Is there a Bayesian justification for the Murdaugh verdict?

Jonathan Falk writes:

I know you’re much more a computational Bayesian than a philosophical Bayesian, and I assume you were as ignorant of the national phenomenon of the Murdaugh trial as I was, but I just don’t quite get it.

Assume the following facts are true:
(1) Two people were murdered: a mother and son
(2) The husband and father, after having denied he was anywhere near the scene, was forced into admitting he was there shortly before they were killed by incontrovertible evidence.
(3) He is/was a drug addict and embezzler.
(4) There is literally no other evidence connecting him to the crime.

Starting from a presumption of innocence (not sure what that means exactly in probabilistic terms, but where p is the prior probability of guilt, p<<.5) how do you as a Bayesian get from (1)-(4) combined with the prior to get to a posterior of "beyond reasonable doubt?" (Again, leaving precise calibration aside, surely p>>.9)

People lie all the time, and most drug addicts and embezzlers are not murderers, and while proximity to a murder scene (particularly one in secluded private property) is pretty good evidence, it’s not usually enough to convict anyone without some other pretty good evidence, like a murder weapon, or an obvious motive. I’d be willing to countenance a model with a posterior in the neighborhood of maybe 0.7, but it’s not clear to me how a committed Bayesian proceeds in a case like this and finds the defendant guilty.

Thoughts?

He’s got a good question. I have two answers:

1. The availability heuristic. It’s easy to picture Murdaugh being the killer and no good alternative explanations were on offer.

2. Decision analysis. Falk is framing the problem in probabilistic terms: what sort of evidence would it take to shift the probability from much less than 50% to well over 90%, and I see his point that some strong evidence would be necessary, maybe much more than was presented at the trial. But I’m thinking the more relevant framing for the jury is: What should they decide?

Suppose the two options are find Murdaugh guilty or not guilty of first-degree murder. Picture yourself on the jury, making this decision, and consider the 2 x 2 matrix of outcomes under each option:

– Truly guilty, Found not guilty by the court: Evil man gets away with one of the worst crimes you could imagine.

– Truly guilty, Found guilty by the court: Evil man is punished, justice is done. A win.

– Truly not guilty, Found not guilty by the court: Avoided a mistake. Whew!

– Truly not guilty, Found guilty by the court: Lying, creepy-ass, suspicious-acting drug addict and embezzler didn’t actually kill his wife and kid—at least, not quite in the way as was charged. But he goes away for life anyway. No great loss to society here.

The point is that the circumstances of the crime and the jury’s general impression of the defendant are relevant to the decision. The assessed probability that he actually did the crime is relevant, but there’s not any kind of direct relation between the probability and the decision of how to vote in the jury. If you think the defendant is a bad enough guy, then you don’t really need to care so much about false positives.

That said, this will vary by juror, and some of them might be sticklers for the “beyond reasonable doubt” thing. From my perspective, I see the logic of the decision-analysis perspective whereby a juror can be fully Bayesian, estimate the probability of guilt at 0.7 or whatever, and still vote to convict.

P.S. For those commenters who haven’t heard of the Murdaugh trial . . . You can just google it! It’s quite a story.

62 thoughts on “Is there a Bayesian justification for the Murdaugh verdict?

  1. I haven’t followed the trail – actually I usually avoid that kind of media circus.

    However, your rundown of jury outcomes is missing a critical factor under the “Truly not guilty, Found guilty by the court” heading:

    “Truly not guilty, Found guilty by the court, actual killer goes free, may kill other people”.

    Also, knowing nothing about the crime except what’s in this post, although drug addicts and embezzlers might not be murderers, they may have debts to people who are, and those people might come looking for them at their secluded homes.

    Another consideration for the jury: if there is no death penalty, they can take comfort knowing that, while a wrong decision might cost a few years behind bars, it can eventually be fixed if new evidence comes to light.

    • David:

      This relates to my point on the availability heuristic. There are lots of examples where there is no readily available alternative to a given hypothesis, but it turns out the hypothesis happens to be false, as the actual alternative was just something that people were not anticipating.

      I’m not saying that your argument regarding the lack of available alternatives is empty. I’m just saying it’s not always so easy to imagine compelling alternatives, but that doesn’t mean that such alternatives are not out there.

      • My point is that if you are doing a Bayesian analysis, you have to consider the alternatives. We don’t just test the null hypothesis. Jonathan Falk said his posterior probability is 0.7. So, where is the rest of his probability?

