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The 2025 Papal Conclave

A Secret History of Gambling on the Papacy

From Renaissance Rome to crypto prediction markets; why people have always bet on the choice of Pope.

“The Pope, please God, will be chosen in the conclave… and not in the marketplace”.
— Venetian ambassador Matteo Dandolo, 1550

The Eternal Wager

Each time the seat of Saint Peter falls vacant, the world holds its breath.

Inside the Sistine Chapel, cardinals in crimson file beneath the vaulted majesty of Michelangelo’s ceiling. The doors close. The locks turn. Tradition, prayer, and centuries of ritual converge in one of the most solemn and secretive acts on Earth: the papal conclave.

Outside, the scene is no less intense. Pilgrims gather in the square. Reporters set up their cameras. And in quieter corners of the city, and now across the internet, a very different kind of preparation unfolds: people begin to place their bets.

They consult experts. They study rumours. Some rely on geopolitics, others on gut instinct. In a practice dating back over 500 years, they wager not out of irreverence, but fascination, a desire to glimpse what comes next in an event shrouded in silence.

Because while the cardinals listen for the whisper of the Holy Spirit, the world outside listens to the murmurs of markets.

Why has this sacred event, one so rooted in divine discernment, long been mirrored by something so human as speculation? Why, across centuries, have people placed money, hopes, and reputations on predicting the next pope?

In this article, we’ll explore that question by tracing the remarkable history of papal betting:

  • How Renaissance Romans gathered around statues and apothecaries to trade odds on the next Holy Father;
  • How fake deaths, forged leaks, and panicked markets once engulfed the Eternal City;
  • How popes, furious at the intrusion, responded with arrests, bans, and eventually excommunication;
  • And how in 2025, long after papal betting was thought dead, it has returned in a new form: sleek, digital, and global, driven by cryptocurrency and blockchain-based prediction markets.

This is the story of the eternal wager, not a tale of blasphemy, but of enduring human curiosity. A story of how, when the Church chooses its next shepherd, the world cannot help but watch… and wonder.

The Origins: Where Politics, Providence, and Wagers Met

Long before the invention of modern financial markets, before stock exchanges or political polling, there was already a thriving culture of speculation in Europe, and some of its earliest targets were the popes themselves.

By the early 15th century, betting on the pope’s life or death had become so widespread that it drew official condemnation. In 1419, the Republic of Venice, no stranger to markets or intrigue, passed legislation banning all wagers on the pope’s health and went so far as to void any existing bets. The Venetian Senate feared not only the moral hazard of profiting from death but the dangerous intersection of faith, rumour, and finance.

Other city-states soon followed. Barcelona (1435) and Genoa (1467 and 1494) outlawed a specific financial instrument: life insurance policies on popes and other high-ranking Church figures. These policies were often framed as legitimate tools for merchants and creditors, especially those whose fortunes were tied to Vatican debts, but they increasingly became vehicles for gambling, often traded second-hand, with buyers hoping to cash in on a pontiff’s failing health.

The motivation wasn’t always cynical. In the deeply unstable world of 15th-century papal finance, a new pope could mean immediate changes in political alliances, appointments, and debt settlements. Businessmen who had lent money to the papacy during one reign might lose everything under the next. Betting on a pope’s death or replacement wasn’t just a game, it was often a hedge against financial catastrophe.

But as with all markets, where hedging exists, speculation follows. By the late 15th century, people weren’t just betting on when a pope would die. They were now placing wagers on who would succeed him.

The first clearly documented instance of gambling on a papal election comes from the September 1503 conclave, following the death of the infamous Pope Alexander VI (Rodrigo Borgia). Rome was tense, factions were deeply divided, and speculation was rampant.

In an early case of market insight beating conventional wisdom, Roman bettors correctly backed Cardinal Francesco Piccolomini as the likely winner. Despite political headwinds, he emerged as the compromise candidate and was elected Pope Pius III.

Though his papacy lasted a mere 26 days, the bet had been sound. The market had intuited what many inside the conclave soon realised: that amid bitter factionalism, the safest option was a neutral, aging, and respected figure. The betting public, perhaps tipped off by courtiers, or simply well attuned to Roman political dynamics, had read the situation with uncanny accuracy.

This moment marked a turning point. It wasn’t just a fluke win for gamblers, it was the beginning of a new cultural dynamic, one in which public speculation on sacred decisions became a regular, if unofficial, part of the papal process.

