Apr 282020

Nothing epitomises the world’s stunned unpreparedness for the fearsome escalation of the coronavirus pandemic better than the lingering dispute about the appropriateness of mass testing.

Until recently, the main objection to mass testing had been a practical one: a scarcity of RT PCR test kits, combined with the complexity and length of the testing procedure, meant that their use needed to be rationed and supervised, with priority given to identifying as many infections as possible, starting from people who showed specific symptoms and were therefore more likely to be infected in the first place.

This was always a weak argument, and it became increasingly surreal as any amount of costs and efforts of getting more tests done paled into insignificance compared to the gargantuan social and economic costs of all other measures enacted around the world. In any case, the point is now being superseded by the appearance of an increasing number of simpler and faster tests, which greatly extend testing capacity. With supply constraints on the way to being removed, a widespread consensus is finally developing about the need to extend testing beyond symptomatic cases, first to healthcare workers and other people more exposed to the risk of infection, then to people with milder symptoms or no symptoms at all, and ultimately to whoever wants to be tested.

There is still little focus, however, on taking advantage of virtually unconstrained testing resources to fulfil the need for randomised testing aimed at measuring and monitoring the virus Base Rate. The benefits of knowing the virus prevalence in the general population are hardly missed. But efforts have so far been concentrated on estimating it through epidemiological models – whose varying conclusions depend on a number of uncertain parameters – rather than on measuring it directly by sampling observation.

A firm empirical grip on the virus Base Rate is the necessary foundation on which evidence can be used to test the infection hypothesis (here is a video primer on the key concepts used henceforth).

A first-line source of evidence of infection is given by symptoms: fever, cough, shortness of breath, fatigue, loss of taste and smell, etc. A person with symptoms has a higher probability of being infected than a person without. We say that P(I|S), the probability of Infection, given Symptoms, is higher than P(I), the prior probability or Base Rate of infection. We call such evidence confirmative.

How much higher? This is measured by Accuracy, which depends on two variables: the True Positive Rate TPR=P(S|I) – the probability of Symptoms, given Infection – and TNR=P(no S|no I) – the probability of no Symptoms, given no Infection. In a clinical context, TPR is known as Sensitivity and TNR as Specificity. A natural measure of overall Accuracy is the average of the two: A=(TPR+TNR)/2. Perfect evidence has maximum Sensitivity (TPR=1) and maximum Specificity (TNR=1), hence maximum Accuracy A=1. Imperfect evidence has TPR<1 and/or TNR<1, hence A<1.

A key relation to notice is that TPR=1-FNR and TNR=1-FPR, where FNR=P(no S|I) is the False Negative Rate – the probability of no Symptoms, given Infection – and FPR=P(S|no I) is the False Positive Rate – the probability of Symptoms, given no Infection. Hence maximum Sensitivity has FNR=0 – no False Negatives – and maximum Specificity has FPR=0 – no False Positives. Notice A=0.5+(TPR-FPR)/2. Also, simple maths shows that evidence is confirmative if TPR/FPR>1 or, likewise, FNR/TNR<1.

Symptoms are confirmative evidence of infection, but they are quite inaccurate. Sensitivity is inherently low: FNR>0 – this is indeed a key issue with the coronavirus: there is a high number of asymptomatic infections. And, in most cases, Specificity is also low: FPR>0 – a fever or a cough do not necessarily imply an infection. Admittedly, the more specific the symptoms, the lower is FPR and the higher is the probability of infection. In the limit, an accumulation of symptoms – fever and cough and cold and shortness of breath etc. – can amount to a Smoking Gun: evidence so specific as to exclude False Positives and provide conclusive evidence of infection. But remember that conclusive evidence is not the same as perfect evidence: absence of pathognomonic symptoms does not prove absence of infection. Accuracy needs high Specificity as well as high Sensitivity.

This point is often missed: it is no use evaluating evidence by its Sensitivity alone or by its Specificity alone. Think of a parrot always shouting: Infected! It would have maximum Sensitivity – no False Negatives – but zero Specificity. Likewise, a parrot always shouting: Healthy! would have maximum Specificity – no False Positives – but zero Sensitivity. More sensibly, think of an airport hand luggage scanner that always beeps, or of an equally useless one that never does.

Symptoms are usually not accurate enough to prove or disprove the infection hypothesis. That’s why we need tests. Tests are not perfect either: most of them produce False Negatives and/or False Positives. But these can be properly measured. Like a good hand luggage scanner, a good test minimises both and optimises their trade-off.

A good scanner needs to have high, ideally maximum Sensitivity, as to avoid False Negatives: it cannot let a gun go through. A perfect scanner would also have maximum Specificity – no False Positives: it would only pick up the bad stuff and never give false alarms. Failing that, however, we obviously prefer Sensitivity to Specificity – we want to make sure that every explosive device is picked up, even if most suspect objects turn out to be innocuous. We tolerate less Specificity to ensure maximum Sensitivity. At the same time, however, we want Specificity to be as high as possible – inspecting every piece of luggage that gives a false alarm would result in massive chaos and missed flights.

Likewise, a good virus test needs to spot every infection, even if that means scaring some people with a false alarm. Such was the test in our story: FNR=0% and FPR=5% – no False Negatives and a small percentage of False Positives. There we saw that the probability of infection, given a positive test result, depends on the Base Rate: despite high accuracy, a low Base Rate implies a low probability – that is why, by the way, we are not flustered when we hear an airport scanner beep. And we saw that with a low Base Rate there is a simple way to deal with alarms: repeat the test. One positive result is no reason for concern, two positives draw our attention, three positives are bad news. On the other hand, we have seen that a negative test result at any stage gives us complete peace of mind: maximum Sensitivity means that the probability of infection, given a negative result, is zero, irrespective of the Base Rate.

How good is the standard RT PCR test in detecting the coronavirus? To my surprise, its accuracy does not seem to be a well-known, well established and agreed-upon number. Worse, it is hardly ever a point of discussion – as if the test were just assumed to be perfect. Well, it isn’t. According to some measures, its Sensitivity – the most important side of accuracy – may be as low as 70% or lower. (A horrific story has it that Dr Li Wenliang, the ophthalmologist who first warned about the Wuhan outbreak in January, tested negative several times before dying from the infection a few weeks later). On the other hand, the test seems to be highly specific: a positive result implies an almost certain infection.

Let’s then assume that’s the case and say FNR=30% and FPR=0% – some False Negatives and no False Positives. This is the mirror image of the maximum Sensitivity test in our story. With maximum Specificity, the probability of infection, given a positive test result, is 100%, irrespective of the Base Rate. On the other hand, with Sensitivity at 70% the probability of infection, given a negative test result, is not zero, but depends on the Base Rate. Namely, if the Base rate is low, say 0.1%, the probability is practically zero. But if the Base Rate is higher, it is well above zero. Let’s say for instance that the Base Rate is 50% – a reasonable assumption for the prior probability of infection in a symptomatic person. Then the probability of infection following a negative result is 23%. This is well below the prior probability – the test is confirmative – but is certainly not low enough to exclude infection. To do so, a second test is needed, which would prove infection in case of a positive result, and would lower the probability of infection to 8% in case of a negative result. Hence, for peace of mind we would need a third test, which again would prove infection if positive, and, if negative, would lower the probability of infection to a comfortable 2.6%.

At this level of accuracy, therefore, the RT PCR test is like an enhanced version of an accumulation of specific symptoms: a Smoking Gun that will certainly spot an infection if there is one, but will not prove absence of infection if there isn’t one, unless repeated several times. It follows that, if the hallmark of a good test is to let no infection go undetected – zero False Negatives – a maximum Specificity test is not as good as a maximum Sensitivity test.