        • Let’s say 25% Friend of Mallory Beach (divided between 40 friends in some fashion, including hit man hired by one or more of them) 4% some other enemy of the Murdaugh family, 1% surprised armed intruder

          I’m not saying that any of this is the result of some explicit calculation. My real question is to philosophical Bayesians who regard Bayesianism as the *only* legtimate rational process of synthesizing information. What do *they* do in this circumstance?

        • If my posterior is 0.7, then I don’t vote “Guilty”. I haven’t looked at the details of the trial. But, I don’t find it hard to believe that you could get a posterior in the 0.95 range, and so vote “Guilty”. I’m not claiming that all jurors weigh the evidence in a reasonable way (I’ve been on a jury). But, you can analyze the evidence using probability.

        • Interesting thought experiment: what probability indicates “beyond a reasonable doubt?” Clearly you think 0.7 is not sufficient. I’d personally agree, but I suspect we are in a range where agreement will not be universal. I start thinking of the reactions to the 2016 Presidential Election and polls. Polls that put Clinton’s probability of winning at 80% were deemed to have failed by many people. So, it isn’t clear to me how varied those individual judgements would be. Since there are 12 jurors, I’d imagine that there are a variety of answers. I wonder if asking jurors to provide their probabilities of guilt in a case would help reach better decisions (my gut feeling is no, but that might make for interesting discussion).

    • As I say below, an obvious alternative is that they were killed by some friend of Mallory Beach, for whose death the son was soon to be tried. But there may well be others.

  2. “but where p is the prior probability of guilt, p<>.9)”

    I think that here Jonathan is saying “beyond reasonable doubt” twice somehow. To me, it looks like you should choose what “beyond reasonable doubt” means: a) it’s very unlikely he’s guilty (hence prior p <>0.9). In spirit, it looks like we should go with option b), but a) also captures the logic. Doing both seems a bit tangled to me.

    • I’m rewriting because it looks like I got re-interpreted by html tags:

      “Starting from a presumption of innocence (not sure what that means exactly in probabilistic terms, but where p is the prior probability of guilt, p≪.5) how do you as a Bayesian get from (1)-(4) combined with the prior to get to a posterior of “beyond reasonable doubt?” (Again, leaving precise calibration aside, surely p≫.9)”

      I think that here Jonathan is saying “beyond reasonable doubt” twice somehow. To me, it looks like you should choose what “beyond reasonable doubt” means: a) it’s very unlikely he’s guilty (hence prior p≪.5); or b) you have to be super sure that he’s guilty to convict (hence posterior p≫.9). In spirit, it looks like we should go with option b), but a) also captures the logic. Doing both seems a bit tangled to me.

  3. My initial reaction was that it was hard to believe they could convict him – and, even more, that it took the jury so little time to decide. Then, as I heard more details (I did not follow the case closely despite all the media attention), it became more reasonable to understand. Regardless of the evidence, I think it is wrong to cast this in terms of quantitative probabilities. The prior need not be p<>.9. There is a reason why the terms “not guilty” and “beyond a reasonable doubt” are not quantitatively defined. I suspect that a jury will vary on assigning numbers to what the terms mean. Some people might wish that the legal system be based on precise numbers for these, but it is not.

    The decision analysis approach should really not apply here – if it does, then I think the judge did not do their job instructing the jury. You (Andrew) frame it as “What should they decide?” While the jurors, being human, may think that way, isn’t the judge’s role to steer them away from that question and to the question of whether the state has met its burden of proof? Factors such as how bad a guy he is, should not be the basis for the decision, and the judge should be careful that the jury does not think that way (certainly, some will regardless of the instructions, but this is where needing a unanimous verdict of the jurors should protect against that).

    The evidence in the case involved a number of phone texts/videos and lies that he told about where he was that night – circumstantial but highly incriminating. The more I heard of the case, the more I could understand the verdict despite my initial surprise. The most puzzling aspect to me was the prosecution’s claim that the motive for the killings was to distract from his financial problems. That seemed like an odd motive for murders, but my understanding is that the real purpose was to get testimony and evidence admitted concerning his substantial financial problems, which it did. In the end, after reluctantly hearing some of the case details, I was not so surprised at the verdict, but still a little surprised it was reached so quickly.

  4. Alternative explanations play a key role in bayesian probability. This is a much more important distinction than the use of priors.

