Bryan Caplan  

How to Tell the Truth With Statistics

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From Robert Gordon's "Everyday Life As An Intelligence Test":

Even defense attorney Alan Dershowitz was guilty of faulty probability reasoning when he correctly pointed out that fewer than 1 in 1000 wives who are abused by their spouses, as Mrs. Simpson had been, are later killed by them... A mathematician replied to Dershowitz, using the relevant conditional probabilities, that "if a man abuses his wife and she is later murdered, the batterer is the murderer more than 80 percent of the time. [references omitted]
Since I believe in the preponderance standard of proof for all offenses*, Gordon's factoid has some interesting implications for me. In a Caplanian world, this base rate would, by itself, have been a prima facie case against O.J. Simpson. To avoid a conviction, it would have been virtually essential for him to take the stand to explain, loud and clear, why we should believe that he was the exception that proved the rule.

* Assuming the charge ought to be an offense in the first place. I'd nullify if I thought it shouldn't.


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COMMENTS (12 to date)
Kurbla writes:

Courts actually work the way you suggest, just probability that determines "guilty beyond reasonable doubt" is not fixed number - because relevant statistics are only exceptionally available. Otherwise, the notion of alibi wouldn't have a sense - the logical possibility of alibi refutes the proof that one is guilty just as well as actual alibi.

KipEsquire writes:

Since the overwhelming majority of violent crime is committed by men, by your reasoning all the prosecution need do in a violent crime case is prove that the defendant is male.

And whatever happened to the (non-linear) tradeoff between Type I and Type II errors?

KipEsquire writes:

Make that: "by your reasoning no jury in a violent crime case could ever convict a woman"

Jeff H. writes:

I don't think so, Kip.

You're also confusing things here. In criminal cases, the prosecution must present prima facie evidence of each element of its case. In Capland (I want credit for this term!), the statistical probability alone in OJ’s case would be enough to start legal proceedings.

Your example doesn’t work because it assumes that the only evidence the prosecution submits is the fact that the defendant is a woman. In this case, the court would never even hear the case, because the defense would rightly point out that because the probability is so low, the prosecution has failed to make out a prima facie case.

Gavin Andresen writes:

Where does the "80% of..." probability number come from?

If it comes from prior convictions, then in "Capland" the legal system has a feedback loop that will tend to judge more people guilty over time:

+ You start with an 80% chance of being guilty.
+ You are found guilty.
+ The next defendant now starts with a slightly higher chance of being guilty (and a higher hurdle to jump over to prove their innocence).

You can avoid this if you have a more reliable way of determining innocent/guilt independent of the court system... but if you do, then why not just replace the court system?

Richard Akers writes:

Kip,

The preponderance standard requires that the evidence show that there is a > 50% chance that the specific suspect committed the crime. So, if you picked a man at random to be the suspect of a violent crime, you'd have to divide the percentage of violent crime committed by men by the population of men to get the chance that that man committed the crime. I don't feel like looking up the actual numbers, but I know that less than 100% of violent crimes are committed by men and the population of men is greater than 3 billion, so the random man has less than 0.00000003% chance of being the culprit.

Bryan,

I really don't think I want to live in Capland (HT: Jeff H.). If we assume a uniform distribution of chance of the convicted being guilty between 51% and 100%, around 1/4 of the prison population will be innocent of the crime that put them there. I would think that prosecution would tend to proceed as soon as enough evidence was gathered to gain a conviction, so I actually expect most convicts to be much closer to 51% likely to be guilty than 100%, pushing the innocent prison population towards 50%.

Now consider the quite common prosecutor mentality of getting a conviction at any cost. Some of this is a result of guilty people going free due to the reasonable doubt standard, so it would lessen somewhat. But some of it is due to other factors, such as the fact that we tend to measure prosecutor effectiveness on their conviction rate. If enough of that mentality survived, I could easily see the innocent prison population hit higher than 50%.

Snark writes:

Richard,

…so I actually expect most convicts to be much closer to 51% likely to be guilty than 100%, pushing the innocent prison population towards 50%.
With a prison population mean of 51% guilty (based on a .95 confidence level), Capland™ DOJ officials might finally dispel the notion that prisons house only the innocent.
Stewart writes:

You are abandoning the presumption of innocence?

michael gordon writes:

1) The balance of probabilities --- another name for the (empirical) relative frequency approach to probability (or preponderance of evidence in US law) --- is in fact used in civil casesNot, however, in the US (or Britain) in criminal cases, where the "presumption of innocence" is part and parcel of a 1000 year tradition of common law.

