#### Million to one chances don't crop up 9 times out of 10?

Police are just going to have to revert to their original style of proving guilt, then:

"It was you what done it, wasn't it?"

A judge in a (sadly unnamed) British case has decided that Bayes' Theorem - a formula used in court to calculate the odds of whodunnit - shouldn't be used in criminal trials. Or at least, it shouldn't be relied upon as it has been in recent years: according to the judge, before any expert witness plugs data into the theorem to …

Police are just going to have to revert to their original style of proving guilt, then:

"It was you what done it, wasn't it?"

The other thing is that you are comparing the realtive likelihood of a double cot death (in itself rare) to that of a double murder (also a rare event). So as I understood it from reading Bad Science (by Ben Goldacre) you have to think in terms of which is more likely - double murder or double cot death, i.e. given that one of two very unlikely things have happened what are the chances it was one or the other.

If I recall correctly the upshot was that it was twice as likely to be a double cot death than double murder.

It isn't that difficult to understand. Shouldn't somebody on the defence team have known this?

One time in my life I actually wished I was on a jury.

> you have to think in terms of which is more likely - double murder or double cot death

That sounds fishy.

This would mean your sample space is the space of "two dead people in a row". This doesn't sound well-defined. Those two dead people - are they grouped according to the same court case? Did they die in the same house? Of the same murder weapon? Did they have the same age? Doesn't really make sense.

The question is "what is the probability of double baby death by natural causes given that there is no other indication of foul play like baby skewered by a fork".

Your sample space is the set of families in which at least one baby died of apparent natural causes that look like cot death.

Turns out that in that set, a family has a high probability of seeing another cot death.

Bayesian stats are misinterpreted for the double cot death vs double murder scenario.

Cot deaths are not stochastically independent. Bad heredity is bad heredity, so is bad environment and so are unfortunately bad parenting habits.

If you feed _ALL_ factors into Bayesian stats you will see quite correctly a very high probability of second cot death. If you do not, the second death becomes very unlikely and the probability of double murder seemingly exceeds the probability of cot death.

Coming back to Bayesian stats, the judge in this case may have more clue than we think (or has read more than we think). Bayesian stats require very _CLEAN_ data which has not been contaminated by a deterministic bias. If your data is biased (no pun intended), you Bayes will be way off because it will be revealing the bias in your data which you may interpret wrongly.

> That sounds fishy.

>This would mean your sample space is the space of "two dead people in a row". This doesn't sound well-defined.

*sigh* - yes there is a bunch of stuff missing from my post because I a) didn't have all the details to hand, and , b) assumed that intelligent people would go digging with the details I'd given to get the full picture*.

To clarify, head on over to badscience.net and search for "The Prosecutor's Phallusy" to get all the info. To quote, what I believe is the main point in this, I'll quote from the original post:

"Two babies in one family have died. This in itself is very rare. Once this rare event has occurred, the jury needs to weigh up two competing explanations for the babies’ deaths: double SIDS or double murder. Under normal circumstances – before any babies have died – double SIDS is very unlikely, and so is double murder. But now that the rare event of two babies dying in one family has occurred, the two explanations – double murder or double SIDS – are suddenly both very likely. If we really wanted to play statistics, we would need to know which is relatively more rare, double SIDS or double murder."

I will point out that my initial statement of twice as likely was wrong - it appears to be between 4.5:1 and 9:1 in favour of it being SIDS.

* I've deliberately tried to be a bit vague as, I assume, Tim Worstall would have given the details if he could/felt it appropriate.

That case the BBC is raising is indeed madness. What is the world coming to when the default view is that someone must have committed some crime if we cannot otherwise explain a sequence of events?

There's already a couple of criminal offences (purgery and perverting the course of justice) on the books that are woefully under applied when it comes to considering the care with which some experts have assisted the legal system. With expertise comes responsibility, especially in relation to understanding the true limits of their own knowledge. If the scientific experts involved in the legal system can't be relied upon to remember that most important of scientific tenets, perhaps the thought of facing criminal charges might focus their minds somewhat.

For example, imagine you have performed no research on the exact question at hand (e.g. can the environmental / biological factors causing SIDS persist in the family home?). Imagine further that you have no peer reviewed work to back up a statistics-backed assertion that you're about to make and are not formally qualified as a statistician. How hard is it to stop, think a bit and say "I don't know."?

