 people (including Lawrence) who would have the same matching blood DNA to Lawrence. If none of the others could be ruled out as the source of the blood on the jacket then the probability of H given E is not one in a billion as stated by the media but 6 out of 7. This highlights the potential enormity of the fallacy.

Even if we could rule out everyone who never came into contact with Dobson that would still leave, say, 1000 people. In that case the probability of H given E is about one in a million. That is, of course, a very small probability but the point is that it is a very different probability to the one the media stated.

The main reason why the fallacy keeps on being repeated (by the media at least) in these kind of cases is that people cannot see any real difference between a one in a billion probability and a one in a million probability (even though the latter is 1000 times more likely). They are both considered 'too small'.

Finally, it is also important to note that the probabilities stated were almost meaningless because of the simplistic assumptions (grossly favourable to the prosecution case) that there was no possibility of either cross-contamination of the evidence or errors in its handling and DNA testing. The massive impact such error possibilities have on the resulting probabilities is explained in detail here.

Nonsensical probabilities about asteroid risk

There is an article in today's Evening Standard in which, rather depressingly, someone who should know better (Roger Highfield, Editor of New Scientist) reels off a typically misleading probability about asteroid strike risk (this is in response to the news today that an asteroid was within just 200,000 miles of Earth).

Quoting a recent book by Ted Nield he says (presumably to comfort readers) that

Our chances of dying as a result of being hit by a space rock are something like one in 600,000. 

There are all kinds of  ambiguities about the statement that I won't go into (involving assumptions about  random people of  'average' age and 'average' life expectancy) but even ignoring all that, if the statement is supposed to be a great comfort to us then it fails miserably. That is because it can reasonably be interpreted as providing the 'average' probability that a random person living today will eventually die from being hit by a space rock. Assuming a world population of 7 billion that's about 12,000 of us. And 12,000 actual living people is a pretty large number to die in this way. But it is about as meaningful as putting Arnold Schwarzenegger in a line up with a thousand ants and asserting that the average height of those in the line-up is 3 inches tall. The key issue here is that large asteroid strikes are, like Schwarzenegger in the line-up, low probability high impact events. Space rocks will not kill a few hundred people every year as implied by the original statement, just as there are no 3-inch tall beings in the line-up. Tens of thousands of years pass between them killing any more than a handful of people. But eventually one will wipe out most of the world's population. 

What Ted Nield should have stated (and what we are most interested in knowing) was the probability that a large space rock (one big enough to cause massive loss of life) will strike Earth in the next 50 years.

Indeed, I suspect that (using Nield's own analysis) this probability would be close to the 1 in 600,000 quoted (given that incidents of small space rocks killing a small number of people are very rare). You might argue I am splitting hairs here but there is an important point of principle. Nield and Highfield avoid stating an explicit probability of a very rare event (such as in this case a massive asteroid strike) because there is a natural resistance (especially from non-Bayesians) to do so. For some reason it is more palatable in their eyes to consider the probability of a random person dying (albeit due to a rare event), presumably because it can more easily be imagined. But, as I have hopefully shown, that only creates more confusion.

Bayes and the Law: Nature article

My commentary piece on the role of Bayes in the Law has just appeared in Nature. A pdf of an extended draft on which it was based is here.

Bayes and the Law: Guardian article

There is an article in the Guardian today that is the result of an interview I had with the journalist Angela Saini. It is about the issue of Bayes and the Law following the RvT ruling and it includes a reference to the consortium that we are putting together to improve the situation.

Another specialist risk assessment company gets bought out

Algorithmics, a company specialising in risk software for financial institutions, has been sold to IBM for $387m.

Agena partnered Algorithmics during the period 2003-2005 when there was a clamour for so-called 'advanced measurement approaches' to operational risk assessment. The Basel 2 accord specified that banks which used a validated advanced measurement approach to calculate their operational risk exposure could set aside a lower percentage capital allocation. This meant there was a major financial incentive for banks to develop such approaches. Most banks looked at Bayesian networks as a potential solution and, indeed, a number wanted Algorithmics to provide such a solution to integrate with the existing credit and market risk software that Algo provided. Algorithmics had started work on their own Bayesian network platform for OpRisk, but decided that AgenaRisk was superior. Hence we partnered them in projects with some major banks, with Agena providing the underlying Bayesian network technology and modelling skills and Algorithmics providing the reporting infrastructure.  Here is a paper we wrote that gives a feel for the BN approach to operational risk that we developed.