This makes little difference if the Base Rate of infection is low. With a negative result, a maximum Sensitivity test guarantees a zero probability of infection whatever the Base Rate, but a maximum Specificity test is almost as good: one negative result is sufficient to reduce the already low Base Rate to almost zero. This is still not good enough if our aim is to avoid a bomb on a plane. But we can live with it if, despite media hype, we accept that a few undetected infections are not as dangerous.

It makes a big difference, however, if the Base Rate is high. In this case, a negative result in a maximum Sensitivity test still guarantees a zero probability of infection, but in a maximum Specificity test it only reduces the probability to what might still be an uncomfortably high level, which could only be lowered by repeating the test several times.

Yet, since the start of the epidemic, RT PCR tests have been targeted on symptomatic cases – people for whom the prior probability of infection was already high before the test. There was a good reason for it: the priority in the early stages was to confirm suspect infections, and isolate and treat the infected. But how many infected people have been ‘cleared’ after one negative test result, and went about infecting others?

RT PCR tests have been used on the wrong targets. They are more appropriate for asymptomatic cases, where the prior probability of infection is low, than for symptomatic cases, where the probability is high. The more specific the symptoms, the higher is the probability of infection. What is the point, then, of testing a symptomatic case just to prove for certain what is already quite likely, while running a high risk of missing a large number of False Negatives?

The most appropriate test for a symptomatic case is not a Smoking Gun, where a positive result proves that the infection hypothesis is true. It is a Barking Dog, where a negative result proves that the hypothesis is false.

Little is known about the degree and type of accuracy of the numerous tests currently being evaluated under the EUA protocol. Ideally, we would like to see both maximum Sensitivity and maximum Specificity tests. Used in conjunction, they would yield a certain answer to the infection hypothesis, irrespective of the Base Rate of infection. Failing that, however, estimating the Base Rate of infection in the general population is a crucial step for a correct interpretation of the test results.

Once we know the test accuracy, as defined by TPR and FPR, the Base Rate BR can be easily derived from

where P(+) is the probability of a positive test result. Hence:

For instance, let’s say we test 10,000 people and 595 of them test positive, hence P(+)=5.95%. If the test accuracy is TPR=100% and FPR=5%, as in the maximum Sensitivity test in our story, then BR=1%. Similarly, if accuracy is TPR=70% and FPR=0%, as in our assumed maximum Specificity RT PCR test, and 70 people test positive, then P(+)=0.70% and again BR=1%.

Notice by the way that this is a general result, valid for any level of accuracy. Say for instance we only have a horribly inaccurate, disconfirmative test, with TPR=30% and FPR=60%. Nevertheless, if we observe that 5970 people test positive, then P(+)=59.7% and again we can conclude that the Base Rate of infection is 1%.

A test with a known level of accuracy is all we need to derive the Base Rate of infection. Crucially, however, this will be the Base Rate of the tested population. Hence, if tests are only performed on symptomatic cases, there will be many more positive results, and the derived BR will be much higher – in fact equal to P(+)/0.7, i.e. 43% higher than the percentage of positives cases, under the assumed accuracy of the RT PCR test. As we saw in the previous post, taking such number as an estimate of the prevalence of infection in the general population would therefore be a gross miscalculation. It would be as if in 2016 Brexit support had been estimated by polling UKIP voters, or Trump support by polling NRA members.

A correct estimate of the true Base Rate of infection can only be obtained by testing a randomly selected, representative cross section of the general population of interest.

The message is finally getting across.

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Mar 192020

With Italy in lockdown and London about to follow, let’s see what we can say in our framework about the coronavirus pandemic.

Funnily enough, the Blinded By Evidence paper starts with a virus. You hear about it on TV and worry you might have it. So you take a test that will tell you with 100% certainty that you have the virus if you actually have it – False Negative Rate (FNR)=0% – and with 95% certainty that you don’t have the virus if you actually don’t have it – False Positive Rate (FPR)=5%. The test comes back positive and you panic, until you are shown that the probability that you have the virus, given that you tested positive, is not near 100%, as you feared, but less than 2%. The reason is that the Base Rate of the virus – its frequency in the population, giving you the probability that you had the virus before you took the test – is 0.1%. And the reason why you were so off the mark is what in our framework we call the Prior Indifference Fallacy: blinded by the test result, you ignored the Base Rate, until reminded of its importance for a correct interpretation of the evidence.

So what’s happening with the coronavirus?

A major difference between our neat stylised story and the messy reality of coronavirus is in the Base Rate. The Base Rate in the story is a known given number – one in a thousand. But what is the Base Rate of the coronavirus? Nobody knows. All we know is that the virus is highly contagious and is spreading. But how many infected people are out there at any point in time? How are they distributed? How can we spot them? We just don’t know. We only know how many have been spotted, as a number of suspect cases – people exhibiting specific symptoms – have been tested and some of them have come out positive. But what about the others – the infected people who have not been tested because they haven’t shown any symptoms, don’t even know they are carrying the virus and go happily about infecting other people? We have no idea. We can only infer that there must be a positive relationship between spotted and unspotted cases – the good old cockroach theory – but what is the multiple? How many unspotted cases are there for each spotted case? We don’t know.

But that’s what we would like to know. As sorry as we are for the known number of spotted cases, and relieved that they are being identified, isolated and treated, it is the unspotted cases that we worry about. How many are they? How fast are they growing? What is the probability that we will get infected by one of them and join their number? What is the Base Rate of the coronavirus?

Such basic questions, but no answers. And, worse, little interest in finding out. Unlike in our story, the coronavirus Base Rate is unknown. But, just like in our story, we fail to recognise its importance for the purpose of finding a correct answer to our questions.

The reason is the same: we are blinded by evidence.

In the story, our question is: what is the probability that we are infected, given that we tested positive? Blinded by the test result, we neglect to account for the small Base Rate and end up with a gross overestimation of the posterior probability.

With the coronavirus, we would also like to be tested. But we can’t, since the RT PCR test that is being used to detect the virus has been confined to suspect cases and is not available to the general public. Unable to take the test on ourselves, our question becomes: what is the probability that we are infected, given that a number of other people tested positive? As in our story, without a test we are naturally drawn to looking for the virus frequency: how many infected people are there as a percentage of the population we interact with? What is the probability that one of them will infect us? Is it small, like the one in a thousand in our story? Or is it “at least 50%”, as yesterday my friend Enzo warned me it is in Milan, begging me not to go there?

No one tells us. So we try ourselves. We look at the data, and what do we see? One horrible figure: the total number of spotted cases, ominously growing day by day. From there, we infer that the number of unspotted cases must be growing at the same pace if not faster, and that it is an unnervingly unknown but surely large multiple of the spotted cases. And, like the character in our story, we panic. We are blinded by evidence. In the story, the panic is caused by Base Rate neglect. With the coronavirus, it is caused by Base Rate inflation.

Let’s see why. The number of spotted cases is the number of people who tested positive out of the number of people who got tested. Clearly, the more people get tested, the larger is the number of spotted cases. So we look at their ratio. This would be a good estimate of the Base Rate if, and only if, the tested people were a random sample of the population of interest. But they aren’t. The tested sample is mainly composed of suspect cases – people who are tested because they show specific symptoms or because they have been in contact with spotted cases. As such, it is far from being random: the prior probability that a suspect case is infected is much higher than if he was picked at random. Hence the ratio of the number of positives over the number of tests is a gross overestimation of the true Base Rate.

