What Bayes is: a test

Note: This is my second blog post in a month-long series on the basics of Bayesian probability theory.

In my previous post, I explained that Bayesian reasoning “is not just a method of drawing inferences from observations or evidence presented; it is also a method of testing the strength or weakness of such inferences.” This testing function of Bayesian methods is crucial: When we are designing, taking, or using a medical test, or a spam-filter test, or any other kind of test, we must cognitively distinguish the test from the underlying hypothesis or event being tested. Put another way, “tests are not the event.” [1] A spam filter tests for spam, but the spam test itself is separate from the event of actually having a spam message, and by the same token, a test for breast cancer, for example, is separate from the event of actually having such a cancer. Or, in the words of one Bayesian scholar:

Even a useful mammography test does not actually change [the underlying reality] whether or not a woman has cancer. She either has cancer or she doesn’t. Reality is not uncertain about whether or not the woman has cancer. We are uncertain about whether or not she has cancer. It is our information, our judgment, that is uncertain, not reality itself.” [2]

Likewise, a legal trial in many ways operates as a “test”–a test of guilt or innocence–and so is separate from the underlying condition being tested, i.e. a defendant’s guilt or innocence. Again, the defendant either has or has not committed a wrongful act (e.g., a crime, a tort, a breach of contract, etc.). [3] Reality itself is not uncertain about the defendant’s conduct. It is we (the jury or the trier or fact) who are uncertain about the defendant’s guilt or innocence. [4] As an aside, this point helps explain the logic of my Bayesian approach to adjudication. A Bayesian or any probabilistic method for trying cases is not designed to tell us with 100% certainty whether a given defendant is guilty or innocent. No method or test could produce such perfect outcomes–a point I will further discuss in my next post–instead, the Bayesian approach is just an alternative method for testing the strength of the moving party’s case, that is, for testing how likely the plaintiff or prosecutor has proven his case.

a branching charge of percentages showing false positives
“Of 10,000 women aged over fifty tested, 36 will be correctly identified as positive whereas 996 will be told they are positive despite not having cancer.” Source: Kit Yates (Science Friday).

[1] See Luke Muehlhauser, An Intuitive Explanation of Eliezer Yudkowsky’s Intuitive Explanation of Bayes’ Theorem (Dec. 18, 2010). See also Eliezer S. Yudkowsky, An Intuitive Explanation of Bayesian Reasoning (June 4, 2006).

[2] Muehlhauser op. cit.

[3] As Krista McCormack has pointed out to me, the legal elements of a wrongful act–that is, what
constitutes a particular crime or tort–are human-constructed elements, since we ultimately define
what elements constitute a tort or crime, such as “battery,” for example. As a result, the underlying
metaphysical reality (e.g., “did this defendant commit a battery?”) might not congruently align with
the human-defined elements of the wrongful act. Nevertheless, there is no reason in principle why we could not apply Bayesian methods to the task of interpreting what behavior constitutes a wrongful act in the first place. That is, we could apply Bayesian methods to predict what interpretation of law is most likely to prevail in a given close case.

[4] Even the practice of science, like litigation, is just a series of tests. There is a test for a given
phenomenon, and there is the event of the phenomenon itself. See Kalid Azad, An Intuitive (and Short) Explanation of Bayes’ Theorem (May 7, 2007).

About F. E. Guerra-Pujol

When I’m not blogging, I am a business law professor at the University of Central Florida.
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