The conclave, sealed by tradition and prayer, now had a shadow twin: the street-level betting market.

The Roman Betting Boom

By the mid-1500s, Rome had become the beating heart of papal speculation. While the cardinals met in cloistered chapels to pray and vote, the people of Rome, especially those in the commercial hub of the Banchi, were running their own parallel conclave.

The Banchi was Rome’s financial district: a lively warren of banks, spice shops, pawnshops, and apothecaries, nestled just across the Tiber from the Vatican. It was here that the city’s sensali – brokers and bookmakers, many of them Florentine – set odds, took wagers, and tracked papal news like stock tickers.

Every papal illness, delay, or whispered alliance triggered a flurry of speculation. Bets were placed not only on who would be elected pope, but on a wide array of related ecclesiastical events:

  • Which cardinal would ascend to the throne of Peter;
  • How long the conclave would last;
  • How many ballots it would take;
  • When the pope would appoint new cardinals;
  • Even whether the Holy Father would make a rumoured trip to Bologna.

Bets were written up on printed tickets called polize or cedole, many of which were dated and stamped in Florence. These slips became de facto financial instruments, bought, sold, and exchanged on rumour and instinct.

Information was the currency, and in a pre-newspaper age, it flowed through handwritten newsletters known as avvisi, the Twitter threads of their day, and the city’s most famous outlet of political mockery: the statue of Pasquino, where satirical verses and gossip were posted for public amusement and scandal.

Wagering fever gripped all levels of Roman society. A tailor might place a small bet on his parish’s favoured cardinal. A nobleman might wager a hundred gold scudi with a banker. Priests, artisans, merchants, apothecaries, and even the servants of cardinals placed bets, sometimes on behalf of their masters, sometimes for themselves. According to records from 1590, at least 26% of those arrested for illegal betting were artisans. One tailor, caught in the act, pleaded innocence, claiming he had four children to feed and no money to gamble.

Even members of the College of Cardinals were not above the action. Venetian ambassadors reported wagers among cardinals involving perfumed gloves, amber rosaries, she-mules, and fine wine, tokens of honour and status as much as indulgence.

Some bettors struck it rich. In 1550, a merchant named Ceuli Banchieri bet big on Cardinal Gianmaria del Monte, soon elected Pope Julius III, and walked away with 20,000 scudi, a sum large enough to purchase an entire Roman palazzo.

But for every Ceuli, there was a cautionary tale. During the volatile 1590 conclave, five Florentine wool merchants wagered heavily on Cardinal Santa Severina. When he lost, all five went bankrupt. Ruins and riches hinged on each puff of smoke from the Sistine chimney.

This culture of betting transformed Rome during a sede vacante. In the absence of a pope, the Eternal City became a feverish information market. Each rumour shifted odds. Each avviso passed from hand to hand like a hot stock tip. For a moment, the power to guess, or fake, what came next felt as potent as the power to vote.

Rumours, Leaks, and Dead Men Betting

The betting was never just idle fun; it had real consequences.

By the mid-16th century, the line between rumour, espionage, and market manipulation had all but disappeared. The conclave was supposed to be a sealed chamber of divine discernment, but in practice, the seal was not secure. And where information leaked, money followed.

The Pole Surge That Almost Became a Coronation

In the 1549–50 conclave, the favourite was Cardinal Reginald Pole, an Englishman and spiritual heavyweight known for his integrity and ties to Queen Mary I. His odds soared so high in the Banchi that many—including cardinals’ own servants—believed his election was inevitable. Betting slips listed him as the overwhelming favourite, and brokers adjusted prices hourly as his name dominated Roman gossip.

So certain was the public of Pole’s election that some conclavists, the cardinal attendants, began dismantling the temporary wooden cells that housed the electors, a symbolic and logistical preparation for the arrival of the new pope.

Just two votes short of election, and on the point of being made Pope by acclamation, Pole insisted on waiting until he won the formal two-thirds majority.

Then, unexpectedly, the tide shifted.

A last-minute political block, led by late arriving French cardinals suspicious of Pole’s English ties, prevented his election by a narrow margin. After weeks of deadlock, it was Cardinal del Monte who emerged as the compromise candidate and was elected Pope Julius III.

For the betting public, the lesson was painful: markets could follow momentum, but they couldn’t predict last-minute intrigue. For those who had already sold their betting slips or spent their “winnings” in anticipation, it was ruinous.