Similarly, if the Court and legal officials don't understand the scientific process, why are they allowed to accept an act on the word of single expert witness? Have they never heard of scientific concensus?

expert witness... "I don't know."

really?

i don't know, but i suspect not

"expert witness... "I don't know."

really?

i don't know, but i suspect not"

Which is exactly what the quake experts in Italy should have said when asked how likely a serious earthquake was instead of saying what they thought based on experience and statistics. We are dealing with 2 different beasties here, people who understand how things work, and the general populace. Oh yes, and I include in "general populace" experts who make comments outside their area of expertise.

My particular dislike of the use of DNA is that everyone is told that DNA is unique, except for identical twins (or clones). What people in court are not told is that they don't sequence the whole DNA (it can't be done, and has never been done).

I think that for the layman, they should say something like; everyone's credit card number is unique to them. We found the credit card number of the person who bought the gun, and that 3 of the 16 digits in the defendants credit card number are the same, so therefore it was him.

>>"My particular dislike of the use of DNA is that everyone is told that DNA is unique, except for identical twins (or clones). What people in court are not told is that they don't sequence the whole DNA (it can't be done, and has never been done)."

Though if people are given probabilities of a random person matching as well as a defendant does, that does at least imply that there isn't uniqueness in the matching process.

And how are you sure what people are told in court ?

Presumably a half-decent defence lawyer could get someone to go into details if they thought it would do some good?

>>"I think that for the layman, they should say something like; everyone's credit card number is unique to them. We found the credit card number of the person who bought the gun, and that 3 of the 16 digits in the defendants credit card number are the same, so therefore it was him."

But that would be highly misleading, since it's close to implying, if not actually implying, that the other 13 digits are different.

Even if someone said the *quite different*

"We found 3 digits from the purchaser's credit card number and the corresponding numbers on the defendant's card match them",

it would still be a fairly poor analogy, since the chance of a random card matching would be 1 in 1000.

slightly pedantic, but AFAIK if your name is/was Henrietta Lacks then your DNA has been sequenced entirely.

As to DNA being unique to a person, try telling that to Karen Keegan and Lydia Fairchild (see http://en.wikipedia.org/wiki/Lydia_Fairchild) who got into legal and medical problems since as Chimeras (see http://en.wikipedia.org/wiki/Chimera_%28genetics%29) they each have two separate sets of DNA (depending on which organ the DNA sample was taken from).

Actually, I was the one being pedantic. There are many long-repeating sections in DNA. You can only section a certain amount in one go, so what you do is sequence lots of bits, and guess that you have sequenced the whole lot; but, and this is a big but, you can't know that you have done it all, and put it all together, because you don't get enough overlap between the different bits to ensure that you've got it all and in the right order.

I know it's bad form to quote Wikipedia, but where I am right now, I can't access much else: http://en.wikipedia.org/wiki/Human_Genome_Project

A colleague of mine used to work in academia doing stochastic calculus and was therefore quite good at stats and had a professor who was even more so. In one prominent cot death case, when this prof heard the reasoning cited in the article (that two or more cot deaths in one family were so unlikely as to be sufficient evidence of guilt), he wrote to the judge and the defence legal team laying out why this was a crazy perversion of logic and offering to act as an independent expert witness at a hoped for appeal/retrial.

Far as I know, neither the judge nor (more shockingly perhaps) the defence ever got back to him on it. That said, the conviction was eventually quashed.

...by Stuart Sutherland.

There's a great chapter (among other great chapters) called "Misinterpreting the evidence." Should be required reading for juries. And everyone else.

and statisics can be used to prove many a dubious point without appearing to lie. There is probably no fundemental problem with Bayesian reasoning, it is when you add lawers that it becomes devisive and dangerous.

However much I agree with the sentiment, maybe you should reference Ben Goldacre for the last half of the article? :)

I know Ben's done good work on this but he wasn't the source. It's been so generally reported over the years (and no, not all using Ben as their source) that identifying any one source doesn't really work.

I wish the crackdown was on all stats, I don't see that much sign that the problem is Bayes' Theorem

Agreed. I don't see any use of Bayes' Theorem in the two-deaths example as the author presented here. That was a straightforward case of the probability of an event and the probability of the event occurring twice, with the accompanying question of whether the two events are independent.

Possibly Bayes was used to compute the probability of the event in the first place - but if so, that's irrelevant to the case as described in the article.

Bayes Theorem says that if you know the probability of B given A, and you know the probability of A and B on their own, you can compute the probability of A given B. It's pretty straightforward, and it's also irrelevant to the probability of a second "cot death" following a first one. Here the thesis proposed by the prosecution is that A and B are in fact two independent occurrences of the same event; thus P(A|B) = P(B|A) = P(A) = P(B). No need for Bayes at all.

