Happy Monday FRians! As always, Monday’s post comes to us from Stephen Hall, and is a continuation of last week’s post. To read Part I, click here. Thank you, Stephen!!
Last week, part I of this article gave some general background into the various standards of proof in the legal system and a description in words and the law school professorial approximations of somewhat equivalent probabilities. Now, one should connect the concept as originally intended with a general overview of the concept of proof itself.
While law professors like to through out probability approximations because it makes their academic perspective appear to be more scientific, in actuality law is an art, not a science, meaning that the results in practice are often far more variable, even to the point of being somewhat arbitrary and apparently capricious, certainly less predictable, than one would hope indicative of an actual systematic approach to the judicial function of government.
As much as a judge may enunciate that a standard is “by evidence tending to show that one side has prevailed in the burden of proof by a preponderance of the evidence standard”, the disheartening truth is that in large part the men and women of the jury try as they might to follow such aristocratic sentiments, really reduce the problem in their own minds as simply, “this is who we believe”.
The U.S. Supreme Court has repeatedly eschewed any adherence to a rigid mathematical formulation in any legal standard, that the burden of proof and the standards of proof are not to be expressed as a mathematical standard.
At its core, and as a mathematician, this is rather disturbing to me. Mathematics is a form of philosophy. It is an expression of the purest of logic and reason. To hold that something cannot be expressed in mathematical terms always sounds to me as an admission of irrationality, of illogic and unreason.
I reject the notions of the Supreme Court, and propose that a proper understanding can be expressed in mathematical terms. It is only because so many lawyers are drawn from linguistic backgrounds rather than scientific backgrounds that such a rejection can be maintained.
“For there is but one essential justice which cements society, and one law which establishes this justice. This law is right reason, which is the true rule of all commandments and prohibitions. Whoever neglects this law, whether written or unwritten, is necessarily unjust and wicked.” ― Marcus Tullius Cicero, On the laws
Or, as I vaguely recall hearing it expressed, “that which defies reason, cannot be law.”
By simply approaching the concept of the burden of proof from an abstract perspective, it would appear fairly obvious to the casual observer that it is a problem easily and properly addressed by the application of a Bayesian analysis with a view of conditional probabilities.
Simple enough, right?
It is an odd thing that this human species has an innate tendency to approach problems completely backwards for some reason making things vastly more complicated and difficult than they ever really need to be.
For instance, what is the probability that in a group of twenty people, at least two of them will share the same birthday? Most people will want to presume that this is a highly unlikely event, when the odds are really more than half, or fifty percent.
One simply has to look not at the likelihood of the event, but rather at the likelihood of the non-event, or rather the odds of them not having at least two people with the same birthday.
Each person, ignoring the leap year for simplicity, and assuming a uniform distribution, can have a birthday on any of 365 days. The first person obviously has a certainty of not having a birthday with the odds of 365 out of 365, or 365/365=1. The next person can then only have 364 possible days if his birthday does not match, or 364/365, rather (365/365)x(364/365). The third person may not match either of the first two, and so forth.
The fraction decreases at an increasing rate as it is multiplied by successively smaller fractions, surprisingly fast. The odds of there being two people who match is the compliment of the likelihood of them not matching, or rather (1-P(x)), where P(x) is the probability that no two match. It turns out that (1-P(x)) < 0.5 after only twenty people are thrown into the mix.
Thus, to apply these principles to the law, one should look not at proving a person guilty, but reducing by conditional probability the possibility that they are in fact innocent. Permit me to illustrate, and explain the title of this article in the process.
Every accused in a criminal trial is afforded “the presumption of innocence”. But, what would “the presumption of innocence” mean in mathematical terms if lawyers and jurors were all mathematicians? Well, there are about seven billion people in the world; at the beginning of a trial, there has been no evidence presented; so theoretically, any one of those seven billion people could be guilty. The presumption of innocence is simply the probability of that one accused divided by the entire population of the planet, a one in seven billion chance that we randomly selected the guilty party.
See? In simple and easily understood terms we have expressed mathematically that legal phrase with which everyone is all too familiar, the “presumption of innocence” as a legitimate and clear expression of probability. Just imagine a large circle with seven billion people in the circle, and one of those people in the circle is guilty. Math is logic.
Now, we begin to introduce evidence, or as I like to call it, a Bayesian negative probability factor. Suppose our crime is a burglary. Our hypothetical crime occurred in Lexington, Kentucky. Well, we can eliminate from our circle every person who has never been to the United states, every person who has never been to Kentucky, and every person who has never been to Lexington. They have an alibi, not in the sense that they can prove they were elsewhere, but that they were never there.
Suppose our crime occurred on August 5th, 2017, we can further eliminate from our circle, not just the people who have never been there, but everyone who was not present on that day. We have introduces our second piece of evidence, and eliminated a whole bunch of suspects, based upon simple probability.
Now, with each additional piece of evidence, we multiply our original probability that it was not the accused by an additional factor, our P(x) gets larger with each piece of evidence, making our (1-P(x)) closer and closer to certainty, or 1.
An eyewitness saw that the burglar was a man, thus eliminating women from our circle; that he was an adult, thus eliminating children; that he was under 40, eliminating the older men; that he was over 160 pounds, eliminating the thin people; that he was Caucasian, eliminating all other ethnic groups, etc.
Each and every piece of evidence eliminates people from our circle, our potential criminals. Once they are eliminated, they can’t come back into the circle, so the trail of proof is a one way street getting ever smaller.
If, in the end, after all evidence has been presented, our accused is the only person standing in that circle, and no one else; then we can feel mathematically certain that they are guilty. If there are two or more people in that circle, then the odds remain at least fifty percent likely that the accused is innocent. No one would convict a man on those odds.
However, suppose we have rendered it down to one person, the accused, remaining in our circle of conditional probability; then the evidence has moved the probability from the “presumption of innocence” to proof “beyond a reasonable doubt” with a mathematical, logical progression.
Of course, in reality the evidence is not quite so certain that we have managed to eliminate every alternative possibility. It was dark, and perhaps the witness was mistaken in determining that they were Caucasian. Perhaps a trick of the light or a thick coat made him look heftier than he was. Perhaps it was a woman who just was heavily muscular and looked like a man. Perhaps that DNA really points to the evil identical twin no one knew about.
One cannot ever eliminate all possibility of error. That is why the legal system qualifies the standard of proof with “reasonable” rather than a standard of mathematical certainty. However, there is a logical, mathematically expressible understanding of the process of the legal journey from presumption of innocence to proof of guilt.
Moreover, I like the expression of reason in triumph over the emotional storytelling approach to which too much of our society succumbs. Anyway, that is my perspective, and now you know a little more of the way in which I tend to think.