In late 2005 Algorithimcs actually got taken over by the Fitch Group. Since Fitch already had their own (non-Bayesian) OpRisk solution - which they had massively invested in - the partnership effectively ended then, as it appears did Algorithmics' interest in Bayesian networks. This is a great shame, especially when you consider the mess that financial institutions have made using classical statistics.

It is difficult to determine the extent to which banks are using Bayesian networks but, as we described here, there are plenty of financial analysts who are using fundamentally flawed methods in situations when the Bayesian approach would work.

Transforming Legal Reasoning through Effective use of Probability

Recent reports have highlighted the difficulties faced by the criminal justice system in adequately responding to the dramatic increase in the amount and complexity of forensic science, particularly given its (not infrequently) questionable value. Despite the growing consensus that the role of experts should be limited to making statements about the probability of their findings under competing hypotheses (instead of, for example, making categorical source attributions), and the ability of Bayes’ theorem to encapsulate the proper or normative effect of probabilistic evidence, Bayesian reasoning has been largely ignored or misunderstood by criminal justice professionals.

Proper use of probabilistic reasoning has the potential to improve dramatically the efficiency, transparency and fairness of the criminal justice system and the accuracy of its verdicts, by enabling the value of any given piece of evidence to be meaningfully evaluated and communicated. Bayesian reasoning employs the likelihood ratio (which is the probability of seeing the evidence given the prosecution hypothesis divided by the probability of seeing the evidence given the defence hypothesis), to illustrate the relevance and strength of each piece of evidence. Bayesian reasoning can therefore help the expert formulate accurate and informative opinions; help the court in determining the admissibility of evidence; help identify which cases should and should not be pursued and help lawyers explain, and jurors to evaluate the weight of evidence during a trial. It would also help identify error rates and unjustified assumptions entailed in expert opinions, which would in turn contribute to the transparency and legitimacy of the criminal justice process.

Unfortunately, there is widespread disagreement about the kind of evidence to which Bayesian reasoning should be applied and the manner in which it should be presented. Much of the disagreement over when it should be applied arises from fundamental misunderstandings about the way Bayes’ reasoning works, whereas disagreement over the manner in which it should be presented could be resolved by empirical research. Misunderstandings in the criminal justice system are exacerbated by the fact that in the few areas where Bayesian reasoning has been applied (such as DNA profiling ) its application has often been faulty and its ramifications poorly communicated. This has further resulted in widespread recourse to probabilistic fallacies in legal proceedings.

A dramatic and worrying example of this was a recent appeal court decision ((2010). R v T. EWCA Crim 2439 , see our draft article about this here) which appears to reject the use of Bayesian analysis and likelihood ratios for all but a very narrowly defined class of forensic evidence. Instead of being accepted as a standard tool of the forensic science trade, Bayesian analysis is perceived by much of the legal profession as an exotic, somewhat eccentric method to be wheeled out for occasional specialist appearances whereupon a judge or lawyer will cast doubts on, and even ridicule, its integrity (hence ensuring it is kept firmly locked in the cupboard for more years to come).

Ultimately this represents a failure by the community of academics, expert witnesses and lawyers who understand the potentially crucial and wide role that can be played by Bayesian analysis and likelihood ratios in legal arguments. This failure must be attributed to our inability to communicate the core ideas effectively. Resorting to the formulas and calculations in court is a dead-end strategy since these will never be understood by most lawyers, judges and juries.

Using Bayes to prove Obama did not write his own book?

There have been many questions about the closeness of President Obama's relationship with Weather Underground terrorist Bill Ayers. A whole new angle on the relationship has been raised in Jack Cashill's book Deconstructing Obama. Using information in this book Andre Lofthus has applied Bayes Theorem to conclude that Bill Ayers actually was the ghost writer for Obama's best selling book Dreams from My Father.

Loftus's analysis is based on a) a comparison of  Dreams with one of Ayers's own books Fugitive Days; and b) a comparison of Dreams with a different book Sucker Punch based on similar material to that of both Dreams and Fugitive Days.

Specifically in a) there were 759 similarities, of which 180 were categorized by Cashill  as "striking similarities", whereas in b) Cashill claims there were just six definite similarities, with a maximum of sixteen possible or definite similarities. As Lofhus's Bayesian analysis is not complete I have done my own analysis here. My own conclusions are not as definite. The evidence does indeed provide very strong support in favour of the books being written by the same author. However, if you have a strong prior belief that the books were written by different authors (say you are 99.9% sure) then even after observing the evidence of 180 striking similarities, it turns out that (with what I believe are more reasonable assumptions than made by Lofhus) there is still a better than 50% chance that the books were written by different authors.