Let’s take for example the latest daily Bulletin from the Italian Health ministry:

And let’s look at Lombardy, where the early cases showed up in February and where almost 50% of cumulative total cases (Casi Totali, in orange) are still concentrated. Total cases in Lombardy amount to 19,884, out of 52,244 tested people (Tamponi, in grey). Their ratio, 38%, is the percentage of tested people who turned out positive. Does it mean that that almost one in four of 10 million Lombards are infected? Obviously not. Likewise, the true Base Rate of infections is not 8% in Veneto or 22% in the whole of Italy.

What is it then? We don’t know. In principle, however, estimating the coronavirus Base Rate would be quite simple. Take an unbiased, well stratified, random sample of the population of interest – a routine statistical technique commonly used in opinion polls and market research – and test them. Provided the test is sufficiently accurate, the percentage of positives is a good estimate of the Base Rate.

Crucially, the tested sample would have to be a fair representation of the general population, and therefore include symptomatic as well as asymptomatic people. This is in contrast with the current practice of confining tests to suspect cases – a reasonable procedure when priority must be given to identifying and securing as many infected people as possible, but an erroneous one, as we have seen, when the goal is to estimate the extent of the virus spread.

The advantage of having a detailed, localised and regularly updated map of coronavirus Base Rates should be obvious. It would give us a basic idea of the frequency of infection in different places and its evolution over time, thus helping us – at an individual level as well as at a public policy level – to modulate our response, focusing it more in areas where the Base Rate is higher and growing, and less in areas where it is lower and stable.

At an individual level, it would help our apprehension to know that the Base Rate in our area is, say, 1%, rather than the imaginary multiple perceived by mask-wearing people. Before you say 1% is too low, think that it would mean 100,000 infections in Lombardy – about five times the current number of spotted cases – and more than 600,000 in Italy – about fifteen times the spotted cases. If it is higher we worry a bit more, if it is lower we worry a bit less. But it would benefit our health to know what it is, and that it is far lower than the hyperbolic figures implied by Base Rate inflation.

At the policy level, the benefits of a differentiated approach versus the blanket lockdowns being imposed in Italy and other countries should also be evident, in terms of increased focus where focus is mostly needed and a reduction of the huge social and economic costs currently imposed on everyone.

So the question is: why is not done?

One answer is that the standard RT PCR test requires a complicated and lengthy procedure and does not lend itself to mass testing – hence the priority set on testing suspect cases. But then the South Korean experience has shown us that mass testing is possible, and that it can be very useful. Similar evidence has come from a small town in Veneto. In addition, several companies, including Roche and the Italian Diasorin, have recently developed cheaper and faster tests.

Another objection is that random testing would produce volatile results, as e.g. one negative case today may turn positive tomorrow. But that is in the very nature of all testing, where variability is dealt with by averaging results on properly sized randomised samples, which do not have to be very large to represent much larger populations with a small margin of error. It is just like any poll, say a Leave/Remain Brexit poll (remember?). In fact, making sense of that variability is the very reason why polls are taken and retaken over time.

A third objection is the one in our story: if the Base Rate is small, even very accurate tests can produce a large number of False Positives and False Negatives. But we know the answer to that: repeat the tests – one positive is unreliable, two positives is dependable, three positives almost certain.

So my answer is: Base Rate testing should be done, and I echo WHO Director-General’s ‘simple message for all countries: test, test, test’.

But randomise.

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Jul 112019

By the time I started writing my DPhil thesis, I had pretty much come to the conclusion that academic life was not for me. So I decided to try and see what it was like to work in the City, and got a summer job at James Capel. Subsequently bought by HSBC, James Capel was then a prominent UK stockbroker and one for the few to pioneer into European equity research. So it was that, overnight, I became their ‘Italian Equity Strategist’.

I wanted to dip a toe in the water – I got a breath-taking full-body plunge into the wide ocean. In no time I was talking to all sorts of ‘clients’ about all things Italy – a true life shaping experience. I still remember – or was it a nightmare? – being in front of a big shot from the ‘Danish Pension Fund’, trying to answer as best as I could his full cartridge of very detailed questions.

It didn’t last long. First, being at work at 7am was definitely not my thing. Besides that, I soon realised I wanted to be on the other side – the buy side, not the sell side. A fund manager, not a stockbroker. So when my friend Bruno got me an interview at JP Morgan Investment Management, where he was working as a company analyst – ‘I’m there at 9am and I can manage my time quite flexibly, as long as I get the job done’ – I was all for it.

But before leaving James Capel I wrote my final piece for their European Equity Strategy publication. It resurfaced recently in a house move. Reading it again after such a long time (yes, the London phone code was 01) made me laugh out loud:

The Italian stock market has gone up 9% by the end of July since the beginning of 1988. This relatively poor performance can essentially be ascribed to fundamental market uncertainty on the critical issues of political stability and fiscal policy, which constitute both the primary target and the key test for the new coalition government headed by the Christian Democratic leader Mr de Mita.

A global reform of the institutional and administrative apparatus of the Italian state is another major concern of the de Mita government. The aim is to make legislation a less lengthy and cumbersome process and to increase the efficiency of the Public Administration.

Political uncertainty – which has kept foreign investors out of Italy for two years – is certainly among the key factors which explain the poor relative growth of the Italian market and the low level of current valuations relative to the performance and prospects of the Italian quoted companies.

As Bruce Hornsby had been singing a couple of years earlier, ‘That’s just the way it is – Some things will never change’.

Since the launch of the Made in Italy Fund, now more than three years ago, I have been banging on this point. Viewed from a top-down, macro perspective, Italy has always looked like an unattractive place to invest. Unstable governments, inefficient public services, bulky debt, higher bond yields and, before the euro, a chronically weak currency. Add for a good measure a few evergreens, such as corruption, the South backwardness and organised crime. And, from a stock market point of view, a limited number of quoted companies – currently about 350, against more than 800 each in France and Germany – mainly concentrated in banking and finance, utilities, oils and a few consumers. The whole lot worth about 600 billion euro – less than Apple. Who would want to invest there?

So common is this ‘country’ way of thinking that it takes some unlearning to realise how fundamentally wrong it is.

Investors do not buy countries. They buy companies – companies that happen to be based in a certain country and are therefore, in most cases, quoted on that country’s Stock Exchange.

But what does that mean? Is Microsoft a US company? Is Nestlé a Swiss company? Yes, that’s where they are headquartered and quoted. But no, not in the sense that their performance is related in any meaningful way to the performance and vicissitudes of their country of origin. What is the relationship between LVMH and the growth of the French economy? Or Ferrari and the stability of the Italian government?

The national dimension of equity investing is largely a remnant of a long-gone past, when most businesses were predominantly domestic. This is clearly not the case today, and not only for the big global corporations, but also, and increasingly so, for smaller firms selling their products and services around the world. To think that there is any direct link between these companies and the economic conditions of their country of origin is lazy at best.

There are still of course many companies whose business is mainly domestic. For these, the linkage to the state of the national economy may be stronger – but it is far from being linear, stable or reliable. Indeed, for some companies a weak economy may create opportunities to gain market share from competitors or to introduce new products and services.

So it is never as simple as economy=stock market. This is so in general, but it’s especially true for Italy, where the sector composition of the market bears no resemblance to the country’s economic reality.

Then what’s the point of the Made in Italy Fund? Isn’t its very name meant to evoke the same national dimension that I am saying makes no sense?

No. The Fund does not invest in Italy as a country. It invests in Italian companies with a market capitalisation of less than one billion euro, quoted on the Milan Stock Exchange.

