Bayesian Punishment

In which it is argued that taking into account the non-binary nature of uncertainty of guilt or innocence could lead to a more just equilibrium with respect to false positives vs. false negatives in the justice system.

There are many arguments to be made for and against the death penalty, but one strong argument against it is that our justice systems simply aren’t robust enough, that, because the innocent are sometimes condemned, we cannot risk execution, as executing the guilty necessarily requires the risk of also executing the innocent.

This argument suggests that punishment should be influenced not only by the severity of the crime, but also by the confidence we have in the verdict. Our inability to determine with 100% certainty that any individual is guilty of the heinous crimes of which they are accused argues that we should not then condemn them to death. This inability is evidenced by the fact that sometimes those who are sentenced to death are later shown to be innocent, for instance with DNA evidence.

Clearly the strength of this argument must also be influenced by the degree of uncertainty that we have in the guilt of the accused. If we find that ¹⁄₃ of those sentenced to death are later found to have been innocent of their crimes, this is a strong argument against the execution of any prisoner. Whatever the supposed benefits of execution over some alternative form of punishment, they are unlikely to outweigh the deaths of so many innocents. If, on the other hand, we had good reason to support a credence of innocence of ¹⁄₁₀¹²then this argument would lose its force: yes, there would be some chance that an innocent would be sentenced to death, but the chance would be small enough that we could expect that it would never happen over the course of the next thousand years. It’s unlikely that such a credence could be justified, but if, in theory we were confident in this way, then the argument would lose much of its force. As such, we can see that the probabilities matter here.

Our willingness to execute those convicted by the justice system should be influenced by our confidence in the verdict that the justice system hands down. Yet this confidence clearly differs from case to case. While a guilty verdict may represent a case that meets some minimum standard of evidence, it cannot be true that all cases meet exactly that standard. Some will go far beyond it while others barely pass its threshold. The binary decision of the justice system “guilty” or “not guilty” masks a continuum of possible bayesian credences in the guilt or innocence of the accused.

We may have a standard of a credence, for instance, of 0.99 to deliver a guilty verdict, and anything below this could result in a verdict of innocence. But there is a real and meaningful difference between 0.99 and 0.999 or 0.9999. As we saw with the case of the death penalty we all have an intuitive understanding of the importance of this difference, and with large populations this injustice takes the form of the suffering of those innocents who are wrongly convicted.

A more fair justice system would take this difference into account.

I suggest that a reasonable way to take into account the non-binary nature of credences of guilt is in the varying of punishment. Just as we may want to avoid the death penalty because we see the false positive rate as too high, this also suggests mitigating terms of incarceration based on credences.

A jury could give a verdict not of “guilty” vs. “not guilty”, but rather of “We take the prosecution’s case to have met the burden of proof for a guilty verdict, and give a credence in this result of 99.9%”.

It may seem like an odd thing for a jury to do, to first lay down a guilty verdict and then admit that it has some uncertainty in the verdict, but this is good on several fronts. First, it’s simply true. All verdicts have some degree of uncertainty, and a justice system that admits this uncertainty is more honest and more fair than one that tries to obfuscate it. Second, this uncertainty is a powerful part of the psychological process that any juror goes through during a trial. They know that they aren’t certain either of guilt or innocence, but this uncertainty affects different people in different ways, based on temperament, ideology and other views. Each member of a jury will worry about convicting the innocent. They will also be reluctant to acquit the guilty. But each may have a different threshold beyond which their credence in the guilt of the accused will allow them to give a verdict of guilt. By spelling out their actual view of the chance that they are wrong, we give them a chance to be more clear about what the decision being made actually is, and this should lead to a more standardized decision making process within juries. While the assigning of credences is still a subjective process, making clear that this is the process that is actually being asked for will improve the reasoning around it. This has been shown, for instance, in the forecasting work of Philip Tetlock. It may be possible to attempt to improve the accuracy of these credences by requiring jurors to bet on some possible future outcome related to the verdict—for instance on whether or not future evidence will overturn the case, or perhaps on whether a second jury reaches the same conclusion. But these suggestions are more fitted to the topic of a separate discussion about improving the accuracy of trial outcomes. Here I am mostly focused on how a continuum of possible verdicts (credences in guilt or innocence) can improve the justice of sentencing given any specific regime of accuracy.

Judges could then take into account these credences during sentencing. Cases more closely approaching certainty in their conviction would be more justified in handing down steeper sentences, whereas cases meeting a less rigorous standard could be given reduced sentences, including even things like fines.

From the perspective of justice, then, this framework seems to make sense. It allows us to do two things: the first is to avoid the worst punishments for the innocent, or at least to greatly reduce them. A much higher standard of guilt could be required for extreme punishments like long prison terms. On the other side, it also allows us, if this were seen as useful, to impose smaller punishments on more of the guilty. The first point is mitigating the damage done by false positives, but the second also has the potential to mitigate the damage done by false negatives. If we think that many of the guilty are being released, then this proposal allows us to lay down small punishments on those accused who are very likely to be guilty but where the evidentiary standard doesn’t meet the requirement for guilt in the current system. If, for instance, we currently require 99.9% certainty to find someone guilty, this new system might give a weaker sentence to those where the certainty reached only 99%.

It may be suggested that currently our actual credences in guilt are so low that anyone below this threshold should not be punished at all. This is an argument, then, for giving lower punishments at the current threshold and for several degrees above it. It suggest that the false negative rate is low, but that the gain in the false positive side can be much higher than in another scenario where the false negative rate is higher. Regardless of the current equilibrium this bayesian punishment regime can improve outcomes on one side or the other. Likely, however, we’re not at the exact point where gains come only from one side, and we can gain both by giving small punishments to more criminals who would otherwise have got off with none and by giving less onerous punishments to those innocents who are wrongly convicted.

If you think that too many of the guilty go free, this regime allows more of them to be punished, without imposing too high costs on the innocent. Deterrence is strongly affected not only by the strength of punishment, but also by the likelihood of being punished at all, and this regime allows us to apply punishment to more of the guilty. To some extent the use of plea deals also has this effect, but it does so by bypassing our usual due process systems and with much less rigor or clarity than the system being proposed.

It is also potentially possible that the clarity that comes from thinking in this bayesian way could actually prevent some innocents from being wrongly convicted and some guilty from being wrongly acquitted. I don’t see this as a strong argument in favor of this proposal, but it is a possibility that I’m hopeful of.

Footnotes:

1. A second jury?! This need not require a second trial, as these two juries could be kept separate. The second could also be smaller, only view the trial remotely, etc. The point is to have some objective measure against which the first jury could measure their confidence. Regardless, this point is tangential to the thrust of my argument.

2. If this is not the case, then our justice system is almost certainly failing us. There is a relationship between the rate of false positives and false negatives, and if we do not have false negatives then it is because we have too many false positives. The opposite is also true. The point here is that this proposal allows us to find a better equilibrium between these two than a binary regime allows.

3. We are almost certainly no where near these figures. If you like, change this to 90% and 80%, or whatever figures you think most closely approximate the actual case.