The “Death” of Carafa

Things turned darker during the 1555 conclave. Cardinal Gian Pietro Carafa, known for his reformist zeal and staunch anti-Spanish politics, was one of the leading contenders. Then, one morning, he failed to appear for Mass.

That was all the Banchi needed.

Within hours, brokers were circulating the rumour that Carafa had died inside the conclave. His odds collapsed instantly. Traders who had wagered heavily on him scrambled to sell their slips for pennies on the scudo. In response, money rushed to other candidates perceived as “safely alive”.

But Carafa wasn’t dead, only indisposed. Days later, he was elected Pope and took the name Paul IV.

The backlash was immediate. The false rumour hadn’t just distorted the markets, it had threatened the integrity of the conclave itself. Riots nearly broke out in parts of the city, and the papal guard arrested several brokers accused of deliberately manipulating public perception for profit.

The scandal revealed how deeply intertwined the financial, political, and religious ecosystems of Rome had become. Betting was no longer a curiosity, it was a form of soft power.

The Paleotti Panic of 1590

Perhaps the most dramatic example came during the conclave of 1590, when the betting markets crowned a pope that never was.

Cardinal Gabriele Paleotti, a respected Bolognese prelate and veteran of the Council of Trent, suddenly surged in the odds. According to avvisi and eyewitness accounts, he reached 70% in the betting markets, a heavy favourite.

The response was chaotic and theatrical:

  • Candles were lit at St. Peter’s and other major churches;
  • His coat of arms was posted in the streets;
  • Militia were stationed outside his house in anticipation of celebrations;
  • News was even dispatched to other cities reporting his victory.

But Paleotti had not been elected.

When the truth emerged, the College of Cardinals reacted with alarm. They ordered the conclave walls to be physically reinforced, to prevent communication with the outside world. Conclavists were made to swear new oaths on the Bible, promising not to send news out of the chamber.

It turned out that two influential cardinals, Montalto and Sforza, secretly agreed to join forces in support of Niccolo Sfondrato, making fortunes betting on him at odds of 10-1 the day before he was elected as Pope Gregory XIV. Cardinal Alessandro Peretti di Montalto may well have needed the money – reputedly, he had an extravagant lifestyle, including keeping a permanent staff of musicians on call to play solos at his palace!

This episode showed how fragile the conclave’s secrecy was, and how much power the Banchi’s betting culture had accumulated.

When Markets Become Weapons

These weren’t isolated incidents. They were part of a pattern: a city that watched the conclave like a horse race, and a marketplace that profited from destabilising sacred events.

In a way, this was a forerunner to the modern concept of “information warfare”. The rumours didn’t just reflect public misunderstanding; they were deliberately planted, exaggerated, or withheld to shape expectations and control outcomes.

Some leaks were real, coming from friendly conclavists hoping to tip the odds in their patron’s favour. Others were fabricated entirely, spread by those who stood to win big if the market swung. Either way, the goal was the same: influence the betting to maximise returns.

By the end of the century, the papacy had had enough. These episodes were among the final catalysts for the crackdown that came in 1591, a crackdown that would criminalise betting on conclaves for the next three centuries.

The Crackdown: When Betting Became a Mortal Sin

By the close of the 16th century, papal authorities had lost patience.

The betting markets of Rome, once tolerated as colourful, even inevitable, outgrowths of civic enthusiasm, had become too bold, too visible, and too destabilising. Rumours of papal deaths, premature celebrations of non-elected cardinals, and market manipulation inside and outside the conclave had shaken public trust in the spiritual process.

The tipping point came in the aftermath of the chaotic conclaves of 1590, during which false leaks, dangerous misinformation, and open gambling had nearly turned the election of a pope into a civic panic.

In response, Pope Gregory XIV, a devout reformer and former canon lawyer, acted decisively. On 21 March, 1591, he issued the apostolic bull Cogit nos, a sweeping and unambiguous condemnation of all forms of gambling related to papal succession.

The bull declared that anyone who wagered on:

  • The outcome of a papal election,
  • The duration of a pope’s reign,
  • Or the promotion of cardinals

…would incur automatic excommunication and perpetual banishment from Church institutions.

But this wasn’t just spiritual warning, it came with teeth.