Why only those and why only there? Two reasons:

  1. It is a good place for finding pearls – companies with high growth prospects, strong and sustainable profitability and attractive valuations. Many of them are smaller companies, leaders in specific market niches, where good management and Italian flair allow them to build and maintain a solid competitive advantage in Italy and abroad. Of course, there are many good companies elsewhere. Buy in Italy they tend to be cheaper. Why? Precisely because investors snub Italy as a country! This is clearly true for many foreign investors, indolently clinging to their ‘country’ way of thinking. But in the last few decades it has been increasingly true also for domestic investors, who in a post-euro, pan-European world have been shedding a sane home-country bias in favour of a snobbish xenophilia.
  2. Soon after I joined JPMIM after James Capel, I started managing the Italian slice of their international equity and balanced portfolios. This was – hard to believe – thirty years ago. Since then I have done many other things, but my involvement with the Italian stock market has hardly ever stopped. I am – I fear to say – a veteran. As such, I like to believe that my experience, together with my ‘Italianness’ – in language, culture and mores – make me especially suited to spotting Italian pearls and, as importantly, avoiding Italian pebbles and duds.

Italy is my country. Like most Italians, I have a complex love-hate relationship with it. Di Maio or de Mita, its politics has always been messy, its public finances rickety, its international credibility regularly in the balance. In my thirty years as an Italian fund manager, I have never been able to build a credible top-down investment case for Italy as a country (incidentally, can one do so for France or Germany or any other developed nation?). But when I flip it around and look bottom-up at Italian companies, especially the smaller ones that form the backbone of the Italian economy, I have no hesitation. In a universe of around 280 companies with less than one billion market cap – now steadily increasing through a sustained flow of new IPOs – I have no trouble selecting thirty or so to include in the Made in Italy Fund. If anything, the problem is to keep track of all the opportunities.

So my attitude to chronic Italian bears is, with Bruce Hornsby: ‘Ah, but don’t you believe them’. Country allocation should not be about countries. It should be about finding pots of value around the globe, and focused managers able to extract them.

P.S. I invite subscribers who haven’t yet done so to also subscribe to the Bayes Investments website, where they will find information and updates on the Made in Italy Fund.

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Mar 172019

Earlier this week I gave a presentation to the Investment Management Club at the London Business School. I hope you find it useful.

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Sep 012018

(Sorry – a few complicated months)

Speaking of best books I have ever read, a more recent one is David Wootton’s The Invention of Science.

Wootton does a marvellous job explaining mankind’s transition from a worldview based on authority to one based on evidence.

As reprised in Steven Pinker’s latest book (p. 9), a typical well-educated Englishman in 1600 believed in demons, witches, werewolves, magicians, alchemy, astrology and other nonsense (p. 6). But a mere century and a quarter later his whole perspective had changed:

Between 1600 and 1733 (or so – the process was more advanced in England than elsewhere) the intellectual world of the educated elite changed more rapidly than at any other time in previous history, and perhaps than at any time before the twentieth century. Magic was replaced by science, myth by fact, the philosophy and science of ancient Greece by something that is still recognizably our philosophy and our science, with the result that my account of an imaginary person in 1600 is automatically couched in terms of ‘belief’, while I speak of such a person in 1733 in terms of ‘knowledge’. (p. 11-12).

Commonly referred to as the ‘Scientific Revolution’, this transition is not easy to understand. The images we have in mind are of sinister cardinals persecuting Galileo and of barmy philosophers refusing to look into his telescope. In the same vein, Wootton quotes Joseph Glanvill, an early advocate of the revolution, who derided the view that telescopes and microscopes were

all deceitful and fallacious. Which Answer minds me of the good Woman, who when her Husband urged in an occasion of difference, I saw it, and shall I not believe my own Eyes? replied briskly, Will you believe your own Eyes, before your own dear Wife? (p. 74, Italics and bold in the original).

(I find this particularly funny, wondering about how essentially the same joke found its way down to Richard Pryor, through Groucho Marx’s Duck Soup. An equivalent joke my friend Peter told me many years ago is that of the English aristocrat, which I used here).

Obviously, such hilarious caricatures leave much to explain. Educated people in 1600 and earlier were no dimwits. So why did they hold what to our eyes seem such outrageously weird beliefs? This is a focal theme in the Bayes blog. Hence I was intrigued to find out that Wootton’s book is centred on the same key concepts.

Following Aristotle, a seventeenth century educated person was taught to think deductively: draw necessary conclusions from undisputable premises. It would be a mistake, however, to imply that he ignored evidence. As we have seen, there is no such thing as a priori knowledge, independent of evidence. Knowledge cannot but be based on some form of evidence – empirical, as it is plain to our eyes; or axiomatic, as it was common before the Scientific Revolution, all the way back to ancient Greece and beyond. Episteme was absolute, irrefutable, self-evident knowledge. And even the wackiest myths and legends of primordial peoples were not haphazard fantasies but elaborations of authoritative evidence, perhaps in the form of dreams by elderly sages and wise men, who interpreted them as divine revelations they were called upon to proclaim and propagate.

Aristotelian principles were self-evident truths. Such as: All bodies move towards their natural place. Therefore, as stars rotate around it and every object falls towards its core, the earth must be the centre of the universe. Or: Heavier objects fall faster than lighter ones. Therefore, a two-kilo bag of sugar falls faster than a one-kilo bag (Wootton, p. 70). Or: Hard substances are denser and heavier than soft substances. Therefore, ice is heavier than water (p. 71).

These are what we call extreme priors: beliefs that are seen as so obviously self-evident that it is considered pointless to test them through menial experimentation (p. 319). As obviously, however, they are – they cannot but be – the product of evidence. I see stars rotate around the earth and objects fall towards its core: therefore, I infer that all bodies move towards their natural place. I see that a two-kilo bag of sugar falls faster than a one-kilo bag: therefore, I infer that heavier objects fall faster than lighter ones. I see that ice is heavier than water: therefore, I infer that hard substances are heavier than soft ones. The evidence is all wrong, hence the inferences are wrong. But how do I know that? Remember: the closer our priors are to the extreme boundaries of Faith, the stronger must be the evidence required to change them. And, as with Glanvill’s husband, little it matters if the evidence is right in front of our eyes. It is plain to see, for instance, that ice floats on water and, as Archimedes – whose writings had been translated in Latin since the twelfth century – had found out in 250 BCE, this is only possible if ice is lighter than the water it displaces. But hey, who is a mere mathematician compared to the supreme father of natural philosophy? Aristotle had figured out that hard substances are heavier. So there must be another reason why ice floats. Well, it is because of its shape: flat objects cannot penetrate water and therefore remain on the surface. Galileo would patiently prove this was nonsense (p. 315), but philosophers remained unimpressed. In the same vein, when Galileo asked his philosopher friend and colleague Cremonini to look at the mountains on the moon through his telescope, Cremonini refused, not because he was a blockhead – far from it: he was a highly respected professor of natural philosophy for sixty years and earned twice as much as Galileo – but because he did not trust the evidence: he did not regard it as strong enough to dent his Aristotelian belief that the moon was a perfect, unblemished sphere.

