What is the probability that the Bayesians died by chance?

Written by Martin Neil and Norman Fenton on September 2, 2024. Posted in Current News

Mike Lynch was the founder and CEO of the UK Cambridge based company Autonomy. It’s a company we know well because their flagship software is rooted in Bayesian probability

In 2011 Autonomy was sold to Hewlett Packard for $11 billion. As Mark Steyn reports, in 2018 the US Department of Justice indicted Lynch and Stephen Chamberlain (who was Lynch’s partner and financial head of Autonomy) for “conspiracy” and “fraud” over the company’s sale to Hewlett Packard.

Lynch and Chamberlain were extradited to the USA. Finally, after a long legal battle on 6 June 2024 a San Francisco jury found Lynch and Chamberlain not guilty of all charges.

As Steyn reports, this in itself was quite a rare event, since 95 percent of such trials in the USA that go to court end in a guilty verdict.

The unusual deaths of Lynch and Chamberlain and the inevitable question of whether they could have happened by chance alone

Less than three months after their acquittal and return to the UK, both Lynch and Chamberlain were killed in separate freak accidents. On 17 August Chamberlain was hit by a car in Cambridge while out on a morning run and was declared dead on 19 August 2024.

On the very same day Lynch was presumed drowned after his luxury yacht “Bayesian” (which was supposedly designed to be unsinkable) sank off Sicily under strange circumstances which have been very widely reported.

Lynch had actually taken a party of family, including his wife and daughter, and friends, including his trial lawyer Christopher Morvillo and his wife, on the yacht for a Mediterranean cruise to celebrate his acquittal.

Of the 21 people on the yacht 14 survived. Lynch, his daughter, and Mr and Mrs Morvillo were among the seven presumed dead.

In the wake of the unusual deaths of both Mike Lynch and his co-defendant, along with their lawyer, on the same day, many people suspect foul play. Hence, people have been asking whether this sequence of events occurred by chance as opposed to by foul play.

In one article John Leake specifically asked for a Bayesian estimate of this ‘Black Swan’ event.

Normally, when presented with such questions we argue that many apparently incredible coincidences are not so incredible after all.

For example, as explained here, whereas the probability that a particular person, say Joe Bloggs, wins the 6 from 49-ball jackpot lottery on their first ever attempt is incredibly small (about 1 in 14 million), the probability that someone, anyone, on any given draw wins the lottery on their first attempt is actually very high.

Indeed, 35 million lottery tickets were sold for the first U.K. National Lottery draw on 19 Nov 1994, meaning it was almost certain that at least one first time lottery ticket buyer would win; in fact, there were seven such winners who shared a prize of £6 million.

Similarly, the probability that the Canadian woman, Ann Lepine, would win the jackpot lottery twice within the space of four years (which she did) might seem infinitesimally small, but once we take account of the number of lotteries around the world and number of people playing regularly, the probability that at least one person somewhere will win twice within a four-year period is highly likely.

Also, as explained here, the same kind of arguments apply to supposedly incredible clusters of unlikely deaths all occurring in a specific hospital when a specific nurse is present.

While the probability of this occurring by chance in a specific case is indeed tiny, the probability of such a cluster occurring in at least one hospital in the UK by chance, over a period of say 10 years, is very high.

Do the same kind of analyses and arguments apply to the case of Lynch and Chamberlain?

Assuming that we accept the official reports that they are indeed both dead – some conspiracy theorists are already claiming they aren’t, the answer is a (a qualified) yes, they do.

First, we must properly formulate the problem given the context. First, let’s ignore:

• the ‘coincidental fact’ that they were acquitted in a fraud case brought by a multinational corporation
• the fact that the luxury yacht Bayesian was claimed to be ‘unsinkable’.

We will take account of these facts later in the Bayesian analysis. Before that we need to establish the appropriate ‘prior probability’ for the pair of freak deaths on the same day. This is the probability, regardless of the evidence at hand about this specific case, based on prior experience and historical data.

So, instead of asking:

“What is the probability that Lynch and Chamberlain would both die on the same day as a result of separate freak accidents by chance?”

We need to first identify the frame of reference, assuming that the scope here is some relevant population of people resident in both the USA and Europe and ask the question.

“What’s the probability that within, say, a 10-year period two senior business people from the same company would both die in separate freak accidents purely by chance”

To answer this question we first need to estimate the probability that a ‘senior business’ person dies in a freak accident on any given day. This is difficult because there are many different types of ‘freak accident’ that can cause death.

This class of event does not just include being hit by a car while out running or drowning when a luxury ‘unsinkable’ yacht sinks. Plane and helicopter crashes are freak accidents that cause hundreds of deaths every year and ‘senior business people’ are disproportionately among the victims given they are more likely to be able to afford these means of transport.

Being accidentally shot, burnt, electrocuted, decapitated etc are also what we might call ‘freak accidents’. If you have seen the Final Destination series of movies you can perhaps imagine many more examples, but you should get the gist of the argument being made here.

The key point is that, had Lynch and Chamberlain succumbed to any of these types of deaths on the same day, we would be asking exactly the same questions about whether this could have happened by chance alone.

Establishing the prior probability for the two freak deaths on the same day

It is difficult to get accurate statistics on deaths by ‘freak accident’, but the CDC reports that in 2021 there were 227,039 ‘unintentional injury deaths’ in the USA. so, given there are 340 million people in the USA and 750 million in Europe.

If we assume a similar rate of such deaths in Europe then that is approximately 500,000 such deaths. Let’s be conservative and assume overall there are 700,000 such deaths in America and Europe combined each year.