If anything, in that case, it's the defense that should have brought up Bayes, after explaining independence and correlation and other basic concepts.

I haven't read the decision, but I suspect the real issue at hand is the abuse of Bayesian Inference, which is an aspect of the Bayesian interpretation of probability theory. Essentially it's a way to answer the question "how likely is this interpretation of the data to be true, based on our initial probability estimates and the data we've collected since?". That makes more sense, in the context of the first example, where the defense would want to challenge a Bayesian interpretation that misstated the actual posterior confidence.

As Gareth says, it was mainly a failure to apply Bayesian statistics that led to the wrongful conviction in the case we aren't going to mention.

I've always thought the X in a million chances of a DNA match was one of the dodgiest statistics around.

DNA is not randomly distributed. So to say "picked at random from the popualtion there is a 1 in X chance of a match" is not the same as "the chance of DNA picked at random from the town in which you and your ancestors have lived in for over 1000 years is 1 in X".

... in that cot death case is that the probability of the first death is unity. Without it there would be no case. It is a given.

Another good book on the subject of stats and randomness:

http://www.guardian.co.uk/books/2008/jul/12/saturdayreviewsfeatres.guardianreview4

It even worse than that. My belief is that, because of the problems of cross-contamination, the DNA of anyone officially involved with the investigation is excluded from analysis. Hmmmm.

There's also the story of the Dutch(?) nurse that was convicted after seven people died on her watch. There was no evidence, just the statistical unlikelihood of seven people all dying while one person was on duty.

The allegation in the Norris case is slightly different, but still an interesting abuse of statistics. It seems that someone spotted a correlation between when his shifts were and when some of the people died. They then got a list of his shifts and looked for other deaths that might be considered suspicious at those times. Then, they claimed that it could not be a coincidence that all the suspicious deaths had occurred when he was around. This is, of course, not valid unless they had looked equally thoroughly for suspicious deaths at other times and found none.

From http://www.bbc.co.uk/news/uk-scotland-15127072 :

"The BBC has uncovered evidence of other similar cases of hypoglycaemia which occurred in the hospital where Norris worked but while he was off duty.

His lawyer, Jeremy Moore, believes there were serious flaws in the investigation and the convictions need to be quashed.

He said: "It seems that they trawled through hospital records looking for evidence of patients that might have died suspiciously but it seems they only cherry-picked those cases when Colin was on duty and ignored any others that might have occurred in the hospital."

"A DNA match to one in a million does not mean that it's a million to one against the bloke 'aving done it, m'lud. Rather, it means that in a population of 65 million that 65 people, based purely on the DNA, could have done it. Our DNA tests thus mean that we now have to go and exclude those other 65, or at least regard them as the prime pool of suspects, not convict our man in the dock purely on the basis that one in a million is beyond that reasonable doubt. Yes, these sorts of mistakes are made in the chain of reasoning."

Yes, they are.

THAT'S WHY WE USE BAYES' THEOREM. It's ONLY by applying Bayes' Theorem that you obtain paragraphs like the above.

Bayes' Theorem is not an "option", it is a necessity, otherwise travesties like Professor Sir Roy Meadow will happen constantly.

So what is the author arguing?

>>"THAT'S WHY WE USE BAYES' THEOREM. It's ONLY by applying Bayes' Theorem that you obtain paragraphs like the above."

No, it's perfectly simple to go from "1 in a million" to "there should be roughly 65 matching people in the UK population" by simple logic and extremely basic maths.

Bayes' theorem might be an expression of that, but the underlying logic would be there with or without any theorem, and for a court case, it would seem better to describe something simply in English than start chucking formulas around.

>>"Bayes' Theorem is not an "option", it is a necessity, otherwise travesties like Professor Sir Roy Meadow will happen constantly."

But surely the first problem there was an expert making the *medical* claim/assumption that cot deaths happen at random.

Given that assumption, if the assumption was wrong, wouldn't *any* maths be bound to give the wrong answer?

There was definitely an issue with the stats in the SIDS case cited in Bad Science. However, it was nothing to do with Bayesian inference. In fact a prerequisite in Bayesian inference is that all observational evidence is independent. The 1 one in 73 million probability came from a 1 in 8,543 probability taken from the occurrence in a population, squared. The is bog standard probability theory.

Bayesian inference relies on a prior which is the probability of a particular hypothesis without further evidence. This is then iteratively modified by the ratio of evidence given the hypothesis to the probability of the evidence without an hypothesis (which actually gives a likelihood ratio not a probability or 'odds').

I work with Bayesian inference and applied correctly it is atonishingly good at predicting likelihood. This ruling is a classic case of baby and bath water.