The idea that Aristotle had it all figured out and that all ‘natural philosophy’ logically descended from his principles was at the core of the seventeenth century’s worldview. As Wootton puts it (reprising Borges), Shakespeare had no real sense of progress. He treated his characters in the Roman plays as if they were his contemporaries. ‘History did not exist for him’ (p. 5). The governing assumption was that, as in Ecclesiastes (1:9), there was ‘nothing new under the sun’ (p. 63). The event that triggered a seismic change in this view and initiated the Scientific Revolution was the discovery of America at the end of the fifteenth century. That’s where Wootton places what he expressively calls ‘the discovery of discovery’ (Chapter 3). There is arguably no better way to convey this concept than through Hamlet’s immortal words to Horatio, which Wootton does not quote, probably because they are so well-known and overused – although he hints at them in the title of Part One. So I will do it for him: ‘There are more things in heaven and earth, Horatio, Than are dreamt in your philosophy’ (Act I, Scene V).

The discovery of the New World showed mankind that in fact there was plenty new under the sun (including black swans, although for those we had to wait until the end of the seventeenth century) and gave rise to an explosive search for new evidence, which continues unabated, in fact accelerating, to our days. Over the following two centuries, curiosity – which theologians, reigning supreme above philosophers in the hierarchy on medieval science, regarded as a sin – became the mighty fuel of progress that it still is.

From their perspective, theologians were right: as long as knowledge is anchored to the two extreme boundaries of Faith, it remains impervious to evidence. Episteme above Doxa, truth above opinion, knowledge above experience, demonstration above persuasion. The discovery of discovery changed all that: it instilled in the minds of educated people ‘the idea that experience isn’t simply useful because it can teach you things that other people already know: experience can actually teach you that what other people know is wrong. It is experience in this sense – experience as the path to discovery – that was scarcely recognized before the discovery of America’ (p. 81).

This is the true sense of experience: exposure to the peril of being wrong. As curiosity compelled people to leave the secure shores of Aristotelian self-evidence, it encouraged them to embrace Cromwell’s rule, which we might as well rename Glanvill’s rule: Believe Your Own Eyes. This was no blanket surrender to evidence at face value. People remained wary – as we are – that evidence can be deceitful. But they opened their mind to the possibility that, in the right amount and shape, it might be capable of changing and even overturning their prior beliefs. Like Cremonini, they still suspected – and rightly so – that eyes can lie. But, unlike him, they gave them a chance: they were ready to answer Popper’s question.

This was the task that natural philosophers – as they were commonly known until the nineteenth century, when William Whewell coined the term ‘scientist’ (p. 28) – set out to accomplish: accumulate enough evidence to prove hypotheses true or false. They did so through carefully crafted experiments, which – precisely because they were well aware of the fallibility of evidence – they persistently reproduced, shared and challenged, provando e riprovando (p. 300), with the ultimate goal of devising the experimentum crucis (p. 381) which, by yielding conclusive evidence (p. 194), could allow them to proclaim a consensual winner of the evidential tug of war. Thus Truth, until then the preserve of infallible self-evident axioms, became a destination, to be travelled to through fallible empirical evidence. Prior Faith became posterior Certainty.

Reverend Thomas Bayes was born in the midst of this journey and lived through it a quiet and secluded life. He was by no means a protagonist of the Scientific Revolution – so much so that he doesn’t even earn a mention in Wootton’s book. Yet he was very much a man of his time, and his theorem encapsulates so well the ethos of the revolution that we can surely call the journey’s destination ‘Bayesland’.

(Wootton does mention Laplace’s dictum, attributing it to The Logic of Port-Royal, which ‘had acknowledged that the more unlikely an event the stronger the evidence in favour of it would have to be in order to ensure that it was more unlikely that the evidence should be false than that the event should not have occurred’ (p. 465)).

Bayesland is where we live and where we have always lived – Archimedes and Aristotle, Galileo and Cremonini, Shakespeare and Groucho Marx, you and I and all living creatures. We learn by experience, updating our beliefs through a multiplicative accumulation of evidence. We all are and have always been Bayesian.

This has been the Scientific Revolution’s greatest achievement: to show mankind that the way we have always learnt in practice was also valid in theory. Progress started when we stopped wasting time thinking we were doing something else. The effect of such a seemingly simple conceptual clarification has been breathtaking:

Of course, it was far from simple – as Wootton brilliantly shows. His book is a pleasure to read from beginning to end, including his thick jungle of notes. I warmly recommend it.

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Dec 102017

Back to the riddle.

We have seen where the word Science comes from: scire means to cut, split (as in scissors), separate, decide true from false. We, like other living creatures, do so on the basis of evidence – what we see there is. We use evidence to update our beliefs. We are all Bayesian.

Despite Kant’s grand attempt to salvage some of it, there is no such thing as a priori knowledge. What may appear to us as transcendent knowledge, emanating from pure reason independent of evidence, is and can only be based on notions – concepts, principles, axioms – that we regard as self-evident.

Such notions are the subject of Metaphysics. The word came about, apparently, to denote the collection of Aristotle’s treaties that his late editors arranged to place after (meta) his Physics. Whereas Aristotle himself had not called them Metaphysics, actually referring to them as ‘first philosophy’, dealing with concepts that came before Physics in importance and generality.

Be that as it may, we can think of metaphysics as the area we enter once we start running out of answers to our Why questions. Answers are local explanations built on our own hard evidence or, most often, on soft evidence emanating from trusted sources. We learn to accept local explanations and live with them, but every answer begets new questions, in a seemingly endless why-chain whose infinity we find impossible to accept. Explanations cannot go on forever. At some point, even the cleverest dad succumbs to the urge to end his child’s relentless barrage of whys with a resounding last answer: ‘because that’s the way it is!’

But, to the undaunted child, dad’s last answer turns into the ultimate question: What is the way it is? Once we set out to answer this question we have entered the land of metaphysics. Metaphysics is mankind’s effort to establish the absolute, unquestionable and irrefutable episteme that stands firm above Physics. Episteme is knowledge that does not need evidence because it is self-evident, certain without experiment and secure from the perils of experience.

How can we achieve such knowledge? Clearly, we can’t reach it from the side of experience, whence we can only expect an infinite regress of explanations. So it must come from the other side. But what’s on the other side? Clearly, we know nothing about it – if we did, we would have already gone past the answer we are looking for. As Immanuel Kant put it, noumena are on the other side – things-in-themselves, absolutely unknowable and irremediably inaccessible to our mind. All we can know are phenomena – things as they appear to us in the light of evidence.

Metaphysics is the boundary between phenomena and noumena – a boundary that mankind would love to cross but can only push forward, unfolding and accumulating new and better explanations of phenomena. Such is the love at the root of philosophia – the ever-burning, insatiable desire for sophia, the supreme wisdom in whose full light we would finally be able to contemplate the way it is. But the light of philosophy is the same light that illuminates phenomena. Metaphysics is and can only be on the side of phenomena – the side of experience and evidence. In the words of Arthur Schopenhauer:

Metaphysics thus remains immanent, and does not become transcendent; for it never tears itself entirely from experience, but remains the mere interpretation and explanation thereof, as it never speaks of the thing-in-itself otherwise than in its relation to the phenomenon. (Will, Volume II, p. 183).

Metaphysics is not and cannot be a priori knowledge, independent of evidence. Its value does not rest on its being beyond evidence, but on being based on notions that we regard as self-evident. Like mathematics and geometry, metaphysics is an axiomatic system – true insofar as its axioms are true. An axiom is that which is thought worthy, weighty, and thus bears authority – a concept interestingly close to the original meaning of probability. Axioms are statements assumed to be self-evidently true, thus requiring no proof or demonstration. Given the axioms, the theorems built on them using truth-preserving rules of inference are demonstrably true.

