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October 30th, 2006 (Permalink)

The Prosecutor's Fallacy

Ben Goldacre has a new Bad Science column discussing a court case involving two distinct mistakes in reasoning about probabilities:

1. The mistake of treating probabilities that may not be independent as if they were. To determine the joint probability of two independent events, you multiply together the probabilities of the individual events. For example, suppose that you toss two coins, what is the probability that both come up heads? Of course, the probability of one coin coming up heads is 1/2. The two events are independent because whether one comes up heads doesn't affect whether the other does, so the probability of both coming up heads is 1/2 X 1/2 = 1/4. However, in the case discussed by Goldacre, the probability of a family having one case of SIDS may well not be independent of the probability of having another, so it is a mistake to use the multiplication rule to determine the joint probability of two such cases in a single family. This type of error doesn't have a name that I'm aware of, though it probably ought to.
2. The "prosecutor's fallacy" is the mistake of confusing the direction of a conditional probability. For example, the probability that you will get lung cancer if you smoke cigarettes is a conditional probability; so is the probability that you smoke given that you have lung cancer. But these are two distinct probabilities, the latter being a much higher probability than the former. Similarly, in the case in question, the probability of an accused person's family having two unexplained infant deaths if they are innocent of murder is different from the probability that the accused person is innocent if there are two such deaths in their family. Suppose, for instance, that there are ten cases of families with two unexplained infant deaths, nine of which are due to SIDS and one is a case of murder. Then, the chance of having two such deaths in a family is very low, namely, ten out of however many total families there are. In contrast, given that a family has two such deaths, the probability that they are due to murder is only 1/10.

   Confusing these different probabilities is an error similar to the propositional fallacy of commuting a conditional―as well as all of the other formal fallacies involving conditional statements, such as affirming the consequent―and perhaps has the same psychological origin. I suppose that it got the name "prosecutor's fallacy" by occurring in some prominent court cases, such as the one discussed by Goldacre, but it might mislead people into thinking that it is only committed by prosecutors.

If anyone knows of other names for these two mistakes, please let me know.

Source: Ben Goldacre, "Prosecuting and Defending by Numbers", Bad Science, 10/28/2006

October 24th, 2006 (Permalink)

An Infinitely Frightening Puzzle

The Boojum is a frightening monster. It is the most frightening monster of all. In fact, it is infinitely frightening. One of the reasons why the Boojum is so frightening is that it attacks people in their sleep, frightening them to death. The only way to keep the Boojum away is to pray to it just before you go to sleep.

I won't blame you if you're skeptical about the Boojum's existence, but you must agree that either it exists or it doesn't. Supposing that the Boojum really does exist, and you don't pray to it, then you risk an infinitely awful Boojum attack. Now, the chance of a Boojum attack may be very small, but it's not zero, and a small chance of an infinite loss is still an infinite loss―this is because an infinite value multiplied by a very small probability is still infinite. On the other hand, if the Boojum does exist and you pray to it, then you lose very little: just a minute or two of your time. Also, if the Boojum doesn't exist and you pray to it, then you'll still lose very little: in addition to some time, maybe a little embarrassment. Finally, if it doesn't exist and you don't pray to it, then you'll come out even.

These possibilities are summarized in the following table:

So, you're slightly better off if you don't pray to a nonexistent Boojum, but infinitely worse off if you don't pray to it and it does exist. However, if you do pray to it, then all you suffer is a small loss in either case. Therefore, for a small loss in time and self-respect, you can avoid the risk of an infinitely frightening Boojum attack. Think of it as insurance: for a small premium, you can be protected from an infinite loss. The only reasonable conclusion is: You should pray to the Boojum!

What's wrong with this argument?

Solution

Solution to the Puzzle (11/1/2006): Several puzzlers pointed out that the puzzle is similar to an argument known as "Pascal's wager". The philosopher and mathematician Blaise Pascal argued that you should "bet" on God's existence because you stand to gain an infinite value if He exists, while only losing a finite amount if He doesn't. The Boojum argument presents a negative version of this argument, where you stand to lose an infinite amount if the Boojum exists and you don't pray to him, but only lose a finite amount if you do pray. It wasn't part of the puzzle proper, so the following readers deserve extra credit for pointing out this similarity: Dan Adams, Lee Randolph, Matt Turner, and Kyle Wilkinson.

So, what's wrong with the argument? Christopher S. Moore identified the fallacy committed:

The logical fallacy of false dilemma―also known as falsified dilemma, fallacy of the excluded middle, black and white thinking, false dichotomy, false correlative, either/or fallacy and bifurcation―involves a situation in which two alternative points of view are held to be the only options, when in reality there exist one or more other options which have not been considered.

Christopher also deserves extra credit for giving the fallacy some aliases that I hadn't heard before!

John Congdon explains why the argument commits the fallacy:

"…[Y]ou must agree that either it [the Boojum] exists or it doesn't" assumes that, if the Boojum exists, it exists only in the manner described. This ignores a number of possibilities: that the Boojum exists but that praying to it won't help you, that the Boojum exists but is purely benign, that the Boojum exists but praying to it will only help if you pray to it sincerely and not merely out of fear or calculation, or―irrespective of the Boojum's existence―that praying to it will certainly offend the Snark, against whose retribution the anger of the Boojum is a mere bagatelle. Since the calculation of possible outcomes does not indicate the full range of possibilities, it is therefore useless as a guide to what action to take.

Dan Adams sums it up nice and concise:

The problem is that only two possibilities are presented: Boojum exists and the claims are true; Boojum doesn't exist. There could be a third option: Gorjack exists and detests prayers to Boojum and will torture all who pray to Boojum, which changes the face of the whole gambit. There could also be this option: Boojum exists and he hates being prayed to and will torture all who pray to him.

In other words, the chart oversimplifies the situation in the following ways: The "Boojum exists" column conceals many alternatives in addition to the one listed, including that the Boojum exists but doesn't like to be prayed to, the Boojum exists but an anti-Boojum monster also exists who hates people who pray to the Boojum, etc. Similarly, the "Boojum Doesn't Exist" column should include the alternatives of an anti-Boojum existing, or a jealous Yahweh who punishes all those who pray to false gods, etc. One needs to take into account the full range of possibilities since they affect the bottom line value of the decision. The possibilities of a Boojum and an anti-Boojum cancel each other out, leading to the correct conclusion that we really have no good reason to pray to the Boojum. When applied to Pascal's Wager, this is known as the "many gods" objection.

Other readers who correctly identified the mistake are: James Knobbs, Mike McKay, and Wade Wesolowsky. Congratulations to them all!

A few readers suggested, not surprisingly, that the argument was an appeal to fear, or more generally, an appeal to consequences. However, it isn't a fallacious appeal, because it doesn't conclude that the Boojum exists, or even that you should believe in it. Rather, the conclusion is that you should pray to the Boojum. It isn't necessarily fallacious to use fear of bad consequences to motivate people to change their behavior, though it may often be better to use a carrot rather than a stick.

Resource: Alan Hájek, "Pascal's Wager", Stanford Encyclopedia of Philosophy

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