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Your essential point is well received: the jury had an opportunity to consider the evidence pointing towards innocence, and we should take the jury verdict and do Bayes updates on evidence which they already considered, and we should also remember that post trial, the probability of any evidence surfacing is strongly dependent on the direction is pointing, 'not only is he guilty, but we have found evidence for another 100:1 update towards that conclusion' does not make for a terribly exciting podcast.
However, I would set my threshold a bit lower than you. You basically say to update away from the 'guilty beyond reasonable doubt' verdict, you would require evidence for 'innocent beyond reasonable doubt'. I think this is a bit of a high bar. If new evidence pointing to innocence 10:1 is surfacing after the trial, then that might already call the verdict into question, because juries tend not to give out numerical probabilities (not that they would be qualified to do so), and reasonable doubt is not a fixed probability value either.
Personally, I have stronger faith in jury verdicts reached using forensic evidence. I think juries have a tendency to over-update on eyewitness accounts. Here, two key pieces of evidence are not only eyewitness accounts, but hearsay. (Permissible hearsay, but hearsay still.) I don't like that at all. Not only have you all the usual unreliability of eyewitnesses, but one of the key checks on outright fabrications by witnesses, the fact that such fabrications will land you in prison for a lengthy stay, is completely missing. If you claim that Alice shot when you in fact saw Eve shooting, you have to consider the possibility that other evidence will surface which will prove you wrong. "He confessed to me" is the ultimate he-said-she-said situation. Even if evidence later surfaces which clears the accused beyond any reasonable doubt, he might still have confessed to you, people lie all the time. Nailing you for that will basically be impossible.
The second account, plus the knowledge of the damning details of the crime make a malicious witness unlikely. If any shenanigans were going on, it would have to be on the investigative or prosecutorial side. That is unlikely, but not impossible. Plenty of people here seem to believe that defense attorneys who try to get guilty clients acquitted are immoral. From that, it is not a huge distance to 'prosecutors who know the defendant is guilty should subvert due process to get a guilty verdict'. I hope that these attitudes are less common among lawyers, though.
Without diving deeply into the case, I will however not say that the verdict was incorrect.
IIRC, the Guardian mentioned that a prosecutor wanted the case reopened, which would be unusual because prosecutors are rarely anti-death-penalty activists. However, given their usual spins, it likely just means that the prosecutor wanted to identify the DNA on the weapon before the execution, and the outcome 'it got there due to bad evidence handling' was embarrassing but acceptable and they were fine with the execution proceeding.
We're pretty far outside of ordinary "Defense Attorneys" and a decent distance from ordinary Prosecutors as well. Once you get deep into appeals, you're dealing with ideologues on the defense side who dedicate their lives to this kind of activist anti-death penalty work. These kinds of people are basically anti-death penalty, and often anti LWOP, extremists who take "an unjust law is no law" to heart. They will actively try to overturn what they view as an inherently unjust sentence by any means necessary.
This particular prosecutor is running for congress. Not that this means he's wrong, or cynical, but it does mean he has different motivations for public advocacy than most DAs.
Now, to engage with the meat of the post:
Not exactly, because you're confusing the logic by trying to smuggle in Bayesian probabilism to a binary based in religious morality. "Beyond a reasonable doubt" began as essentially a theological formula: it was the point of certainty at which even a mistake by the jury that lead to the death of an innocent man would not harm their souls' salvation. Beyond a reasonable doubt is a defense you offer to the Lord on judgment day "Hey look I had doubts but they weren't reasonable;" which hypothetical then becomes a standard for taking action "I'm comfortable with this verdict because I would be comfortable defending this action on judgment day."
I can update toward innocence or a jury mistake in my heart of hearts all I want, the question is at what point we make a binary switch from his legal guilt to his legal innocence (or a legal mistrial, but we'll ignore that for the moment). Moving back from the death penalty for reasons of doubt about his guilt isn't really an option within the legal system, nor should it be, the penalty and guilt phases are separate, either you free him or you kill him. We're not talking about updating Bayesian probabilities, we're talking about taking action. Either killing a murderer or freeing him.
So you agree with me that the standard for reversing a verdict should be reversed in strength from the standard for reaching it. The burden has shifted.
Now as I keep emphasizing these are not probabilities, but using probabilities to make the point: if you set your decision points at >95% to reach a guilty verdict, and at <50% certainty you overturn the verdict on appeal, any update after the verdict that doesn't get you to 49% is irrelevant. I can update all I want between 95 and 51, the verdict stands, he fries.
Where I object is when people try to smuggle in reasonable doubt after the verdict, so that the standard is >95% to convict, and if at any time afterward any other person involved (judge, prosecutor, etc) gets to <95% the conviction should be overturned.
Thanks for the background on reasonable doubt.
You are correct, in the absence of numerical odds given by the jury and given that updates after the verdict are very costly, we should have some hysteresis built in for the verdict. Say we don't want to execute some man who is innocent with a probability p_k (perhaps 10%, or 5%, or 1%). What we should do is require a higher standard for conviction verdicts, perhaps p<p_k/x (Where x might be 5, perhaps). Then after the verdict, we have some slack to not revert the verdict even in the case of evidence with an odds ratio 1:x in favor of innocence.
Of course, an extremist view would be that we should never overturn verdicts and just accept the deaths of innocents as already priced in, statistically, in our p_k threshold (as long as the juries are well calibrated). However, not using all the available info (with exception of the exclusionary rule in court) seems indefensible (and I see neither you nor anyone here arguing for it). The reasonable doubt standard on judgement day likely applies on a case-by-case base, the governor who signed an execution order for a man who is likely innocent and just tells God 'well, I found that jury verdicts generally achieve your ordained threshold, and found it to bothersome to update, but look, the other 19 men I hanged were all murderers, so statistically speaking, we are good' would likely be hellbound in most theologies.
Of course, Scott Alexander advocates explicit probabilities:
I'm confused as to what you're saying here, so I don't want to argue about it too hard until you explain it.
I think you're ignoring the systemic value of finality in verdicts. If we constantly allow convictions and sentences to be paused and altered while we sit and consider whether some new piece of evidence alters the probability of guilt to a degree too great to countenance, we'll never get anything done.
Made up probabilities are all well and good for those of us in the peanut gallery, but for decision makers final decisions must be made, and they must be respected with some degree of certainty.
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