As such, the validity of an axiomatic system depends on the weight of its axioms. The more precise, clear, obvious, intuitive, indubitable the axioms, the stronger the system. Take Euclid’s Elements, which, as we know, is built on five axioms (or postulates). As we have seen, one can argue about the fifth. But not about the first: A straight line can be drawn joining any two points. Or the second: A finite straight segment can be extended indefinitely into a straight line. The third: From any straight segment a circle can be drawn having the segment as radius and one endpoint as centre. And the fourth: all right angles are equal. A geometry in which any of these four axioms is untrue is even hard to imagine. They are glaringly, unquestionably self-evident.

Now let’s compare it to Spinoza’s Ethics, which he explicitly wrote along the lines of Euclid’s Elements.

Here is its first axiom: ‘Everything which exists, exists either in itself or in something else’. The second: ‘That which cannot be conceived through anything else must be conceived through itself’. And the third, which we have encountered as the Principle of Sufficient Reason: ‘From a given definite cause an effect necessarily follows; and, on the other hand, if no definite cause be granted, it is impossible that an effect can follow’. And so on. One may or may not agree with any of these statements – provided that he truly understand what they mean. But it would be at least preposterous to regard them as self-evident.

And what about Definitions, which in Elements as well as in Ethics precede the Axioms? Let’s take the first three. In Elements they are: 1) ‘A point is that which has no part’. 2) ‘A line is breathless length’. 3) ‘The ends of lines are points’. Hard to disagree. But in Ethics: 1) ‘By that which is self-caused, I mean that of which the essence involves existence, or that of which the nature is only conceivable as existent’. 2) ‘A thing is called finite after its kind, when it can be limited by another thing of the same nature; for instance, a body is called finite because we always conceive another greater body. So, also, a thought is limited by another thought, but a body is not limited by thought, nor a thought by body’. 3) (we have seen this one) ‘By substance, I mean that which is in itself, and is conceived through itself: in other words, that of which a conception can be formed independently of any other conception’.

Whaaat? Definitions and axioms can only be as clear as the terms that compose them. We all know and agree on what a point, a straight line and a circle are. But what about essence and existence, cause and substance? They are much more complex, vaguer and harder concepts to define and comprehend. It’s no wonder, then, that all the ensuing Propositions in Ethics are, let’s say, less cogent than Pythagoras’s theorem. Take, for instance, Proposition XI, Part I:

God, or substance, consisting of infinite attributes, of which each expresses eternal and infinite essentiality, necessarily exists.

Here is the proof:

If this be denied, conceive, if possible, that God does not exist: then his essence does not involve existence. But this (Prop. VII) is absurd. Therefore God necessarily exists. Q.E.D.

Uhm. And what is Proposition VII?

Existence belongs to the nature of substances.

and its proof:

Substance cannot be produced by anything external (Corollary, Prop. VI), it must, therefore, be its own cause – that is, its essence necessarily involves existence, or existence belongs to its nature. Q.E.D.

Oh well. I spare you Proposition VI and its Corollary. Spinoza was a great philosopher and an admirable man, and his Ethics is a trove of powerful thoughts and ideas. But its metaphysical value can only be as compelling as its murky foundations.

This is metaphysics’ typical pitfall. While usually conceived as the product of pure reason, standing above physics and unrestrained by experience, metaphysics can’t really be nothing else than a more or less coherent inferential system which is in fact so entwined with evidence as to be entirely based on supposedly self-evident foundations.

The trouble is that self-evidence is in the eye of the beholder. And – as we have seen repeatedly throughout this blog – it is amazing what different people, from the dimmest to the supremely intelligent, come to regard as self-evident. Once one is satisfied that he has made all the way through why-chains to answering the ultimate question, and that he finally knows the way it is, it is tempting to invert direction and reinterpret reality in the light of his newfound metaphysical principles.

This, as we shall see, is a recipe for disaster.

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Oct 292017

I had never spent much time thinking about Bitcoin. After reading a couple of articles to figure out what it was, I associated it with the muddled assemblage of Austrian devotees ranting against central banking, fiat currency, ‘big government’, ‘the elites’ and ‘the establishment’, and left it at that.

But then the other day, when my basketball buddy Adam, who is trading cryptocurrencies, asked me what I thought about them, I realised I needed a proper answer. Fighting my flippancy impulse – last year I had lashed out at Ed on Brexit and at John on Hillary Clinton ‘corruptness’ (Bernie Sanders’ flavour, not Trump’s) – I just told Adam that I hadn’t given it much thought. But that was not acceptable either. I had to have a closer look.

Luckily (HT @manualofideas) I soon found this recent post on Aswath Damodaran’s blog, which, in typical crystal clarity, makes all the relevant points. The post, written with Bitcoin at $6,100, should be read alongside an earlier post, written only a few months ago, when Bitcoin was priced at $2,800, and a later post, written a few days ago in response to critics. In a nutshell:

  1. Bitcoin is not an asset, because it does not generate future cash flows. As such, it does not have a value.
  2. Bitcoin is a currency, enabling the exchange of goods and services. As such, it has a price, relative to other currencies.
  3. The relative price of a currency depends on its quality as a unit of account, a medium of exchange and a store of value.
  4. One can invest in assets, based on an estimation of their intrinsic value, but can only trade in currencies, based on the anticipation of their future price movements. Buying Bitcoin is not an investment.

What I didn’t know is how many cryptocurrencies there are beside Bitcoin: 1221 of them at the last count – with fancy names like Ripple, IOTA, Qtum, Stellar Lumens – for a total market cap of $169 billion! Each has its own website, detailing how different and better they are compared to the others, and each can be traded on dozens ‘exchanges’ – with other fancy names like Bithumb, Coinone, YoBit, Poloniex. Most of them have explosive price charts, and Adam feels very good about it – he’s been buying more beer rounds. But what will be the dollar price of IOTA a year from now? Like Damodaran, I am not saying it will be zero. But I can’t see how anybody could have any idea.

I will let Adam ponder upon Damodaran’s analysis. As an addition to his considerations, I see his table contrasting the Pricing Game and the Value Game as a striking illustration of the ruinous influence of the Efficient Market Theory.

By collapsing Value into Price, the EMT turns an honourable intellectual pursuit into a vacuous guessing game, where thinking is overruled by action, patience by speed and brains by guts. If prices are always where they should be, and only new information can change them, then success is determined by how quickly one is able to collect and react to news. High-frequency, algorithmic and other types of ‘quant’ trading are a direct offspring of the EMT. And so is home-made online trading, as well as its mirror image, index funds. They all make a mockery of the noble art of investing.

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Aug 272017

On the way back to London from Italy earlier this month, I decided to stop in Basel. It was mid-way and it had long been on the list of cities I wanted to visit. Why it was on that list started to surface as I picked a hotel on Trivago. Euler Hotel – definitely. We arrived in the evening and the boys were keen to get back home. So I only had half a day the following morning.

Basel’s old town centre is quite small and its main landmark is the Münster, a Romanesque church with a long and interesting history. As we waited for its doors to open at 10, I started touring the adjacent cloister. One of the highlights of the place is that Erasmus was buried there in 1536 – a sudden death following an attack of dysentery. But while looking for the grave in the cloister, wandering among tombs and commemorative plates of the city’s notables, one of them gave me a jolt:

Jacob Bernoulli, of course. He was born and lived in Basel his whole life, and died there on 16 August 1705 – morbo chronico, mente ad extremum integra – at the age of 50 years and 7 months.

Jacob – the eldest scion of the prodigious Bernoulli family – is one of my heroes. The author of the greatest masterwork in early probability theory, Ars Conjectandi, he is also credited as the first to discover the relationship between continuous compound interest and Euler’s number e, the base of natural logarithms. There – I suddenly realised – was a big piece of my subconscious attraction to Basel. Enchanted by my discovery, I asked my second child to pose for a photo next to the tombstone – my elder son was wandering somewhere else, supremely bored and impatiently waiting for lunch and departure.