--- On the Continent of Europe, the presumption of innocence is supposed to apply in criminal cases, but here we are dealing with Roman (and Roman-Dutch) legal traditions, which emphasized traditionally state-security . . . not the rights of individuals. Hence, in France and elsewhere in West Europe, the accused is supposed to cooperate with the investigating magistrate --- prior to a trial if it occurs --- to establish the "truth of his behavior." There is no right not to cooperate. If the accused is then brought to trial, the investigating magistrate reports on whether the accused has cooperated fully or not. If not, the presiding trio of judges --- with or without a jury (it depends on the kinds of criminal cases) --- is supposed to consider that as part of the evidence in favor of guilt.


2) As Akers notes, the preponderance of relative-frequency approach to "guilt" in a wider sample than Bryan's --- violent crimes, men commit more than 50%, a random selection from the number of men in the world (more accurately: in the relevant jurisdiction . . . which could be the US in a murder case) --- requires an n(a)/n where n approaches infinity.

3) In Bryan's case, the relevant n(a)/n would be = n (wife-murderers)/n (all wife-batterers in the jurisdiction).

Among other things, though the n (wife-batterers) is undoubtedly large and gives more confidence with a (subjectively chosen) margin of statistical error or reliability, there is no reliable way to ascertain that n (wife-batterers). Why? Because obviously most wife-battered women do not report the battering to the police. Victim surveys might do better, but then you will run into other frequency and sample problems --- such as defining the threshold of battering. Is it a mild slap on the arm? A mild slap on the face? Maybe a harder slap on the face by a husband in response to his wife's slap on him etc.

That is why we use a much harder standard of guilt in the US (and UK) of a unanimous jury deciding the case as being beyond all reasonable doubt. The EU Continental countries do not require such unanimity even if there is a jury (on which, believe it or not, the trio of judges will sit and give legal expert views).

4) On a technical level, to infer from a relative probability trend of n(a)/a = 80% that a specific individual is very likely to be guilty, moves the statistician from a balance of probabilities (empirical relative frequency) to a Bayesian subjective approach. In such an approach, of course, experts will have different priors and may disagree markedly.

That is the nature of the problems of probability reasoning.

5) Bryan tries to get around this problem, it seems, by insisting that Simpson testify and prove (or show convincingly) that he wasn't guilty.

In effect, he is asking for a switch from the US-UK common-law traditions --- which he obviously favors as a libertarian in all property matters and cases against governmental interference --- to the Roman-legal statist criminal codes that prevail in West Europe on the Continent. In this case, that the accused must cooperate with the investigating magistrate to ascertain the truth of what happened in matters of his behavior pertaining to the crime.

A query prompts itself here: Would Bryan like to switch to such a legal system in matters of property rights?

-- Michael Gordon, AKA, the buggy professor http://www.thebuggyprofessor.org

michael gordon writes:

A quick follow-up, very concise . . . the web manager here, Lauren Landsburg, threatening to boot me into a trash bin if I don't cut the words I use to set out a rigorous argument. (His, or rather Bryan's and Arnold's, right, no?: it's their blog and they set the rules, yes?)

Namely? The difficulty of convicting accused criminals in the US and UK legal systems, compared to those in the EU Continental, is the major reason plea-bargaining is used so extensively in the US system. (In the UK, where violent crime --- as it is virtually all over West Europe --- is much worse than in the US, the Labour government tried to change the law of unanimity to a lesser majority --- maybe 9/12 --- in order to convict more frequently. It met too much opposition, that proposed change).

Interestingly, the French courts have recently decided to allow plea-bargaining, but mainly to reduce the clogging number of cases in the pipeline.

Michael Gordon, AKA, the buggy professor: http://www.thebuggyprofesso.org

RL writes:

Why this anecdote is an example of Dershowitz making an error versus Dershowitz lying in court to benefit himself or his client (something those who follow his writings on political subjects have no difficulty believing he would do) is unclear.

michael gordon writes:

"...In Bryan's case, the relevant n(a)/n would be = n (wife-murderers)/n (all wife-batterers in the jurisdiction)." michael gordon

After a moment's reflection, I have to apologize for my stupidity here. The correct probability would read:

n(a)"guilty murderers known to be wife-batterers" / (n) "all accused murderers known to be wife-batterers."

-- Michael

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