Thursday, October 01, 2009

Random Thoughts about "Mathematical Analysis" of Evidence -- about "Trial by Mathematics"

There has been an academic hullabaloo in some academic quarters about "mathematical analysis of evidence" in trials. This hullabaloo began after the exchange between Laurence Tribe and Michael Finkelstein in the early 1970s about the possible use of probability theory and Bayes' Theorem in trials. I pejoratively call this discussion and debate a hullabaloo because some of this discussion seems to rest on naive notions about the possible uses of numbers.

Some of the attacks on "mathematicization" refer to probability-based equations as "algorithms." But it is important to keep in mind that not every number is an algorithm. For example, first :-) , numbers can be used to count things, enumerate them, tally them. Second, not every mathematical equation or expression purports to be a description of how things work in the world; i.e., not every mathematical equation or expression is a "model" of some part of the world, or some process. (Is "5 +7 = 12" a "model" of the world or some part of the world? Cf. Immanuel Kant and his Critique of Pure Reason. See also Plato.) Third, not every probability -- e.g. .7 -- "stands for" or "represents" a thing or quantity "in the world"; i.e., not every probability is based on or makes use of statistics; i.e., not every probability is or purports to be "stochastic."

Sometimes (but not always, of course) probabilities are used to represent our thoughts about uncertain propositions; i.e., sometimes probabilities represent "credal states" and probability theory is used in an attempt to make our own thinking about our own (uncertain) credal states logical.

Nothing found above shows or suggests that probability theory should be routinely used in trials and none of the above tells us when probability theory should be used in trials. (It should be noted, however, that when litigated issues involve certain kinds of random natural processes -- e.g., radioactive decay -- it is nearly impossible to avoid the use of probability theory.)

I have found -- and some other people have found -- that fiddling with probability expressions can help me avoid basic mistakes about my judgments about the implications of uncertainty -- e.g., about the significance or possible implications of the proposition "80% of all wrongful convictions involve eyewitness identification evidence." Probability theory helps me think through the possible answer to the question, "So what?" (Often only probability theory allows one to understand the many ways in which the implications of statistics can be greatly exaggerated.) It does not follow, of course, that probability theory will help jurors understand uncertain inference. (But probability theory may help a lawyer figure out how to make an argument about uncertain inferences in a way that a jury can understand. [Caveat: Today many juries have members who have much greater mathematical sophistication than almost any trial lawyer does.])

The long-running debate about "trial by mathematics" is in large part much ado about almost nothing.

N.B. Standard probability theory is not the only mathematical system that purports to deal with uncertainty. See, e.g., the theory of fuzzy sets and systems that rely on ordinal rather than cardinal numbers.

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