After leaving the cloister, unable to come up with anything intelligible to say about Bernoulli, I told the kids about Erasmus and Paracelsus – another illustrious Basler. At 10 we visited the church – Erasmus’s grave is inside – and shortly after I realised my time was up – the children would have killed me if I had proposed any more ‘history stuff’. So we walked back to the Euler Hotel – Leonhard Euler was born in Basel two years after Jacob Bernoulli’s death. He was the first to use the letter e for the base of natural logarithms, apparently as the first letter of ‘exponential’, rather than of ‘Euler’. He also established the notation for π and for the imaginary number i, all beautifully joined together in Euler’s identity e+1=0.

On the road to London, I kept thinking with delight at my semi-serendipitous encounter with Bernoulli. Then it struck me: I had seen that tombstone before. Back home, I checked. I was right: it was in one of the best books I have ever read, Eli Maor’s e: The Story of a Number.

As I reopened the book, it all came back to me: the Spira Mirabilis.

The logarithmic spiral is the curve r=ae in polar coordinates (r is the radius from the origin, θ is the angle between the radius and the horizontal axis, and a and b are parametric constants). Bernoulli had a lifelong fascination with the self-similar properties of the spiral:

But since this marvellous spiral, by such a singular and wonderful peculiarity, pleases me so much that I can scarce be satisfied with thinking about it, I have thought that it might be not inelegantly used for a symbolic representation of various matters. For since it always produces a spiral similar to itself, indeed precisely the same spiral, however it may be involved or evolved, or reflected or refracted, it may be taken as an emblem of a progeny always in all things like the parent, simillima filia matri. Or, if it is not forbidden to compare a theorem of eternal truth to the mysteries of our faith, it may be taken as an emblem of the eternal generation of the Son, who as an image of the Father, emanating from him, as light of light, remains ὁμοούσιος [consubstantial] with him, howsoever overshadowed. Or, if you prefer, since our spira mirabilis remains, amid all changes, most persistently itself, and exactly the same as ever, it may be used as a symbol, either of fortitude and constancy in adversity, or, of the human body; which after all its changes, even after death, will be restored to its exact and perfect self; so that, indeed, if the fashion of imitating Archimedes were allowed in these days, I should gladly have my tombstone bear this spiral, with the motto, Though changed, I rise again exactly the same, Eadem numero mutata resurgo.

This is the full quote from a paper by Reverend Thomas Hill published in 1875 (p. 516-517), from which Maor’s book takes an extract (p. 126-127), taken in turn from another book. Hill did not quote the source, but the original Latin quote can be found here (p. 185-186, available here), with the indication that it comes from a paper published by Bernoulli in the Leipsic Acts in 1692, which should be found here.

Bernoulli’s enthusiasm is easy to understand and to share. The logarithmic spiral is found in nature and art. The Golden spiral, whose growth factor b is the golden ratio, is a special case, and so is the circle as b tends to 0.

At the same time, it is difficult not to laugh at the manner in which Bernoulli’s wish was finally granted. Perhaps confused by the reference to Archimedes, the appointed mason cut an Archimedean spiral at the bottom of the tombstone, which has none of the properties Bernoulli so admired in the logarithmic spiral. And, to add insult to injury, he missed the word ‘numero’ from the motto. Bloody builders – always the same…

Bernoulli’s considerations made an impression on me when I first read Maor’s book. The spira mirabilis as a symbol of fortitude and constancy in adversity, or of the human body restored to its perfect self even after death. But after reading the passage in its entirety, I find it even more beautiful and inspiring. And how about taking a picture of my son – simillimus filius patri – next to the spiral, before any of this had come back to my mind?

By the way, my son’s name is Maurits, like (but not named after) M. C. Escher.

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Aug 252017

When I arrived at the Drayton Arms, he was already there. He had contacted me a few days earlier and we had arranged to meet for a drink. He worked for a head hunting firm, focused only – he was keen to specify – on investment management. After the introductory chit-chat, I made it clear that I was not interested in a job offer, and he made it clear that his purpose was to present his services to my firm’s potential hiring needs. With that out of the way, the conversation moved on amiably, flowing from market conditions to value investing, Brexit and other world affairs.

Until at one point – I can’t remember how and why – we veered towards terrorism, and from there to 9/11. “Of course” – said Sandeep, with the casual air of someone who is sharing the obvious among world-savvy, knowledgeable people, “it was clearly an inside job”.

“What? What do you mean?” – I looked at him straight in the eyes.

“What? You don’t think so?” – Sandeep was genuinely taken aback by my sudden change of tone. Which, I agree, requires some explaining.

I have a Spinozan tolerance for freedom of opinion. It is the essence of Bayes: different priors, different information, or different interpretations of the same information, can give rise to different conclusions. This is obvious, and there is nothing wrong with it. But of course it doesn’t mean that anything goes. It means that, even when I have a strong view, I hold on to Cromwell’s rule and remain open to the possibility that, however high in my mind is the probability that I am right, I may be mistaken. As we know, hypothesis testing is the result of a tug of war between confirmative and disconfirmative evidence, which accumulates multiplicatively, leaving the possibility that, however overwhelming the evidence may be on one side, it may be annihilated by even one piece of conclusive evidence on the other. Another consequence of this framework is that, while I strive for certainty, I am comfortable with uncertainty: if neither side is strong enough to win the tug of war, there is nothing wrong with accepting that a hypothesis is only probably right, and therefore also probably wrong.

It is important to remember, however, that this only works insofar as one makes sure that evidence accumulation is as thorough as possible on both sides. This is easy to understand: there is no point gathering a lot evidence on one side while neglecting to do it on the other. One side will win nothing but a rigged game. But it is far from easy to do it in practice, as it requires fighting our natural tendency to succumb to the Confirmation Bias. The easier one side seems to be winning, the stronger should be our urge to reinforce the other side. It is by winning an ever tougher tug of war that we can aim to approach certainty.

This is an aptitude I have learned to nurture. The more I am convinced about something, the more I like to explore the other side, trying to distil its best arguments. If this succeeds in lowering my confidence, so be it: I feel richer, not poorer. And if it doesn’t, I am richer anyway, as I have built a clearer picture of what the other side stands on. This, after all, is what understanding means – distinct from justifying and, more so, from agreeing. The better one understands an argument, the easier it becomes to dismantle it and, perhaps, convince people on the other side to change their mind.

This is where sometimes I fail to keep my composure: when I face a conviction based on a pile of one-sided arguments, typically soaked in hyperbolic language, which blatantly misrepresents, disregards or belittles the other side. But what really gets on my nerves is a dirtier trick: when the balance of evidence is overtly on one side, the only way to overturn the verdict is to find – or, failing that, make up – a conclusive piece of evidence on the other side. This is the standard trick employed by conspiracy theorists: I call them Conclusionists, and the pit they fall into a conclusive evidence trap.

That’s what happened with Sandeep.

“Of course I don’t think so!” I replied. “How can you say such a … thing?” I asked, working on resisting my own adjectival overpouring. He looked at me with candid disbelief. How could I be so naïve? The web is full of information about it – he said. And when I asked him to give me an example, he explained: “Of course it is not in the usual places. You need to know where to look”.