Always remember that the average man has less than two legs. Paris because...just because.

"and applied correctly it is atonishingly good at predicting likelihood"

I think the point the judge was making though is that it wasn't being applied correctly and thus erroneous conclusions were being drawn.

Sorry, are you related to amanfromMars, I didn't understand a word of what you said :-)

I always remember Bayes' theorem as the 'probability of an event B happening/not happening given that event A has happened/not happened, where A and B are independent events'

What a lot of people do is that they attach meaning to statistical results that don't exist e.g. the probability of the numbers 1 2 3 4 5 6 being the winning numbers in the national lottery is the same as any other random 6 numbers, but people attach meaning/ probability to the sequence 1 2 3 4 5 6.

perfectly rational statistics can also give the wrong impression, for example saying 40% of all sick days are taken on a Monday and Friday give the impression that there are excessive sick days taken on Monday and Friday, but Monday and Friday comprise 40% of the working week.

Having calculated a likelihood ratio, the scientist in R -v-T translated that likelihood ratio into an “expression of support”, using a standard scale:

Likelihood ratio within range 1 to 10 = “Weak support”,

10-100 = “moderate support”

100 to 1000 = “moderately strong support”

1000 to 10,000 = “strong support”

10,000 to 1,000,000 = “very strong support”

>1,000,000 = “extremely stroing support”

The judgment has caused a minor panic amongst “police” forensic scientists because they have been calculating likelihood ratios and translating those likelihood ratios into expressions of support even in cases where there is no objective data on which to base calculations.

For example, a scientist might guesstimate that the probability of observing “lots of blood” on a defendant’s clothing (as opposed to “small amounts of blood”) given that defendant is the attacker as 0.75. The probability of observing this finding if he was not the attacker, but merely came to the aid of the victim after attack, might be guesstimated as 0.25.

The process will be applied to different - hopefully, but not always, - independent findings (eg. “lots of spattered blood”, “lots of blood on the cuffs”). A likelihood ratio is then calculated.

This LR is translated into a phrase using the table: “the scientific findings provide strong support for the view that Mr Defendant attacked Mr Victim rather than Mr Defendant having helped Mr Victim after the attack”.

The Court of Appeal judges criticised this process for its lack of transparency. Sometimes it's a scientifically rigorous approach that supports and documents an expert opinon. All too often it's pseudo-scientific claptrap.

What is very worrying is the idea that you can convict something based on probability. Improbable things *do* happen. That is why they are improbable, not impossible.

And considering how many people there are in the world, if it is improbable that an event happens to one person, it is much more probably that it happens to *a* person.

What if someone next to you gets struck by lightning, and you are convicted of assault because it's more likely that you assaulted them than them getting struck by lightning? Bad example, but justice? Pah.

>>"What is very worrying is the idea that you can convict something based on probability. Improbable things *do* happen. That is why they are improbable, not impossible."

Thanks for that lecture on probability. I'm sure we all needed it.

It seems you think that 'reasonable doubt' is wrong and courts should only convict based on absolute certainty.

Well, I guess that *would* save a lot of money in the judicial system.

Though on the other hand, when people start taking the law into their own hands, compared to the numbers of people wrongfully convicted, I wonder how many innocent people would get hurt in escalating vendettas, or as a result of wrongful accusations?

But all those examples, like the cot death, are from NOT using Bayesian.

Given that the chance of cot death isn't independent in two children with the same parental genes you need to use a Bayesian approach.

Using "proper" statistics is like saying that 1:4 people are chinese, my mum,dad and brother aren't - so I must be!

No, statistics would say there is a probability of 1 in 4 that you are chinese.

Similarly statistics might say that there is a 1 in 1 million chance that you will be killed in an airplane crash i.e. if you take 1 million flights you _will_ be killed in an airplane crash, however there is nothing in that statistic that say it won't on the first of those 1 million flights.

although I believe the odds are significantly more than a million to 1 against.

"i.e. if you take 1 million flights you _will_ be killed in an airplane crash"

An appropriate comment for an article about the abuse of statistics. In fact, if the risk is one in a million, the probability that you will be killed after a million flights is about 63%. You calculate this by raising the odds that you will NOT be killed on a particular flight (999999:1000000) to the power of the number of flights, and subtracting from one. (0.999999)^1000000 = 0.368.

It's easier to see it with a smaller number. If the odds of rolling a six on a dice is 1 in 6, the odds of rolling a six at least once in six rolls is 1 minus the odds of _not_ rolling a six, six times in a row - 1-(5/6)^6, or about 66%.

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