06 December 2010

P(A Person Has Committed A Crime | The Person Has Been Convicted Of That Crime)

What is the probability that a person has committed a crime given that the person has been convicted of that crime?

With a prison population of 88,000, if the answer to that question was 99%, 880 people would be in prison who had not committed the crime for which they were incarcerated.

The question forces us to think about the fiction of conviction and the fact of committing. One is a reality constructed in our minds and supported by violence, the other a reality that is independent of our minds. If someone commits an offence and we don't convict them, it doesn't mean that they haven't done it; similarly, if someone is convicted of an offence, it doesn't mean that they did it.

Unfortunately, this distinction is easily forgotten.

Ask people whether or not to bring back capital punishment; ask people whether or not it is acceptable to conduct medical experiments on prisoners; ask if prisoners should have rights, and the fiction of conviction becomes the reality of 'they committed the crime'.

Does this matter?

To the people who have been convicted but have not committed an offence it matters. But to society as a whole?

To society as a whole it is just as important to convict people who have committed a crime as it is to acquit people who have not committed a crime. Due to the imperfections of the process, for the greater good, we must accept some injustice.

Acceptance of injustice is difficult enough, especially to the victim, but what about the nature of the injustice. Being a victim of circumstance is different from being a victim of others.

What happens when people are prosecuted not because it is thought that they have necessarily committed the crime but because they are vulnerable, less able to defend themselves. What happens if policing is done such that people are pursued not because it is thought that they have committed the offence in question but because they are not especially able to defend themselves.

Some appalling examples of wrong convictions easily spring to mind, such as Stefan Kiszko.

I can't see into the hearts of the police officers who pursued Stefan Kiszko; I don't know whether they held the belief that he had committed the crime, but I can imagine similar circumstances, perhaps less extreme, where those who convict have no interest whatsoever as to whether the person found guilty has committed the crime. It is rumoured that some prosecutors get a greater satisfaction in convicting a person knowing that he hasn't committed the crime rather than knowing that he has committed the crime.

Not only can one imagine instances of people being convicted of a crime that they didn't commit due to unfortunate circumstances such as bad luck; due to incompetence or spite of police and prosecution services, one can also envision people being convicted for political reasons.

Handing over the baton to Gareth Pierce we have,

"It is not difficult to achieve a conviction of the innocent. Over many decades several common factors have been identified, [...] achieving the co-operation of witnesses by means of a combination of inducements and fear of the alternative (the tried and tested method of obtaining evidence for the prosecution on which many US cases rely); the provision of factual information by scientists where there is no proper basis for it (a recurrent theme in UK convictions as well as in the US); reliance on ‘identification’ evidence which is no such thing. Add to that the political will to achieve a prosecution, and the rest is easy. Fabrication demands outright dishonesty, but it isn’t always necessary, or necessary in every aspect of an investigation: the momentum of suspicion, and a blinkered determination to focus on a particular thesis and ignore evidence pointing to the contrary, is a certain route to achieving the desired end."

The more we consider the question, "What is the probability that a person has committed a crime given that the person has been convicted of that crime?", the more difficult it will be to convict innocent people.
Update, 24th Dec 2010. The following paper may be of interest,

A Derivation of Probabilities of Correct and
Wrongful Conviction in a Criminal Trial
Henrik Lando
Copenhagen Business School and Lefic
March 14, 2006

Update 28th February 2011
Listened to Radio4 last night. Gerry Conlon (innocent of Guildford 4 bombing yet 15 years in jail) said 20,000 miscarriages of justices p.a "

Update 7th April 2011. I've had a couple of thoughts about putting a number to the question I asked at the beginning of the post: the probability that someone has committed a crime given that they've been convicted is 0.5.

Firstly, either the person did commit it (p=1) or he didn't (p=0), the average of which is ((1+0)/2) 0.5.

Secondly, we know that people do commit crimes and that of those who do some are convicted, in this case p=1. Unfortunately, we know that some people are convicted on the basis of lies (Birmingham 6, etc), incompetence or mistake, which takes our value of p=0. Taking these three values and finding their average, ((0.5 + 0 + 1)/3) doesn't shift us from 0.5. Until we know the proportions of people who definitely did do it and the proportions who definitely did not do it, we can't shift from this 0.5 probability level.