Within days, the Governor of Rome ordered police raids on the betting shops of the Banchi. Dozens of brokers were arrested, their records seized, their patrons questioned, and some even tortured to extract information about high-ranking clients. Investigators discovered that many of the city’s most respected merchants, bankers, and even servants of noble households had placed bets, some on behalf of cardinals themselves.

One newsletter writer at the time wryly noted that the crackdown would entangle and embrace many lords and several illustrious cardinals”, implying that Gregory’s net had caught far more than just street-level speculators.

Some of the arrested brokers fled the crackdown by taking refuge in noble sanctuaries, the only places in Rome where papal authority could not enforce arrests without permission. First, they hid in the palace of Cardinal Francesco Sforza, then in the houses of the powerful Orsini and Colonna families, before finally retreating to a vineyard owned by Paolo Sforza near Monte Cavallo. Even this choice had symbolic sting: Sforza himself had ordered their arrest.

These evasions were not just acts of survival; they exposed the contradictions of a system where spiritual authority, aristocratic privilege, and money collided.

While some betting continued underground, perhaps whispered in taverns or settled privately among friends, the public, organised, and institutionally tolerated system of papal gambling in the Banchi was gone.

The penalties were not just limited to Catholics in Rome. Cogit nos was a universal document of Church law: it applied to any Catholic, anywhere in the world. And though rarely enforced outside the Papal States, the threat of excommunication cast a long shadow. The bull remained in force for over three centuries, never formally rescinded until the sweeping reforms of Pope Benedict XV, who codified a new legal framework for the Church with the 1917 Code of Canon Law.

By then, the betting markets of the Renaissance had become a distant memory, replaced by other forms of speculation, but never entirely forgotten.

Resurrection: Modern Markets and Media Spectacle

The crackdown may have driven papal betting underground, but it didn’t stay buried forever.

By the 19th century, a new form of public speculation began to surface, not in the alleyways of Rome, but in the lotteries of a newly unified Italy. When Pope Pius IX died in 1878, Italians didn’t openly wager on his successor, but they flooded the state lottery with number combinations tied to his age, reign, and namea fusion of superstition, numerology, and low-stakes gambling that blurred the line between veneration and chance.

The fascination with betting on the pope’s death or replacement was no longer a uniquely Roman habit; it had become a European curiosity. By the early 20th century, British and Italian newspapers were publishing speculative coverage of papal elections. In 1903, following the death of Leo XIII, London bookmakers quietly accepted bets on who would succeed him, with Cardinal Rampolla and Cardinal Sarto (the future Pius X) among the favourites. Gambling is also recorded on the papal conclave of 1922.

Bookmaker odds for the 1958 conclave, while not widely promoted, show Cardinal Angelo Roncalli the 2-1 favourite. The odds were justified when Roncalli became Pope John XXIII.

Still, formal papal betting remained relatively small scale, and generally frowned upon, until the late 20th century.

In 1978, two popes died within months of each other. The world watched in awe, and so did the British betting industry. For the first time in decades, bookmakers openly offered odds on both conclaves: first following the death of Paul VI, and again just weeks later after the sudden death of John Paul I, whose 33-day reign shocked the Catholic world.

The response from the Church was swift, if limited. The Archbishop of Westminster, Basil Hume, issued a warning to British Catholics, urging them not to place bets on the election of the next pope. He saw it as inappropriate, even sacrilegious. But the warning had little effect—if anything, it stoked media interest and public curiosity.

In the event, both Pope John Paul I (Alberto Luciani) and John Paul II (Karol Wojtyla) were longshot winners.

The Internet Era: Betting Goes Global

Then came the internet.

In 2005, during the conclave following the death of John Paul II, papal betting was no longer a fringe curiosity, it was big business. Irish bookmaker Paddy Power accepted more than the equivalent of more than £200,000 in wagers, advertising odds on dozens of cardinals from every continent. The bookmaker called it “the biggest non-sports betting market of all time”.

At first, betting focused on the usual papabili—popular, media-savvy, politically connected cardinals. One of them, Joseph Ratzinger, began the race at 12–1. As the conclave neared, his odds surged to 3–1, fuelled by media speculation, insider analysis, and a growing sense that he was the natural heir to John Paul II’s intellectual and theological legacy.

When Ratzinger emerged from the conclave as Pope Benedict XVI, the markets exhaled. For once, the bookmakers had got it right.

But lightning didn’t strike twice.

