On Reasonable Doubt and Probability

One of the lingering philosophical questions in law generally but in criminal law in particular is whether the various burdens of proof, and the burden of proof beyond a reasonable doubt in particular, are, in some underlying sense, probabilistic. We interpret everything probabilistically these days; sophisticated websites offer projected outcomes for an upcoming baseball season expressed in tenths of a win, explicitly to make the point that they're probabilistic averages, not predictions of a specific final win total. But we are naturally a little uncomfortable convicting someone on that basis, a discomfort rooted in the old phrase "to a moral certainty," the forebear of "beyond a reasonable doubt." Is a 10% probability of the defendant's innocence "reasonable doubt"? Is 1%? Do we really think that we convict people with 99% probabilities that they're guilty? Do we really want to start thinking about this? But isn't it the correct way to think about it, in at least some senses?

And I've just had an idea about a slightly different way to phrase this question, if not to answer it, based on the facts of a case I'm just starting to read for my Evidence class. In this case, which bears the awesome name of People v. Mountain, a criminal defendant was convicted based, among other things, on evidence about blood type. There was type A blood at the crime scene, and the defendant was type A. Type A blood is found in roughly one-third of people, a little more in this country if we don't distinguish between positive and negative. Now, it is unambiguously true that knowing there was type A blood at the scene should make us more likely to believe that a type A defendant, any type A defendant in fact, is guilty. This is where Bayesian statistics come in handy; math below the fold.

Suppose that we have, on the basis of all the other evidence, some estimate of how likely the defendant is to be guilty, call it p, and an estimate of his innocence, 1 - p. Knowing that the perpetrator shared the A blood type with the defendant should make our new estimate of his guilt approximately p/(p + (1 - p)/3). You get there as follows: in the universe where the defendant is guilty, which we start out thinking to be the real universe with probability p, we would observe a type A perpetrator 100% of the time. In the universes where the defendant is innocent, we would expect to observe a type A perpetrator about 33% of the time, give or take. So we can intuitively rule out two-thirds of those hypothetical universes, based on our knowledge of the perpetrator's guilt. But now we have a smaller sample of possible hypothetical universes, so we need to readjust the baseline of our probability estimate, meaning that the p universes where he's guilty take up more of the probability. The end result is to increase the ratio between p and 1 - p by a factor of 3. If we thought it was a coin flip whether the defendant was guilty before we learned this, well, now we think it's 3 to 1 that he is. If prior to gaining this knowledge he was, say, just a random guy off the streets of New York City, which I'll call a 1 in 8 million chance of guilt just at random though that's very imprecise, well, now we think it's something more like 1 in 2.3 million. Big whup.

And now here's my question about the meaning of reasonable doubt, and whether we can conceive of it probabilistically. If, in principle, it's just a probability threshold, then it should be possible for something that makes a relatively small difference in the probability estimate by itself to push us over the top. Say we want to require 99% certainty. Then in a case where we have a bunch of evidence that seems to establish a 98% chance of the defendant's guilt, the additional knowledge that both the defendant and the perpetrator are type A would increase that to about 99.3%, and we'd have a conviction. From a 1.3% increase in our estimated odds that we've got the right guy. From a piece of evidence that literally would not distinguish the defendant from over one-third of the random people you pulled off the street, or on the jury.

Is that right? Is it possible that we can divide a reasonable doubt by a factor of three and get an unreasonable doubt?* Because I think that the answer to that question has to also be the answer to whether reasonable doubt is in some sense a concept of probabilities. If you're comfortable letting blood type analysis, using a common blood type no less, push a not-quite-reasonable-doubt case over the finish line, I think you think that reasonable doubt is nothing more than a vague intimation of some kind of probability standard. It means that the jury needs to think there's only a very very small probability that the defendant is innocent. And if you don't think this is right, then I think you think that's wrong, that reasonable doubt is about something more than mere probabilities. But if that's wrong, what's actually going on? That's a question way beyond the scope of this blog post. I just wanted to point out what it means if we adopt the probabilistic interpretation, and conversely what it means if we're not comfortable with the kind of "proof" demonstrated above.

*Note that, if it's possible to do this with a factor of three, it is possible in theory to do it with any factor no matter how small. If I start at 98.95% certainty the defendant's guilty, and learn something which cuts my doubt by a factor of 1.1, I have moved over the 99% threshold. The probabilistic interpretation, then, seems to imply that there can be a case that just misses the standard for conviction, where learning virtually anything else that makes you even a scintilla more likely to think the defendant is guilty would tip the scales. That strikes me as kind of radical.