    We can have a very small unnormalized posterior (where G_M is Murdaugh guilty and E is evidence). The numerator of Bayes rule is then:

    p(G_M)*p(E|G_M) << 1

    The posterior depends on normalizing this to the sum of all the possible explanations, ie the denominator is:

    p(G_M)*p(E|G_M) + sum( p(G_0:n)*p(E|G_0:n) )

    Here G_0:n are all the other possible guilty parties 0 through n. If the second term is much less then the first (regardless of how small the first term may be) then it drops from the equation and the posterior is close to one:

    p(G_M|E) ~ p(G_M)*p(E|G_M)/ [p(G_M)*p(E|G_M)] ~ 1

    In a trial it is the job of the defence to come up with these alternative explanations, failure to do so would be a red flag to me as a juror.

    In science, it is the job of the authors themselves to write a section about this (sometimes peer reviewers add to the list). Once again, failure to discuss any obvious other explanations is a red flag.

  5. I think Falk does not give enough weight to his item 3. First, the spouse of a murder victim is much more often found to be the murderer than is a person selected at random from the population. This fact would go into the prior, raising it above <<.5; I don't know by how much is reasonable. (This is not specific evidence against Murdaugh in this case, which is why I would place it in the prior. Learning that he was the spouse simply tells us which prior to use.) Second, Falk casually dismisses Falk's lying about his whereabouts with "People lie all the time". They do; but not to the police for months on end about critical facts concerning a murder, relenting only in the face of incontrovertible proof presented Perry Mason style in the courtroom. At that point, jurors might reasonably ask, "Why did he engage in this long set of lies?" Murdaugh tried to present alternative theories as to why he lied, but the jury, by abduction, may have found the obvious explanation– he was the murderer– to be the best inference.

    • Gregory:

      Yes, good point. A related point is that it’s not clear how to map “presumption of innocence” into “prior probability of guilt.” To start with, there are 330 million people in the country and only one killer, so this gives a prior probability of 1 in 330 million (setting aside the possibilities such as the killer being from another country, there being two killers, etc.). Then you can add information such as that Murdaugh lives in the same state, that he knows the victims, etc., and the probability he is the killer goes up. “Presumption of innocence” has to map to the idea that you shouldn’t convict the guy without strong evidence, but it’s not clear where to draw the line between what is “prior” and what is “evidence.”

    • I came here to say what Mayer said. But I’ll give it some numbers.. About forty percent of married women who are murdered are killed by their spouse, so my prior probability of Murdaugh being the murderer, given no other information, wasn’t “much less than 0.5” as Falk says it should be wrongly in my view). Instead, it’s somewhere around 0.5. Given the other information it certainly goes up a lot. Falk suggests that somewhere around 0.7 is about as high as he thinks could be reasonable, whereas, given just what Falk says in his summary (which really is everything I know) I would say 0.7 seems unreasonably low!

      I don’t know whether I would have reached the “beyond a reasonable doubt” stage, which I might set quantitatively at something like 99.5%, I agree it seems hard to get way up there. But 90% or more, that doesn’t seem nutty to me at all.

  6. I think this is maybe too constricted:

    > People lie all the time…

    He was lying about facts directly related to the crime. Comparing to “people lying all the time” seems a mismatch to me. It should be compared to “people lying about facts germane to guilt or innocence of murder.”

  7. The Bureau of Justice Statistics report on family violence states “Wives were more likely than husbands to be killed by their spouse. Wives were about half of all spouses in the population in 2002 but 81% of all persons killed by their spouse.” So the prior probability in the Murdaugh case is not < 0.8. Analyzing a crime or a verdict by probability alone fails to take into account the human dimensions of crime.

    • Just because 81% of all persons killed by their spouses were women doesn’t mean that 80 percent (or anything like it) of murdered wives are murdered by their husbands. Maybe 5%?

        • Table 6 is indeed interesting. A few points: 24.1% of women were in fact killed by a spouse, but only 3% of males were killed by a parent. Furthermore, 46% of males weer killed by an acquaintance, suggesting that, on background knowledge alone, the revenge theory is actually favored over the parental theory (if you assume that Paul was the target and his mother was collateral damage)

  8. Applying this sort of a priori expectations to justice seems extremely problematic, as well as the idea that because the guy is shitty, then his right to presumed innocence is somehow reduced. This all leads itself very easily to applying horrible stereotypes and perpetuate injustices. This can easily justify easier guilty verdicts for members of marginalised populations who might be statistically overrepresented in crimes in big part *because* their marginalisation. And, of course, these weak convictions can be then used to further justify future convictions.