In 2013, following Benedict’s shocking resignation, a fresh round of speculation began. Bookmakers offered odds on a field of global contenders: Cardinal Scola of Milan, Cardinal Ouellet of Canada, Cardinal Turkson of Ghana.

But the market got it wrong.

The man chosen by the cardinals, Jorge Mario Bergoglio, the quiet Jesuit archbishop of Buenos Aires, was listed as a distant longshot and ranked 15th or lower on most betting boards. At the opening of the conclave he was trading as long as 55-1 on the Betfair betting exchange. He was barely mentioned in the press, much less picked by pundits.

When the white smoke rose and Cardinal Bergoglio emerged as Pope Francis, bettors were stunned. The markets had misread the conclave entirely, despite La Stampa’s Vatican Insider column highlighting him as a leading contender in the closing stages of the conclave. Specifically, barely 30 minutes after the second outpouring of black smoke, La Stampa’s reporter, the highly respected Giacomo Galeazzi, had effectively named the three final contenders. One of these was Jorge Mario Bergoglio. The betting odds barely shimmered in response. The information was later confirmed as having been accurate. The others named, and more expected, were Cardinals Angelo Scola and Marc Ouellet.

What these two elections revealed is that papal betting is a peculiar mix of political analysis, superstition, and guesswork, and apparently some insider knowledge. Unlike elections or sports, conclaves offer no polling data, no campaign trails, and no open debates. Decisions are made ostensibly in secret, often at the last minute, and frequently driven by factors that are meant to be invisible to the outside world.

Yet people still bet, not because they expect certainty, but because they long to interpret what is believed by many to be unknowable.

2025: Papal Prediction in the Crypto Age

When Pope Francis passed in April 2025, the world paused in reverence, and then began to place its bets.

But this time, it didn’t happen in the backrooms of Roman spice shops or inside Irish betting houses. It happened online.

Within hours, the decentralised crypto-based prediction market Polymarket lit up with activity. Over $3 million was wagered in just the first 24 hours, rapidly rising above $12 million in the next few days, fuelled by speculation, social media, and the thrill of turning divine mystery into digital assets.

The candidates weren’t all unfamiliar, some had featured in 2013 and 2005, but the field was more global and diverse than ever before. The top names early in the market included:

  • Cardinal Pietro Parolin, the seasoned Vatican Secretary of State;
  • Cardinal Luis Antonio Tagle, the charismatic Filipino prelate and longtime favourite among progressives;
  • Cardinal Matteo Zuppi, the media-friendly Archbishop of Bologna;
  • Cardinal Peter Turkson of Ghana, long discussed as a possible first Black pope;
  • Cardinal Pierbattista Pizzaballa, the Franciscan Latin Patriarch of Jerusalem.

But the wagers didn’t stop with names.

Polymarket’s users could also stake on a range of outcome-based propositions, such as:

  • What continent will the next pope come from?
  • Will the next pope be under 70?
  • Will the next pope be under 5’8” tall?
  • Will the conclave end within five days?
  • Will the new pope take the name John, Paul, or something entirely new?
  • Will the next pope be Black?

These were not mere novelty bets, they were serious markets. Traders scrutinised every papal obituary, tweet from Vatican journalists, and even photos of cardinals arriving at the Domus Sanctae Marthae for clues. Influencers and armchair theologians shared their predictions alongside crypto traders and cultural critics.

The smart contracts guaranteed transparent resolution once white smoke rose and the Vatican announced Habemus Papam. But until then, the markets fluctuated, mirroring the energy of a Renaissance avviso passed from hand to hand.

Old Habits, New Tools

What was remarkable, however, is not how new it all felt, but how ancient the instincts remained.

Despite the modern ledgers and decentralised prediction protocols, the betting dynamics of 2025 are uncannily similar to those of 1550:

  • Whispers moved prices.
  • Misinformation surged and genuine information was ignored.
  • A large portion of the public, lacking deep political insight, placed bets on charisma, nationality, or prophecy.

The only real difference was that instead of a Florentine broker scribbling a name on parchment in the Banchi, users now clicked a button and transferred currency into smart contracts backed by crowdsourced liquidity pools.

And just as in centuries past, there will always be winners who cash out at the right momentand losers who watched their tokens burn in the digital wind.

Do the Markets Get it Right?

Yes, sometimes, but certainly not always.

