The Black Swan: Second Edition: The Impact of the Highly Improbable: With a New Section: On Robustness and Fragility: 2 (Incerto)

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In the Venray murder case, the Dutch Supreme Court determined on what grounds courts may reject the alternative explanations offered by defendants and when they should justify their decision to do so. In this case comment I offered an interpretation employing Bayesian probability theory.

So, the Supreme Court’s ruling is about how courts should deal with cases in which the evidence does not ‘refute’ a defendant’s story, but they still wish to reject this alternative explanation. Before moving on to my interpretation of this ruling, I want to discuss both situations in which the court can point to evidence that refutes the defendant’s story and situations in which the court can reject this explanation though no refuting evidence exists. I analyse both situations using a Bayesian framework. 3. Rejecting stories with and without evidence Suppose that the defendant’s story is weak and that the prosecution’s case is strong. This means that if no further evidence or arguments were adduced, the defendant would most likely lose the case and be found guilty beyond a reasonable doubt. So, if the defendant tells a story that is initially improbable, he risks losing the case if no new evidence confirms his story. The defendant may then have a burden to introduce new arguments or evidence that would make the court decide in his favor or he risks losing the case. 9 These benefits of justification also occur when courts justify why they reject an alternative explanation. In such cases the justification gives both the court and the audience insight into why that explanation is improbable enough not to create a reasonable doubt. However, there are cases in which this kind of insight is not required. In particular, some stories that defendants tell are so obviously improbable that we would gain little by arguing against them. For example, take a (real) case in which the defendant pleaded that he was not accountable for the child porn on his computer because his mind was controlled by aliens. 16 It seems fair to say that no reasonable audience would consider the ‘alien’ explanation remotely probable. Furthermore, a defendant who offers such an explanation would either be delusional or insincere. So, it is improbable that arguments would sway him. Hence, the court would (most likely) gain little by justifying why it rejects this alternative explanation, with respect to the parties, legal community and general audience’s understanding of it. According to the Supreme Court, courts can reject an explanation if it ‘did not become plausible’ during the criminal proceedings. 8 The obvious question when interpreting this statement is why some explanations need to ‘become plausible’. The answer to this lies in the proof standard. As mentioned, in the Netherlands the proof standard states that the court should be convinced of the defendant’s guilt based on the admissible evidence. However, in practice, many legal scholars believe that the standard is actually similar to that of common law countries—that guilt has to be proven beyond a reasonable doubt ( Ter Haar & Meijer, 2018, 7.4; Nijboer et al., 2017, pp. 73–74). So, if the defendant hopes to be acquitted by telling an alternative story, that story needs to be good enough to create a reasonable doubt about his guilt (assuming that the prosecution’s case is in itself strong enough).It will surely never happen again – not just the case of a team in such despair as last season reviving themselves to challenge, but bulldozing past not one or two but six footballing giants and beyond to live out the most sensational of storylines. Under the manager who never should have been appointed, they said, Leicester would be relegated. 5,000/1, the bookmakers said. But they can never, ever take this stupid, staggering achievement away – and that’s the greatest feeling of all. P ( H 1 | E ) = P ( E | H 1 ) × P ( H 1 ) P ( H 2 | E ) = P ( E | H 2 ) × P ( H 2 ) Here H 1 and H 2 represent the hypotheses that either one or the other explanation is true. In this version of the formula, whether the evidence skews the prior ratio in favor of guilt or innocence depends on the ‘likelihood ratio’, P(E|H 1)/P(E|H 2). When the likelihood ratio is higher than 1 it means that the evidence raises the probability of H 1 whereas a likelihood ratio lower than 1 means that the probability of H 2 is raised. There have been plenty of other heroes along the way – head of recruitment Steve Walsh has crafted an outstanding scouting network so essential to the building of this history-making squad; fans’ favourite midfielder Andy King, meanwhile, has become the first player to win the top three divisions with the same club, and now prepares for a final-day trip to Stamford Bridge – the ground on which he used to be a ball boy in Ranieri’s Chelsea days.