    • Agreed. I find this line of reasoning particularly disturbing. The job of the jury is to determine guilt/innocence for the particular crime in question, not whether the person was overall good or bad.

      >The assessed probability that he actually did the crime is relevant, but there’s not any kind of direct relation between the probability and the decision of how to vote in the jury. If you think the defendant is a bad enough guy, then you don’t really need to care so much about false positives.

      WTH, this is ridiculous! I bet this is exactly the kind of thinking that prosecutors are looking for during jury selection.

  9. – Truly not guilty, Found guilty by the court: Lying, creepy-ass, suspicious-acting drug addict and embezzler didn’t actually kill his wife and kid—at least, not quite in the way as was charged. But he goes away for life anyway. No great loss to society here.

    The point is that the circumstances of the crime and the jury’s general impression of the defendant are relevant to the decision. The assessed probability that he actually did the crime is relevant, but there’s not any kind of direct relation between the probability and the decision of how to vote in the jury. If you think the defendant is a bad enough guy, then you don’t really need to care so much about false positives.

    Taking this as a normative statement of values rather than a description of others’ values, I think this is a pretty terrible thing to say.

    • Somebody:

      It’s not intended to be a normative statement of values, nor is it intended to be a description. It’s more of a reconstruction of a possible line of reasoning that is consistent with the scenario described by Falk. The original question was, How could the jurors so quickly vote to convict given that the evidence did not seem enough to take the probability to the “beyond reasonable doubt” territory. My response was that the jurors could be considering consequences, or doing an informal decision analysis, given all the factors available.

      • I see now that you’re talking about potential lines of reasoning for the jurors. But I think this phrasing makes it seem like you’re arguing that this is not a fundamentally flawed decision analysis

        Decision analysis. Falk is framing the problem in probabilistic terms: what sort of evidence would it take to shift the probability from much less than 50% to well over 90%, and I see his point that some strong evidence would be necessary, maybe much more than was presented at the trial. But I’m thinking the more relevant framing for the jury is: What should they decide?

        Again, I get that this isn’t actually your thinking. But I’m going to argue with it anyways because I think it’s important

        1. The judicial system does not punish people for being a bad person. Law is not morality. Law is an insufficient substitute for the peoples’ moral compasses and judgements. A society that makes a habit of giving people life in prison for being unpleasant is not a society that the jurors or anyone else want to live in.
        2. The judicial system tries people for specific crimes. If he did a different crime, he should be charged with that other crime and his guilt for that crime should be evaluated. I think white collar crimes are serious and perhaps are not taken seriously enough, but embezzling -> murder would be a massive sentence inflation. A society where you can tack on a specious murder charge to give people life in prison for lesser crimes is not a society that the jurors or anyone else want to live in.
        3. The way it’s stated here, it looks like being a “drug addict” is its own crime in the same way as being an embezzler and a murderer and is an auxiliary case for punishment. I don’t think being a drug addict is a good thing, but there’s a categorical difference in that the primary victim of a drug addict is themself and I don’t see what good it does anyone to have state enforced punishment.
        4. There are a few losses to society — the cost of keeping a person in prison, the investigation stopping and the actual murderer getting away with it to potentially do more crimes, the moral cost of a too-harsh punishment. Murdaugh happens to not be eligible for parole, but were that not the sentence, a non-violent criminal is almost certainly more dangerous to society coming out of prison than going in.

        I think this moralistic reasoning is pretty common and pretty damaging, which is why it bothers me so much.

        • Somebody:

          In principle I agree with all of your points, but I’m not so sure in practice.

          Suppose “Barb” is one of several frequent offenders in your neighborhood. She has robbed several people at gunpoint but no one wants to report the crimes because she and her friends are known for retaliating. One day Barb goes to party and someone is murdered at the party. Barb was not involved with the crime, but because her reputation is well known and some evidence hints she might be guilty, she is arrested, tried, convicted, and sentenced.

          Barb has been wrongfully convicted for for murder but she is a dangerous criminal and one the community justifiably fears. On a scale of 1 (worst) to 10 (best), how well has justice been done? How would you feel if you lived in the community where Barb operates if she was released back on to the streets to rob people at gun point? In a perfect world Barb should not be convicted for a crime she did not commit. But in a perfect world Barb should have been convicted for the crimes she did commit. Which should take precedence?