  • 1503 – Pope Pius III: The Roman betting public correctly backed Cardinal Francesco Piccolomini in the September conclave. Though his reign lasted just 26 days, the market had read the factions well. He was the obvious compromise candidate when everyone else cancelled each other out.
  • 1958 – Pope John XXIII: Cardinal Angelo Roncalli was the 2-1 favourite, followed by Cardinals Agagianian and Ottaviani at 3-1, then Stefan Wyszynski and Giuseppe Siri at 4-1. The odds were justified when Roncalli took the name Pope John XXIII.
  • 2005 – Pope Benedict XVI: Despite early support in the market for Cardinal Arinze, bookmakers saw a surge in late bets on the German cardinal Joseph Ratzinger, whose role as theological heir to John Paul II was underestimated by political pundits but not by bettors paying attention to Vatican whispers. Bookmakers shortened his odds from 12–1 to 3–1 in the final days, and they were right.

In each case, the markets were less driven by divine intuition than by pragmatic analysis: who was electable, who had support across factions, and who could command two-thirds in the end.

Famous Misses:

  • 1549–50 – Reginald Pole: So favoured that attendants began tearing down conclave cells in anticipation of his coronation. He never made it to the throne. Political interference, especially from the French, derailed his momentum at the last moment.
  • 1555 – Gian Pietro Carafa: Brokers falsely declared him dead. His odds lengthened dramatically, until he unexpectedly emerged as Pope Paul IV. The rumour cost some investors dearly and nearly sparked riots.
  • 1590 – Gabriele Paleotti: Reached 70% in betting markets. His supporters began celebrating, lighting candles and posting his coat of arms around Rome. He was never elected. The market had been entirely misled by gossip and misinformation.
  • 1978 – Both John Paul I (Alberto Luciani) and John Paul II (Karol Wojtyla) were longshots in the betting, Cardinal Sergio Pignedoli was the 5-2 favourite in the first conclave before Luciani was ultimately elected as John Paul I. There was no clear favourite in the second conclave of 1978, but of those mentioned in the serious betting, such as Cardinals Ursi, Pappalardo, Poletti, Benelli, and Siri, Cardinal Wojtyla was not among them, eventually being elected after the eighth ballot.
  • 2013 – Jorge Mario Bergoglio: Nearly invisible in international media coverage, although identified as a leading contender during the conclave by La Stampa’s Vatican Insider column, and traded as a longshot with bookmakers and on the betting exchanges. When the white smoke appeared and Francis stepped onto the balcony, most bettors were shocked. La Stampa’s Giacomo Galeazzi was not.

Unlike political elections, there are no polls. No debates. Just cloistered deliberation among men who may change their minds several times in the process. Last-minute compromise is common, and the most obvious candidates, known as papabile, can become victims of their own front-runner status.

As one saying goes, He who enters the conclave as pope leaves as a cardinal”.

Even in the digital age, with livestream punditry and crypto-based betting, the conclave sometimes remains stubbornly unpredictable. Its solemn secrecy and the unpredictable nature of consensus make it, ironically, one of the least efficient betting markets in the world.

And yet, people keep trying.

Final Thoughts: Why We Bet on the Papacy

At its heart, betting on a papal conclave isn’t about irreverence, it’s about proximity. It’s about the deeply human urge to get closer to power, to prophecy, and to a moment in history that feels bigger than us all.

From the spice merchants of Renaissance Rome to modern crypto traders, people have always tried to read the smoke before it rises, to decipher patterns in rumour, nationality, and timing. It’s an act of participation, not in the spiritual process, but in the cultural spectacle that surrounds it.

For some, it’s just a game. For others, a kind of hopeful prophecy. But for all, it’s a reminder that faith and curiosity are not mutually exclusive.

The conclave remains silent, sacred, and sealed.

But outside, on the streets of Rome, in bookmakers’ offices, and across decentralised blockchains, the eternal wager continues.

Further reading on this topic is available here:

Leighton Vaughan Williams and David Paton (2015). Forecasting the Outcome of Closed-Door Decisions: Evidence from 500 Years of Betting on Papal Conclaves. Journal of Forecasting, 34 (5), 391-404.

file:///C:/Users/epa3willilv/Downloads/forcasting%20the%20outcome%20of%20closed%20door%20(1).pdf

When Should We Question Infinity?

Exploring an Ancient Paradox

A version of this article appears in Puzzles, Paradoxes, and Big Questions. By Leighton Vaughan Williams. Chapman & Hall/CRC Press. 2024.

