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In UK civil trials, it is said that

  1. The burden of proof is on the claimant.
  2. The case will be decided ‘on the balance of probabilities’.

Are these assertions consistent? Based on the second statement, one might think that the outcome of a case will depend on whether it is more likely than not (>50%) or less likely than not (<50%) that the claimant’s allegation is correct (based on the relevant evidence). But if that is true, I don’t see how the claimant has any special ‘burden’ that they need to discharge. Sure, in order for the claimant to win, the relevant decision maker (let’s say it is a judge) needs to think that it is more likely than not that the claimant is right. But equally, in order for the defendant to win, the judge needs to think that it is more likely than not that they (the defendant) are right. So the situation is entirely symmetric!

Update: as some have commented, the claimant may lose if the judge thinks that the probability is exactly 50%. If you think, for example, that the judge’s assessments should be restricted to integer percentages (0%, 1%, 2%, etc.), this means that the claimant wins if the probability is 51% or higher, and loses if the probability is 50% or lower. This situation is very slightly asymmetric; but only very slightly! And if the judge can make more nuanced judgments (no longer restricted to integers), the asymmetry shrinks.

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  • 1
    If you say "you damaged my car", and I say "no, I didn't", and none of us shows proof, then because you have the burden of proof it's 100% probability that I didn't do it - as far as the court is concerned.
    – gnasher729
    May 6 at 9:27
  • @gnasher729 Can you explain why the court would be 100% sure (or pretend that they are 100% sure?) that you didn't damage the car in this situation? I am not saying that you are wrong, but this strikes me as a highly irrational way of forming probabilistic judgements. [For one thing, if the court is Bayesian, and assigns probability 1 to this event, then they can never change their mind! This is the problem with "dogmatic priors".]
    – afreelunch
    May 6 at 9:54
  • “But equally, in order for the defendant to win, the judge needs to think that it is more likely than not that they (the defendant) are right.” The complement of x > 0.5 in the range [0,1] is x<=0.5, not 1 - x > 0.5
    – xngtng
    May 6 at 13:40
  • @afreelunch The court would not be 100 percent sure he didn't do it. It would just find that you hadn't carried your burden, without making any probablistic judgement at all. See my answer below for more elaboration on how the courts address this question.
    – bdb484
    May 6 at 14:58
  • Here is a fun fact in the uk there is a 3rd outcome of a trail not proven.
    – Neil Meyer
    May 6 at 18:56

3 Answers 3

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First of all, law is not mathematics and it is futile to try to assign an "exact" statistical percentage or apply Bayes' theorem in any specific case unless the facts of the case are particular enough. In the end, judges or juries (most often) do not apply a statistical model to their cases, what matters is if the evidences (by both parties) presented on one claim convince them that the claim is more likely to be true than not, that is, what they "feel" or "think" about the probability.

More probable than not means > 50%; if the fact finder finds scenarios leading to different outcomes equally likely, the claim is not proven and the party bearing the burden of proof fails.

If you want to analyze it mathematically, the burden of proof does not necessarily means that the probability of a claim favourable to the burdened party to be true in the absence of evidence is zero. Since only two values are possible as an outcome on any particular claim (as far as the burden of proof is concerned), a step function must be used to convert the probability to a judgement. The burden of proof represents the choice of that unsymmetric step function.

The burden of proof also represents the legal (not necessarily "factual" or statistical) assumption (an imposed, unquantifiable prior "probability").

Here I use the word "claim" generally, for a case to succeed there can be multiple elements or claims that all need to be proven to be more likely than not to be true, e.g. if you want to sue someone for medical bills because they crashed into you and you have a broken leg, you need to prove both that they were responsible for the crash and that the medical bills are for the broken leg, and the broken leg is a result of that crash.

The burden of proof means that if you do not provide any evidence that the leg was broken as a result of that crash, or if your evidence is very weak, or if your evidence is not stronger than the other party's counter-evidence, your case must fail; the defendant does not have to prove that the crash did not cause your broken feet.

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  • Where does your step function jump? (At p = 0.5? You say it should be asymmetric…)
    – afreelunch
    May 8 at 10:53
  • @afreelunch Define p. But you seem to ignore the first paragraph of the answer. Law is not mathematics and if you want to use math to model legal theories you will also have to accept legal assumption and logics as axioms.
    – xngtng
    May 8 at 11:08
  • As a consequence, any use of math must accept that the trier of fact is not bound to conduct statistical reasoning (if such thing is even possible), and in the case of jury, is not required to explain their reasoning at all. In this case, if you define p as whatever value the trier of fact assigns to the probability that the plaintiff's claims are true, then yes, p = 0.5, or rather p is of a value such that the finder of fact decides the plaintiff's claims based on the evidence are equally likely to be true or false.
    – xngtng
    May 8 at 11:14
  • p is the probability the judge assigns to the claimant’s being correct
    – afreelunch
    May 8 at 11:32
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These statements are consistent, because you are ignoring a third possibility: a case where the evidence is perfectly balanced, i.e., the evidence makes it neither more nor less than 50 percent likely that the claimant's allegation is true.

So imagine Plaintiff sues Defendant for trespass and presents only one piece of evidence: a hair left behind by the trespasser and matching Defendant's DNA. Defendant stipulates that someone trespassed but presents only one piece of evidence: his identical twin's DNA, which is likewise a perfect match to the sample.

It is now 50 percent likely that Defendant trespassed, and 50 percent likely that Defendant did not trespass. The parties have brought equal amounts of evidence and balanced the scales perfectly. Defendant wins because Plaintiff did not carry his burden, which was greater than Defendant's.

The difference can be seen more clearly in a scenario that is even less likely. Imagine a trial in which neither side presents any evidence at all. Again, the scales are perfectly balanced, but Defendant wins because only Plaintiff had a duty to present any evidence at all.

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  • The latter case is much more likely than the former. It happens all the time for various reasons (e.g. when a plaintiff fails to appear and a defendant choses not to present any evidence or put on a case as a result). I've never seen a case involving identical twins based solely on a DNA test ever.
    – ohwilleke
    May 7 at 6:26
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    Yes, I was thinking about that afterward. I guess I was thinking about it as a deliberate trial strategy, which I hope no one would actually choose.
    – bdb484
    May 7 at 20:16
  • But how likely is it that the evidence is exactly balanced? This should be the case hardly at all, if ever. (See my previous comment for technical elaboration.)
    – afreelunch
    May 8 at 10:56
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    As @ohwilleke noted, it is a meaningfully common occurence, although I sloppily suggested otherwise. I doubt it's a very large share of cases, but it frequently happens that a plaintiff files a case and then stops prosecuting the case, or fails to present any admissible evidence, or is barred from presenting evidence for procedural reasons. In any of these cases, there's no reason for the defendant to present any evidence, and so the evidence is perfectly balanced at 0g apiece. Because it was the plaintiff's job to tip the scales, the defendant wins.
    – bdb484
    May 8 at 14:15
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Personally, I think the concepts refuting and proving is not all that accurate in a law setting.

You only really have to 'prove' your case is more probable than the.opposition. This is not proving in the same way as what the word is used in academic debates.

In law it is acceptable for a guilty man to go free if there was insufficient evidence to convict him. This is ultimately why a verdict of not guilty is given not innocent.

Not guilty means that there was insufficient evidence to convict not that the person did not do it. It is an important distinction to make.

The burden of 'proof' lies on both parties to 'prove' there case beyond a reasonable doubt.

Still proving a case is different than proving a science thoery.

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