          Personally I’d rate Barb’s murder conviction as a 7 on the justice scale. Some form of justice has been done, imperfect though it may be. It is better for everyone that she is convicted of the wrong crime rather than no crime because she *is* a dangerous criminal. The obvious risk is that someone similar to Barb could be convicted when they haven’t committed **any** (or any serious) crime. But in places where crime is high, it’s understandable why people would have greater tolerance for that risk, isn’t it? I think so.

        • Well, first of all, I don’t think this example really resembles the case under discussion at all.

          But in principle, yeah, it’d feel bad to release Barb in your hypothetical, but it feels that way because you’ve constructed a hypothetical where the omniscient and impotent reader knows what Barb has been doing but cannot put her in jail for it. The dramatic irony is frustrating to a reader, but in real life nobody is omniscient and nobody knows the facts; rather, everybody has uncertainty given their perspective and can take actions on that basis.

          Stepping into the perspective of a juror who lives in the area, I don’t really know why I would vote to convict Barb on specious evidence for murder when I could just report that she’s been robbing people. Yeah, she and her friends might be able to intimidate their way out of the robbery charges–but surely then they’d be able to intimidate their way out of the murder charges as well? I just don’t see a world where everyone in the community has at many stages of the justice system decided that getting her on specious charges for murder is easier than getting her on strong charges for robbery.

          That said, I know situations arise where we all “know” that Al Capone is a mob boss for years. But you know, we got him in the end and took down the rule of the Italian mobs in the end. Sure, I can empathize with the desire to ignore the law to punish the bad people that I just “know” are bad, but I don’t think it actually works.

          1. People are at a higher risk of being imprisoned or punished, which is itself a threat to public safety
          2. The law can be abused to threaten people. It’s actually extremely easy for a small, coordinated group to spread rumors in a tight-knit community and give any given person a bad reputation.
          3. It becomes difficult to punish actual perpetrators, given the incentive structures of the police and the DA
          4. People become mistrustful of law enforcement, don’t report crimes, and unrest develops

          It’s easy to think that due process is like a treat we can have in good times, but when the going gets tough, we need to get hardboiled loose cannons to do harsh justice for the right reasons. But Rodrigo Duterte effectively legalizing the murder of suspected drug dealers in the Philippines has not improved public safety or fixed drug addiction; it’s just led to an absurd number of people being killed for specious reasons.

        • somebody –

          Seems to me you might be discussing something akin to jury nullification which, I think, reflects risks running in opposing directions – with fugitive slave laws being a noted historic context.

  10. Thanks to all for your comments.

    First, I agree wholeheartedly with somebody, Elio and jd. My biggest concern about this case was that he was indeed convicted very quickly largely because he’s a horrible person, and I find that extremely disturbing.

    I don’t think people (and this includes jurors) are committed Bayesians, but my original question to Andrew was what should a committed Bayesian do? This gets around the quantification questions in the sense that committed Bayesians are committed to expressing things probabilistically.

    Even if 25% of all murders of wives are committed by their husbands (and this is way too high) that evidence cannot be presented in court (as it obviously prejudicial) so it would have to be incorporated into the prior. But even if you do, that suggests that the prior probability of guilt is indeed far under 50%, so I don’t think my base rate is too low. I think it is fair to say that if the prosecution had presented no additional admissible evidence at all that there is no way to convict Murdaugh. They did present substantial additional evidence, the most critical of which was exploding Murdaugh’s alibi. Again, I have no problem having that fact alone take the posterior over 70%. And I agree that the mere fact that you think an alibi and therefore create a stupid lie about it has *some* evidential value.

    I wrote my email to Andrew before I knew very much more about the trial than what I wrote him. I agree with Dale Lehman that further research revealed some more facts that indeed increase the posterior probability, but the alternatives still trouble me. And there are alternatives that would need to be quantified, including a surprised thief or a different family member or even someone holding a grudge over the death of Mallory Beach who Paul Murdaugh killed and for which his mother’s death was collateral damage. These all get priors as well, right? How many revenge murders are committed? Lots. If I told you that Paul negligently killed Mallory and Paul was murdered while awaiting trial wouldn’t there be an extremely high prior that Paul’s father *wasn’t* guilty? Those facts cited above about spousal murder are not conditioned on the death of an accused murderer at the same time.

    Overall, the comments here seem focused on the defendant’s probability and not the probability of alternatives. It’s not for nothing that this is called The Prosecutor’s Fallacy,

    • And there are alternatives that would need to be quantified, including a surprised thief or a different family member or even someone holding a grudge over the death of Mallory Beach who Paul Murdaugh killed and for which his mother’s death was collateral damage. These all get priors as well, right?