INTRODUCTION

Born in the 5th century BC in Elea (a Greek colony in southern Italy), Zeno of Elea is one of the most intriguing figures in the field of philosophy. Zeno’s paradoxes are a set of problems generally involving distance or motion. While there are many paradoxes attributed to Zeno, the most famous ones revolve around motion and are extensively discussed by Aristotle in his work, ‘Physics’. These paradoxes include the Dichotomy paradox (that motion can never start), the Achilles and the Tortoise paradox (that a faster runner can never overtake a slower one), and the Arrow paradox (that an arrow in flight is always at rest). Through these paradoxes, Zeno sought to show that our common-sense understanding of motion and change was flawed and that reality was far more complex and counterintuitive.

The Achilles and the Tortoise paradox, as one example, uses a simple footrace to question our understanding of space, time, and motion. While it’s clear in real life that a faster runner can surpass a slower one given enough time, Zeno uses the race to craft an argument where Achilles, no matter how fast he runs, can never pass a tortoise that has a head start. This thought experiment forms a remarkable philosophical argument that challenges our perceptions of reality and creates a fascinating paradox that continues to engage scholars to this day.

These paradoxes might seem simple, but they invite us into deep philosophical waters, questioning our perception of reality and illustrating the complexity of concepts we take for granted like motion, time, and distance. In this way, Zeno’s contributions continue to have profound relevance in philosophical and scientific debates, encouraging us to critically explore the world around us.

THE PARADOX OF THE TORTOISE AND ACHILLES

In one version of this paradox, a tortoise is given a 100-metre head start in a race against the Greek hero Achilles. Despite Achilles moving faster than the tortoise, the paradox argues that Achilles can never overtake the tortoise. As Aristotle recounts it, ‘In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead’.

THE UNDERLYING INFINITE PROCESS

This paradox lies in the infinite process Zeno presents. When Achilles reaches the tortoise’s original position, the tortoise has already moved a bit further. By the time Achilles reaches this new position, the tortoise has again advanced. This sequence of Achilles reaching the tortoise’s previous position and the tortoise moving further seems to continue indefinitely, suggesting an infinite process without a final, finite step. Zeno argues that this eternal chasing renders Achilles incapable of ever catching the tortoise.

A MATHEMATICAL SOLUTION TO THE PARADOX

The resolution to Zeno’s paradox lies in the mathematical understanding of infinite series. Using a stylised scenario where Achilles is just twice as fast as the tortoise (it’s a very quick tortoise!), we define the total distance Achilles runs (S) as an infinite series: S = 1 (the head start of the tortoise) + 1/2 (the distance the tortoise travels while Achilles covers the head start) + 1/4 + 1/8 + 1/16 + 1/32 …

By mathematical properties of geometric series, this infinite series sums to a finite value. In other words, despite there being infinitely many terms, their sum is finite: S = 2. Hence, Achilles catches the tortoise after running 200 metres, demonstrating how an infinite process can indeed have a finite conclusion.

PHILOSOPHICAL IMPLICATIONS: IS AN INFINITE PROCESS TRULY RESOLVED?

Zeno’s paradoxes, while they might be resolved mathematically, open a Pandora’s box of philosophical questions, particularly concerning the nature of infinity and the real-world interpretation of mathematical abstractions. How can a seemingly infinite process with no apparent final step culminate in a finite outcome?

The Thomson’s Lamp thought experiment, proposed by philosopher James F. Thomson, provides an insightful analogy. Imagine you have a lamp that you can switch on and off at decreasing intervals: on after one minute, off after half a minute, on after a quarter minute, and so forth, with each interval being half the duration of the previous one. Mathematically, the total time taken for this infinite sequence of events is two minutes. However, a critical philosophical question emerges at the end of the two minutes: is the lamp in the on or off state?

This question is surprisingly complex. On the one hand, you might argue that the lamp must be in some state, either on or off. However, there is no finite time at which the final switch event takes place, given the infinite sequence of switching. Hence, the state of the lamp appears indeterminate, raising questions about the applicability of infinite processes in the physical world. More prosaically, of course, you may just have blown the bulb!

This conundrum mirrors the situation in Zeno’s paradox of Achilles and the Tortoise. Just as the state of Thomson’s Lamp after the two-minute mark seems ambiguous, so does the concept of Achilles catching the tortoise after an infinite number of stages. While mathematics gives us a definitive point at which Achilles overtakes the tortoise, the philosophical interpretation of reaching this point through an infinite process is not as clear-cut.