As far as I'm concerned, this research is screaming to be done since, as is, we're saddled with a criminal punishment system where there is only a 0.5 probability that a person who has been convicted of a crime committed it. Yes, I know that our prejudices tell us differently but we should be able to do better: our system shouldn't be based on faith and prejudice.
Update 14th April 2011, from Mindhacks, How to jail the innocent,
"The Innocence Project has used DNA technology to overturn hundreds of wrongful convictions. Slate has an excellent two part series on the two main reasons why these people were falsely jailed: eyewitness misidentifications and false confessions."

Update 12th August 2011, again from Mindhacks, False Confession Fishing in the Lab.
"During the test, which was filmed by a hidden camera, ten participants actually did cheat. Bafflingly, though, another eight falsely confessed when accused by the experimenter, despite participants having been told cheats would be fined €50 ($72). "
Vaughan also gives us, "The Persuasive Power of False Confessions": yet more scientific studies suggesting that there are a lot of innocent people who have been falsely convicted of a crime.

This time the focus is on the corrosive effect towards the objectivity of investigators upon hearing a confession: once the confession is heard they turn into Procrustes and fit ambiguous evidence to the hypothesis that the 'confessee' did it.

Update 22nd September 2011. Columbia University provides the study, A Broken System: Error Rates in Capital Cases, 1973-1995" and a follow-up study, A Broken System, Part II: Why There Is So Much Error in Capital Cases, and What Can Be Done About It.

The first report provides the following diagram:
which gives us an answer to the question posed at the very top of this post. What is the probability that a person has committed a crime (in this case one which courts the death penalty) given that the person has been convicted of that crime? Based on this work the answer is 32 %; you would've got a better chance from flipping a coin than going through this judicial system.

If that's the answer for capital crimes, what about non-capital offences? Criminal damage, theft, fraud etc.

The methodology used in the report is that of results from the justice system being used to test the justice system. It isn't a case of some sort of scientific analysis being used. This would be better ... as to whether or not any State has a system that could withstand this scrutiny, I don't know.

Update 3rd October 2011. Of course, don't forget that the fiction of the findings of a court are applicable to civil cases as well as criminal ones. Hence in both situations,
"We must not allow [Court] decisions to construct our reality."

Update 5th October 2011. It looks as though my question will be answered soon. Academics at Probability and Law blog have got the funding to answer my question.
"We have already got the support of people at the Criminal Cases Review Commission to look at cases on file to do exactly this."


  1. What mathematical nonsense. Some people are guilty and some are not guilty, and therefore (add 1 to 0 and divide by 2) 50% of people convicted are innocent? Try the argument on something else: some people have curly hair and some people don't, therefore 50% of people have curly hair? How can you possibly expect to be taken seriously?

  2. ^what's your analysis and answer to the question posed?

  3. @Vronsky, let me try again.

    Someone holds up a bank, he's wearing a crash helmet, what are the chances that his hair is curly rather than straight?

    Taking your assumptions (particularly, ignoring that he may be bald) then the argument is as you describe, there's a 50% chance that he's got curly hair.

    But in this case we can go further. We can sample a representative proportion of the population and, if we do this task properly, we can find out that, and I'm pulling these figures out of the air, 5% of the population have curly hair, 95% have straight hair.

    Using these figures, we can go back and give a probability figure of the chances that the robber will have curly hair. The best explanation on the web I've seen for this analysis is by Yudkowsky, so you can work out the answer for yourself.

    Now, let's go back to my original post. The point is that the process of sampling and analysis of the general population to determine the probability rates of curly/straight hair are not done when it comes to the comparable situation of conviction/committed.

    Although we know that there have been people wrongly convicted, I'm complaining because there hasn't been any proper analysis to find out what proportion of the population of convicted people have actually committed the crime for which they have been convicted. Therefore, rationally the best we can say in the absence of any other information, when we hear that someone has been convicted of a crime, is that there is only a 50% chance that he committed the crime.

    Does it make sense this time around?