      I know nothing about the case, but why didn’t the defense bring these up? Or did they?

      • I didn’t follow it either, but Paul’s trial for killing Mallory Beach was brought up several times by the *prosecution* who argued (somewhat nonsensically IMO, but I wasn’t there) that financial pressure on Alex Murdaugh from the upcoming civil suit over the death of Mallory Beach was an important part of the motive.

    • “Even if 25% of all murders of wives are committed by their husbands (and this is way too high)”… no it isn’t. The actual number is about 40%. If all you know is that a married woman has been murdered, your prior probability that the husband is guilty should be around 40%.

    • Jonathan oa –

      > he was indeed convicted very quickly largely because he’s a horrible person, and I find that extremely disturbing.

      That seems excessive to me. Why do you state that so categorically?

      • Also, maybe “he was a horrible person” needs to be unpacked. It could include meaning a person who displayed behaviors that would increase the probability that he might commit murder, it could include proof that he’s a flat out liar, and proof he lied for 20 months about facts material to the case – his alibi (something with the prosecution keyed in on).

        Are those aspects separable from thinking he’s a horrible person?

    • > he was indeed convicted very quickly

      Are you referring to the time that the jury spent deliberating? I don’t know why juries take so long. When I was on a jury, it was clear when the testimony ended what the verdict should be. The jury I was on didn’t deliberate that long, but the time we did take was spent in explaining things to the few jurors who were confused.

  11. “how do you as a Bayesian get from (1)-(4) combined with the prior to get to a posterior of “beyond reasonable doubt?” (Again, leaving precise calibration aside, surely p>>.9)”

    The real issue here is how to quantify the significance of circumstantial evidence. And the short answer is that you cannot do it. For some folks – not pointing a finger at Jonathan Falk, all he did was ask an interesting question – the inability to quantify leads to an inability to process.

    I first noticed this when Tom Brady was accused of using deliberately deflated footballs. The NFL botched the locker room sting operation, resulting in zero deflation being within the confidence interval. But it never mattered because the NFL had text messages in which the guy who handled the balls before the games was jokingly referred to as “the deflator.” Game over. When Brady’s defenders were subsequently confronted with this circumstantial evidence that made their arguments pointless, nothing but crickets.

    It’s the same with those who argue that Covid had a natural origin. Maybe it did. But these folks seem to be unable to handle the fact that while we do not know exactly where the pandemic started or who was Patient Zero, we do know that it originated within three miles of an institute of virology that specialized in bat coronaviruses and dabbled in gain-of-function research. The director was nicknamed “the bat lady.” The last thing she was allowed to say in public was “could it have come from our lab?” A large segment of the population seems to be simply incapable of understanding just how huge the burden of proof is for those who argue for a natural origin. If they mention the institute at all, it is to ridicule those who favor a lab leak hypothesis, but mostly they ignore it. The only way the burden of proof could be met is if the Chinese government completely opened up about everything and made Dr. Shi available for interviews with investigators. And that will never happen because it would almost certainly provide evidence of a lab leak.

    I think the jury did a great job in not getting distracted by the lack of physical evidence when the circumstantial evidence was plenty strong.

  12. Isn’t it just that lying about being present at the scene of a murder is an incredibly risky thing to do? To the point where an intelligent person would only go down that road if their other options were really really bad?

    So, p(X has something devastating to hide about a murder | X lied repeatedly about being there) ~= 1, and
    p(X was the murderer | X has something devastating to hide about a murder) ~= 1.

    • I don’t agree with your final line. It’s not hard to imagine a genuinely innocent person in the same situation choosing to lie if they felt the truth may lead to them getting wrongly convicted. You say it’s risky to lie, and maybe it is, but when caught in the moment of a situation like this you may feel it’s riskier to your freedom to tell the truth – especially if you lack faith in the police and/or justice system. I know I would, at the very least, be reluctant to hand over information that may make me look very bad in a murder case that I didn’t cause.

      The discovery of a lie in witness testimony may lead to useful evidence, but the choice to lie in and of itself can’t be reliable evidence one way or the other in my opinion.