Oh my God. One tends to imagine Conclusionists as showing some exterior signs of dimwittedness. But there he was, a perfectly nice, bright-looking guy, splattering such shocking bullshit. As he excused himself to the men’s room, I tried to collect myself. But failed miserably. “So Sandeep” I asked him as he came back, even before he could regain his seat “Who killed JFK? And what about those moon landings? And the Illuminati? It’s all down to Queen Elisabeth, eh?” I deserved a sonorous expletive. But Sandeep was a gentleman, and perhaps he had regretted his own condescension over his micturating interval. “I see your point” he smiled “I’m not saying that everything you find on internet is true. But…” At which point I grabbed the two seconds void and, after mumbling some sort of apology myself, I cleared the air with a liberating “Anyway…” followed by a question about salaries, as if the whole interlude had never happened. The conversation resumed its cordial tone and carried on for a while, until it was time to go. We departed with the inevitable “Let’s keep in touch”. I never heard from him since.

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Aug 212017

They arrived in the morning, bright and early. The dishwasher had been acting strangely, so I had finally called in the engineers to figure out what was going on. I like to fix these things myself around the house, but this time, after fiddling in vain for a few days, I had given up.

“‘morning, Sir – how can we help?” Doug, the senior of the duo, had the reassuring air of the expert who has seen it all.

“Well, this is what’s happening” I started, hopeful but sceptical that Doug would immediately find an obvious explanation. “The washing cycle does not end properly. As you can see, it stops in the middle, with water still lying at the bottom. It’s not the filter or anything like that” I added, making it clear that I knew my stuff. “Sometimes, after I open and close the door a couple of times, it restarts and goes on to the end. But other times, like today, it just stops”.

“Let’s take a look” said Doug, and with a nod and a whisper instructed his younger mate Trevor to check under the sink. At this point I left, one because the children had woken up and two because watching Trevor puffing and laying his giant tattooed belly on the cold marble floor was a bit too much so early in the morning. “Call me if you need me” I said. But I had hardly greeted the kids that Doug called me back. “Here it is, Sir” – the dishwasher was working again. “It was the connection to the water drain. It is shared with the washing machine and sometimes it can be a bit too much, you know. Anyway, we’ve changed it around and it should not happen again. But remember never to use the dishwasher and the washing machine at the same time”.

“Ow…kay” I said, trying to conceal my puzzlement and following Doug’s invitation to look under the sink at the result of Trevor’s manipulation. I couldn’t see any difference – and I had never used the two machines simultaneously. “Are you sure?” I wanted to ask, but I refrained – Doug looked very sure, and ready to leave. “Thank you very much” was all I could say. “Pleasure, Sir” said Doug, “it should be alright but we’re here if you need us. Have a good day”.

Alas, the little hope I had for a quick solution soon faded away. The dishwasher finished the cycle that Doug and Trevor had managed to restart, but the next one flopped again in the middle, as I found out the following morning. A little door banging helped it to the end, and so it did in next few days. But the whole process soon became increasingly irritating: sometimes everything worked fine, sometimes the machine stopped and restarted by itself as I entered the kitchen, and some other times I had to keep banging the door. A week later I called back.

“Sorry guys” I apologised on the phone as I explained that their fixing wasn’t working. “No worries, Sir. We’ll be there tomorrow early in the morning”. So that evening I started a new cycle, with the intent of showing them the result in the morning and creating the ideal conditions for a new assessment.

I got out of bed as they rang the bell. They came in and we walked to the kitchen. One, two, three: I opened the dishwasher door, ready to show them the usual stagnant pool of water. Et voilà: no water. This time the cycle had ended properly. “No problem at all, Sir” said Doug, helping to alleviate my evident embarrassment. “We’ll put it down as ‘Intermittent Malfunctioning'”.

As they left with what I couldn’t help interpreting as a wry smile of amusement, I started contemplating my life with an erratically faulty dishwasher. Sure enough, the stop and go resumed. But what was the point of calling them again? So I kept going for a while, banging and cursing. Until one day it all came to an end. No banging, no lights, nothing. The machine was completely dead, and an increasingly smelly sludge at the bottom left me no alternative to calling Doug once again, with a view to arranging for a replacement.

This time Doug came alone, and after a few fearful moments in which I was dreading a new mysterious restart, he declared death himself. He took away the wooden bar under the dishwasher and started fiddling with its feet, exploring ways to slide it out of its casing. I left him again, and again he soon called me back. “Here, Sir” – the dishwasher had come back to life. To my befuddlement and consternation, Doug offered a new explanation: “You see, Sir, it all has to do with the alignment of the feet. They have laid the machine on MDF – that’s not the correct way, they should have used a harder material. With time, the feet have sunk a bit into the wood, enough to misalign the door closing. That’s why banging works sometimes. I have now raised the feet a bit so it’s all back in line. If this doesn’t work, the next thing is to replace the door, but I will not do it myself – I tried it once, but the hinges snapped back and I almost lost my finger. Anyway, I don’t think it will be necessary. I believe I figured it out – it’s amazing how one keeps learning after all these years”.

Oh well. I didn’t know what to make of Doug’s new theory, but he had managed to raise my hopes a bit. Once again, I would have the evidence in the morning. But later in the day I received a phone call. It was an electrician, who explained that he had been instructed by Doug to look at the dishwasher’s plug and asked whether he could come in the afternoon for a check. I was confused – Doug had said nothing to me about the plug. But why not? The whole thing was starting to reveal an amusing side.

As the young electrician came in, I gave him an abridged version of the saga. He nodded, quite uninterested, and set out to slide out the dishwasher to reach for the plug, which he had figured out was right behind it. After a few minutes he called me back. “Here, have a look” he said, with a quiet smile. The plug was stuck to the rear of the dishwasher, its plastic back partially melted and fused into it:

The mystery was finally and completely solved. And, as in the best detective stories, the explanation was simple and totally unexpected. The plug, stuck to the back, would intermittently lose contact with the socket due to the dishwasher’s vibration in mid cycle. That’s why door banging helped – it restored contact, as sometimes did just walking back into the kitchen, as floor vibration was enough to produce the same effect. All the electrician had to do was to move the socket to the side panel and reinsert the plug there. A dishwasher that was about to be chucked away is now in perfect shape and flawlessly performing its wonders.

So much for Doug’s theories. He had first tried a routine explanation – one that would probably fit most similar cases – but received disconfirming evidence from me. He then got confirming evidence from his own observation – a treacherous occurrence in many circumstances. Then, when a new piece of disconfirming evidence arrived, he built a new theory around it that seemed to fit the facts. This was as wrong as the first – and even more so, as it lacked generality and was created on the spot.

To his credit, however, Doug was crucial to finding the truth. I don’t know why he didn’t tell me about the plug – maybe it was late lateral thinking, or maybe he had it in mind but didn’t want to spoil his new-fangled theory – or simply, with no Trevor around, he didn’t feel like going through the motions of sliding the machine out.

Be that as it may, Doug was a true scientist. The search for the truth proceeds neither by deduction nor by induction but – in Charles Sanders Peirce’s somewhat awkward phrasing – abduction. We test hypotheses to produce explanations and select those that provide the best explanation of the observed evidence. The key to the process is to be open to revising and possibly rejecting any explanation in the light of the observed evidence. But a true scientist goes further: he actively looks for evidence that would reject his best theory and only stops when he finds conclusive evidence. In our dishwasher tale – a true story – the fused plastic plug was a Smoking Gun: evidence that conclusively explained the dishwasher’s strange behaviour. Hence we say it was the cause of such behaviour. I sent the picture to Doug’s phone but got no reply – I can’t remember, but perhaps, unlike his owner, the phone is not a smart one.

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