Next season the bigger clubs can reclaim their places atop the pile, and even make their plays at trying to buy Leicester’s best players. They might just succeed. What does an explanation with a low prior probability look like? First, it may have parts that do not fit well together. For instance, the explanation might imply that the defendant was in two places at the same time. Alternatively, the defendant may tell a story in which motive and action do not fit well together, such as a story about a robbery where nothing was stolen. Finally, the explanation may consist of a number of independent and individually unlikely events ( Lettinga, 2015, p. 53; Josephson, 2000). Second, an explanation can also have a low prior probability because it does not fit well with our generalizations about how the world typically works. For instance, we may believe that innocent bystanders do not run away from the police, that a suspect cannot cross the city in 10 min or that the police rarely forges evidence. The more an explanation violates such generalizations, the lower its prior probability is. 5. Incredible explanations are told by an unreliable storyteller Just don’t call it a fairy tale. If the phrase has proved a worn cliché over the last six months anyway, now it’s simply not true. The Collins Dictionary defines ‘fairy tale’ as being “a highly improbable account”– but this has happened. This is very real.When a court considers an explanation to be implausible or incredible it must generally justify why it does not believe the defendants explanation before convicting him. However, according to the Supreme Court, some explanations are so ‘highly improbable’ that courts do not have a duty to respond to them. 15

That ‘highly improbable’ should be interpreted as ‘obviously improbable’ is also something that has implicitly been noted by Dutch courts. For example, the Dutch Supreme Court once overturned a decision by a lower court because it had failed to give a justification for its decision to reject the defendant’s alternative scenario. 17 The supreme court argued that even if the lower court thinks that a defendant’s alternative scenario is improbable, it will sometimes have to offer a justification for this conclusion, because not every improbability is ‘evident’. When reasoning about which story to accept, rejecting one story and accepting the other often means finding ‘discriminating evidence’, i.e. evidence that fits better with one story than another ( Van Koppen, 2011, pp. 52–55). In Bayesian terms this means evidence where the likelihood ratio strongly favors one story over the other. Such evidence discriminates between the two explanations because we would expect the evidence much more if one explanation were true than if the other was. If the likelihood ratio is sufficiently much greater than 1, the probability of one explanation will be high and the probability that the alternative explanation is true will be low. In such a cases the court can point to the discriminating evidence as a reason why it rejects the alternative explanation.

All Solutions for IMPROBABLE

They know that much of their success this campaign has been down to other sides failing miserably, and that next season offers them all a shot at redemption; Manchester City and Chelsea with incoming managers, Manchester United too if Jose Mourinho gets his way, Arsenal in what might be Arsene Wenger’s final season at the club, and even Liverpool where Jurgen Klopp now has a full summer of fine-tuning ahead. Mauricio Pochettino’s Tottenham, who ran them so close, are built to last. Feels, for the first time, like we're genuinely moving into 'big game' territory here. I don't like it. #lcfc All this is supposed to be satirical but it has little or no resemblance to life as we know it, without which satire is impossible. Just as I was trying hard to become interested in the advertised state-of-the-nation-satirical-political novel, it suddenly turned thriller, and pretty daft thriller at that: the prime minister’s senior private secretary is murdered because he threatens to blow the whistle; two other people from the political classes are hit on the head, one with a wrench and the other, if I remember clearly, with the butt of a gun. The young woman who was hit on the head with a wrench, Jen Lewis, is the daughter of Myfanwy Davies-Jones, who is, I think, supposed to evoke Molly Parkin. Various grandees such as historian Lord Briskett and the mysterious gay fixer and strategist Alois Haydn pop up. Haydn, who is bankrupt, decides that, although he is the chief adviser of the yes camp and the architect of the conspiracy, he will leak what has happened to the opposition and at the same time short Britain, making himself many millions when the markets crash. So this becomes a financial fraud novel as well. Haydn’s Indian partner leaves him in disgust, perhaps to demonstrate that there is some decency and sense in this world. Vardy broke a Premier League goalscoring record, Mahrez won the PFA Player of the Year award and next season they have the sweet sensation of Champions League football heading to the East Midlands – as England’s Pot 1 representatives, no less. But they know things won’t be like this again.