  13. While using ‘hard science’ probabilities might make abdicating judgement to a decision rule appealing to some, I see a large number of problems with it:
    1. The numbers are incomplete. A lot more crimes are happening than we know (dark field), and of the cases we know about, a lot of the perpetrators are not found. Even if they are found, people are convicted falsely. Just ask the Innocence Project.
    2. The data on crime for the US is full of holes as there is no federal obligation for each of the many thousand law enforcement agencies in the US to report their data to the FBI. We surely have selection bias in the data.
    3. Cross-effects of crime are difficult if not impossible to estimate. Say a person steals a loaf of bread (like in Les Misérable), does this change the probability that a person has murdered someone? Moinseur Thénardier and Javert appeared to believe so, but the whole point of this marvelous book is to point out that they are wrong.
    4. By extension of three, often smaller charges are dropped in a big case because the burden of proof is so high and there is nothing to be gained by adding those smaller charges. Imagine someone trying Pablo Escobar because he had hyppos on his ranch… surely that wasn’t legal, but nobody cared, because the people wanted him for stuff like blowing up an airplane with scores of dead people.
    5. It’s impossible to have a probability assigned to every possible outcome in the world. Therefore, we have to ignore some information as we cannot assign a probability to it which we can use to update our belief on whether a crime occurred or not.
    6. Let’s assume we had correct data and we could estimate cross-crime effects sufficiently well, we face another dilemma: You cannot convict someone on probabilities. There is either guilty, or not. We would need a decision rule with a probability threshold, e.g. 90% upward and you convict. All people below automatically go free, all above get convicted so come what may. Even if our numbers are correct, 10% of the convicted folks are by pure math innocent! On the other side, we have criminals without prior convictions likely just go free, because of fewer ‘misdeeds’ that raise the probability. A remedy would be to multiply the probability with the penalty (e.g. 80%×20 years= 16 years).
    7. If a transparent decision rule was made for a person to be convicted, criminals would start to exploit it, leading to bias in the data. As horrible as it sounds, the intransparency of the justice system increases the perceived cost of committing a crime (using the crude expected utility theory from microeconomics), disincentivising people to commit a crime (if they are even logically thinking, see the next point).
    8. We often assume crime is done by bad people. It is not. It’s just done by people. Often, they act in affect. Something sends them off the rails and they go and do something regrettable. A probabilistic model could not distinguish between a stone-cold criminal and one that just acted ‘in the spirit of the moment.’ The probabilities for crimes in affect and premeditated crimes are sure very different!
    9. For ‘simple cases’ where it’s clear that the perpetrator is guilty we would still need to do a lot of math just to say, “Well, we are pretty sure it was you who punched him in the face, after a dozen people saw you doing it…”

    It just doesn’t add up to quantify crime in terms of probability. The often unjust justice system we have does at least have a shot at getting it right all the time (even though we know it doesn’t). I think that’s worth improving upon, and statistics might help here or there, but I do not see how a mathematical system could be a better system by default.

  14. As for the decision theory approach, can I take a moment to say AAAAAAAAAAAAAAAAAAAAAAAA!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

    Seeing as the other accusations *haven’t gone to court yet*, this approach means that a prosecutor wishing to lock someone up for life need only toss a dozen accusations at the wall and get one of them to stick at 75%.

    And, given the fuzziness, the accusations don’t need to be personal. Why is it ok to lock up a drug addict for a crime he didn’t commit? Because drug addicts routinely engage in antisocial behaviors such as aggressive begging, blocking public spaces, and even outright robbery. Did this man do any of those things? Who cares? He’s a drug addict.

    It’s a small step to “Who cares what he’s done? He’s a [insert ethnic or religious minority here].”

  15. According to the Legal Eagle video, there was other evidence. No single slam dunk, but (for example), Murdaugh “lost” a gun matching the murder weapon. The probability of a mostly-responsible gun owner losing a weapon on a given day is quite low. The probability that a murderer familiar with forensics would dispose of a weapon untraceably is high. There are other hypotheses worth considering, but this is still a significant update, and there’s more like this.

    • Well, he owned dozens of weapons, so losing some isn’t that unlikely. Plus, there were tow different murder weapons, and ther wasn’t any evidence he even owned the second one. (And why two weapons?)

      I realize I’m starting to sound like a Murdaugh apologist here, but I’m pretty confident he’s guilty! I just have doubts that don’t strike me as unreasonable…

      I’m pretty sure I would have been hated pretty quickly if I’d been on the jury, assuming there wsn’t some other piece of underreported evidence that I don’t know about that would have influenced me.