Assuming one percent of the population are “senior business people” (that means we are assuming there are about 11 million ‘senior business people’ in the USA and Europe) then we can assume that there are 7,000 freak deaths of senior business people per year.

That is about 19 such deaths every day. So, the probability that a specific business person dies a freak death on any specific day is 1 in 573,000.

Since the question we are answering assumes that the deaths of the two business people really did happen by chance alone, and since we know that their deaths were the result of very different causes, we can assume that they are ‘independent’ which means that the probability they BOTH day on the same specific day from a freak accident is 1/573,000 times 1/573,000.

That is a probability of 1 in 328,329,000,000 (1 in 328 billion).

However, there are 3650 days in any 10-year period, so the probability two specific business people die from freak accidents on the same day over such a period is (approximately) 3650 times greater than 1 in 328 billion.

This is about 1 in 90 million.

Note: it’s approximated not just because the 1 in 328 billion is approximate but because, the correct answer requires us to calculate one minus the probability that they do NOT die from freak accidents on the same day over such a period and this result is slightly less than just multiplying 36,503 by 1 in 328 billion as explained here.

But the 1 in 9 million number applies only to two specific business people. We are already assuming that there are about 11 million business people in USA and Europe. What we now need to estimate is how many pairs of such business people are ‘related’ in the sense of having worked for the same company.

There are an enormous number of pairs of business people among 11 million. In fact there are over 60 trillion such pairs. That would mean that if we restricted our question to asking the probability that two business people (not necessarily from the same country) in the USA or Europe would die from freak accidents on the same day over a 10-year period the answer is very close to 100 percent certain.

You should not be surprised by this. I have already argued above that about 19 business people die every day from freak accidents in the USA and Europe.

So, on any given day, it is almost certain that at least two business people from somewhere in America and Europe, will die from (independent) freak accidents, which means that over a 10-year period it is even closer to certain that on at least one day we will see two business people die from (independent) freak accidents.

However, we need to narrow our ‘search’ of pairs of business people down to those who have worked for the same company. Even if we assume the probability that a random pair of business people working for the same company is as low as 1 in 10 million then, assuming the 60 trillion pairs of business people, this still means there are about 6 million such pairs.

Since the probability of any specific such pair dying on the same day from a freak accident over a 10 year period is 1 in 9 million, this means the probability that at least one pair of important business people from the same company die in separate freak accidents on the same day purely by chance over a 10-year period is about 67 percent.

So, there really is nothing particularly unusual about it. There is only a 33 percent chance it will not happen.

Using Bayes to take account of the evidence specific to Lynch and Chamberlain

The fact that the prior probability of foul play in the general case is only 33 percent does not mean we can conclude that the probability that the “The deaths of Lynch and Chamberlain were caused by foul play” is only 33 percent?

What it does mean is that, from a Bayesian perspective, we can assume the prior probability of foul play is 33 percent before we take account of the other specific evidence in this case.

For example, the following evidence supports the foul play hypothesis:

• the acquittal in a fraud trial brought by a massive multi-national company
• the deaths within a short time following the fraud trial acquittal
• the fact that it was a luxury yacht that sunk was whilst moored

To take account of this evidence we have to introduce some subjective conditional probabilities. Specifically, we need to consider the following ‘likelihood ratios’:

• How much more likely is it that we would find that the pair of business people were acquitted in a fraud trial if there was foul play than if there was not? Let’s say it is reasonable to assume that it is about 100 times more likely than not. By Bayes’ theorem, assuming 100 times more likely, the posterior probability of foul play jumps from 33 to 98.01 percent.
• How much more likely is it that we would find that the deaths occurred shortly after they were acquitted in a fraud trial if there was foul play than if there was not? Let’s say it is reasonable to assume that it is about 10 times more likely than not. By Bayes’ theorem, assuming 10 times more likely, the posterior probability of foul play now jumps from 98.01 to 99.8 percent.
• How much more likely is it that we would find that it was a luxury yacht that sunk was whilst moored if it was foul play than if it was not? Let’s say it is reasonable to assume that it is about 100 times more likely than not. By Bayes’ theorem, assuming 100 times more likely, and given the previous evidence, the posterior probability of foul play jumps from 99.8 to 99.998 percent.

So, taking account of this evidence, we conclude that the probability of foul play was about 99.998 percent.

Equivalently, we can conclude that the probability that the deaths of Lynch and Chamberlain were not caused by foul play is about 0.002 percent, that is 1 in 50,000.

However, there are also claims that incompetence of the yacht captain caused the sinking of Bayesian, and he has indeed been interrogated on a possible manslaughter charge. This would provide support for the “no foul play” hypothesis.

In that case, let’s say it is reasonable to assume that it is 1000 times more likely we would find the captain’s incompetence caused the sinking if there was no foul play than if there was. By Bayes’ theorem, and given the previous evidence, the posterior probability of foul play now decreases from 99.998 to 98.01 percent.

That would mean that the probability that the deaths of Lynch and Chamberlain were not caused by foul play is about two percent, that is 1 in 50.

Of course, there are those who would argue that arresting the captain on possible manslaughter charges is actually further evidence of foul play by ‘powerful forces’ in that he is being made the ‘fall guy’ for a crime they have committed.

But, we currently have no evidence of that. On the contrary, in the vast majority of cases where accidents like this occur, human error or incompetence is found to have contributed to the outcome.

The key point is that, as more evidence about the case is revealed, we can easily update the results.

See more here substack.com

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