Killer P-values

Our shallow perception of criminal court trials is devoid of any mathematics whatsoever unless it’s the dollar amount of fraud or number of victims. (Yes, I’m including some of you, lawyers too. :P) But as a person who lightens up when a seemingly different matter draws from a concept of pure/abstract mathematics, I was partly in awe and partly cynical of the way a court trial decision is taken given all the caveats of the law and the evidence.

The postmodernism-enthusiast-me would agree with the real me when I say that nothing can be really be distinguished as real or fake.

You saw it “with your own eyes”? You could be dreaming/hallucinating.

You don’t hallucinate? You could be on heavy drugs or could be watching just a very realistic dream.

You don’t do drugs? You can be so much dependent on them that you’re in denial and perennially “under the influence”.

Everyone else saw that? Groupthink!

Everyone else individually and unanimously said they saw something? Someone could be setting you all up. (Just like the moon landing. Just kidding. :P)

Or worse, you could all have been bribed to say that by someone powerful.

Or even worse, can’t you all have Alzheimer’s?

I don’t know. And no one can really know.

The way we perceive the world is more of a “majority wins” approach.

It’s only a matter of how likely the above cases are within the bounds of our understanding of the world. Of course, if we’re all addicted to a substance called oxygen and actually going sober is what we call “death”, none of us can possibly know that. We’re here to hypothesize.

So if the defendant’s gloves are found around the victim’s dead body, what’s the probability that the guy is innocent yet they land up around there covered with blood. Perhaps, someone planted it. If we jail people based on little evidence, most perpetrators will be caught but that would lead to some innocents falling prey too. On the other hand, if our expectations from the evidence are unrealistically high, hardly any innocent will be charged but a lot many criminals will be enjoying a normal life outside prison. The basis of all criminal trials is to avoid false positives as much as reasonably possible even though that means some criminals going scot-free.

Just like any hypothesis test, we start with a null hypothesis – the default case – the status quo and the alternate hypothesis, what we are contesting to check if it’s true or not contrary to what we’ve been believing so far. Then we find the probability that the given evidence could have been there even if the person is innocent. This probability can’t be zero, especially when we’re talking about the past. But if it’s way too low, we can safely assume that it’s not just plain bad luck for the defendant and he is, in fact, guilty of the charges. This probability is called p-value in technical terms.

Even though the person is charged with the crime, at the beginning of the trial (and until the jury declares otherwise) the accused is assumed to be innocent. Only if overwhelming evidence of the person’s guilt can be shown is the jury expected to declare the person guilty-otherwise the person is considered innocent.[1]

Now that we understand the gravity of this p-value. Let me tell you a depressing story which will hopefully make you feel better about your current life, at least that’s why I read depressing stuff :D.

Once upon a time (back in the 90s), an affluent English solicitor named Sally Clark had a baby boy. But the child couldn’t survive longer than a few weeks, dying of “sudden infant death syndrome”, a mere beginning of a series of unfortunate events that shaped her life forever. She and her husband decided to have another baby after a couple of years and were blessed with another son. But as luck will have it, he died of similar condition as well. On both occasions, she was alone with the child at her home and was still coping with the depression caused by losing her first-born when she lost her second son too. And if that is not enough distress for the couple, they were arrested for their sons’ murder. Sally was portrayed as the child-killer, while her charges against her husband were dropped. A pediatrician professor testified against her quoting a chance of two children from an affluent family suffering from cot death as 1 in 73 million. He had assumed these two events as independent events, suggesting no influence of genes or environment for siblings suffering this tragic death. This flawed number suggests it was highly unlikely that the kids died of natural deaths, and must have been killed by their mothers. She was subsequently convicted with a 10-2 verdict against her and sent to prison where she was bullied and she became increasingly depressed.
Her husband left his job, started working as a lawyer, sold their home to fund the trials. He appealed the decision and she was released after 3 years of imprisonment after it was evident that the statistical evidence presented was deeply flawed. However, she never was able to cope up with the trauma and died after a few years of alcohol poisoning (or suicide).

Since then, there have been several research studies on the weaknesses and problems a simple statistic can have when considering matters of life and death for court trials, but provided a lot of us still have a jury system, I am not very confident of how far we’ve come.

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