  16. It is/was a very tough call. If you convict an innocent, the guilty goes free, but practically speaking, if the police consider it 70% likely that the accused was guilty, they will not put much if any more effort into the case if the jury votes Not Guilty due to reasonable doubt. (Based on my extensive reading of the excellent Michael Connelly crime novels.) (There is no point looking for more evidence against the accused after the trial due to Double Jeopardy.) So add to your decision tree that the trial is probably the only chance of holding anyone responsible, to deter similar crimes. (That is not what the judge will tell them, but what would be in the back of my mind, if I thought the accused was probably guilty.)

    As to deliberation time, each juror spends the whole trial deliberating. If they find themselves in agreement on the first vote they have the choice of stalling to make things look better, or going back to their lives immediately.

    I wonder if a reliable lie-detector could be developed, using MRI scans during questioning. Then I realize how Trump would use that to insure loyalty.

  17. “how do you as a Bayesian get from (1)-(4) combined with the prior to get to a posterior of “beyond reasonable doubt?”

    Here’s how. By looking through the priors and focusing on the uncontested brute fact that Murdaugh’s voice was recorded on his son Paul’s cell phone between 1 and 2 minutes before Murdaugh murdered him.

      • Am I understanding this? The accused was at the property when his wife and son were killed, and is known to have been within earshot of the son just minutes before?

        What would it take for you guys to convict someone? Video of them committing the crime? Serious question.

        • > The accused was at the property when his wife and son were killed.

          And lied about his whereabouts for 20 months, until video evidence proved he was lying after which he said he was lying to cover for being an addict.

        • Serious answer: (a) a lack of plausible alternatives; and/or (b) some conceivable motive. Of course, what’s plausible is to some extent subjective here, but let me turn the question around.

          A rich spoiled brat kills a beloved member of the community in a drunken boating accident, points fingers of blame at others and awaits trial. What would it take for you to ignore the possibility that one of them decided to short-circuit justice?

          What we have here is a guy with opportunity and no plausible motive combined with a large set of people with motive and *unexplored* opportunity. (The fact that the police failed to explore this issue is a central part of the defense case, such as it was.) Now if you convinced me that there was no legitimate opportunity for others, I’d concede that your statement is probably enough to convict.

        • J(ao),
          Maybe it comes down to what one considers a “plausible alternative.” As Andrew has pointed out, extreme skepticism can be a form of extreme credulity. To me, it strains the definition of “plausible” to suggest that someone else shot Murdaugh’s wife and son — with a shotgun and a rifle, neither of them a quiet weapon — while he was nearby without him noticing, just minutes before he left the property. Is it _possible_ that some person or people parked at a distance, walked to the property, waited until Murdaugh had stepped away from the location of his wife and son (but not until he had left the property), shot them both without Murdaugh hearing, grabbed the wife’s cell phone and ran with it to the edge of the road where Murdaugh would drive past a couple of minutes later and tossed it there, then hid while Murdaugh drove by? Well, sure. That sort of thing happens in the movies all the time! But it seems very very unlikely. How unlikely, though? 1 in 100? 1 in 1000? 1 in 10,000? It’s hard to put a number on it. I’m not going to say for sure that I would have voted to convict — until I read this blog post and the comments I didn’t know anything at all about the case, and now I only know what I’ve learned here, and by reading two news articles — but at least I understand how a jury could think there is not a “reasonable doubt.”

          As for the lack of a “conceivable” motive…this word, I do not think it means what you think it means. For crying out loud, Johnny Cash shot a man in Reno just to watch him die!

        • Well, sure. That sort of thing happens in the movies all the time! But it seems very very unlikely. How unlikely, though? 1 in 100? 1 in 1000? 1 in 10,000? It’s hard to put a number on it.

          The problem isn’t really that such explanations are unlikely a priori, it is that they are vague. You could explain any kind of evidence with that type of theory, ie the likelihood P(evidence|there_was_a_setup) is small because it can equally well explain pretty much any set of evidence.

          If you can derive a specific prediction to be checked, eg find tire tracks from other cars, that is much more precise. Then it becomes a key question of whether anyone looked for such evidence. Failure to collect evidence that may be inconvenient is a common strategy used to mislead.

          It is all about populating and evaluating the likelihoods, the priors are a minor step that weight the various possibilities.

  18. When I think of how priors influence a jury in a high profile murder case, I think of the O.J. case.

    Why was there such a strong racial signal in public opinions about the jury’s decision? I think priors.

    For many white people, it seemed implausible that the police, even with a racist cop involved, could have rigged so much evidence.

    For many black people, that was quite plausible.

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