The Thomson’s Lamp thought experiment highlights that while we can use mathematical tools to deal with infinities, interpreting these results in our finite and discrete physical world can be philosophically challenging. It reminds us that philosophy and mathematics, while often harmonious, can sometimes offer different perspectives on complex concepts like infinity, sparking ongoing debates that fuel both fields.

ZENO’S PARADOXES, THE QUANTUM WORLD AND RELATIVITY

Zeno’s paradoxes, which have puzzled thinkers for millennia, find surprising echoes in the realms of quantum mechanics and the theory of relativity, two foundational components of modern physics. These paradoxes, originally aimed at challenging the coherence of motion and time, intersect with quantum and relativistic concepts in thought-provoking ways.

In quantum mechanics, the principle of superposition allows particles to exist in multiple states at once until observed. This phenomenon reflects the essence of Zeno’s Arrow Paradox, where an arrow in flight is paradoxically motionless at any instant. This comparison highlights how quantum theory disrupts traditional views on motion, suggesting that at a microscopic level, movement doesn’t conform to our standard or philosophical expectations.

Meanwhile, the theory of relativity introduces the concept of time dilation, where time appears to ‘slow down’ for an object moving at speeds close to the speed of light. This idea provides a modern perspective on Zeno’s Dichotomy Paradox, which argues that motion is impossible due to the infinite divisibility of time and space. Through relativity, we see that motion and time are relative, not absolute, concepts—illustrating a deep connection to Zeno’s philosophical challenges, even after over two millennia.

CONCLUSION: PHILOSOPHICAL DEBATE AND CONTEMPORARY RELEVANCE

Contemporary philosophers continue to grapple with Zeno’s paradoxes, not only as historical curiosities but also as fundamental challenges to our understanding of reality. These paradoxes force us to reconsider how we conceptualise time, space, and motion. They remind us that our intuitive grasp of the world is often at odds with its underlying complexities. In today’s world, where scientific and technological advancements continually push the boundaries of what we understand, Zeno’s paradoxes remain as relevant as ever, reminding us of the enduring power and limits of human reason and the ongoing journey to comprehend the universe in which we live.

The Birthday Paradox

An Exercise in Probability Magic

A version of this article appears in ‘Twisted Logic: Puzzles, Paradoxes, and Big Questions’, by Leighton Vaughan Williams. Chapman & Hall/CRC Press. 2024.

SIZE MATTERS

What is the minimum number of individuals that need to be present in the room for it to be more likely than not that at least two of them share a birthday? This is what the ‘Birthday Paradox’ (or ‘Birthday Problem’) seeks to solve.

For the sake of simplicity, let’s assume that all calendar dates have an equal chance of being someone’s birthday and let’s disregard the Leap Year occurrence of 29 February.

A BASIC INTUITION: ANALYSING THE ODDS

At first glance, you might think that the odds of two people sharing a birthday are incredibly low. In a group of just two people, the likelihood of them sharing a birthday is a mere 1/365. Why is that? We have 365 days in a year, hence there’s only one chance in 365 that the second person would have been born on the same specific day as the first person.

Now, let’s take a group of 366 people. In this case, it’s certain that at least one person shares a birthday with someone else, due to the simple fact that we only have 365 possible birthdays (ignoring Leap Years).

The initial intuition may suggest that the tipping point—the group size at which there’s a 50% chance of two individuals sharing a birthday—is around the midpoint of these two extremes. You may think it lies around a group size of about 180. However, the reality is surprisingly different, and the actual answer is much smaller.

THE CALCULATIONS: UNRAVELLING THE BIRTHDAY PARADOX

To understand the concept better, we need to dig deeper into the probabilities involved. Let’s consider a duo: Julia and Julian. Let’s assume that Julia’s birthday falls on 1 May. The chance that Julian shares the same birthday, assuming an equal distribution of birthdays across the year, is 1/365.

What about the probability that Julian doesn’t share a birthday with Julia? It’s simply 1 minus 1/365, or 364/365. This number illustrates the chance that the second person in a random duo has a different birthday than the first person.

Adding a third person into the mix changes things slightly. The chance that all three birthdays are different is the chance that the first two are different (364/365) multiplied by the probability that the third birthday is unique (363/365). So, the probability of three different birthdays equals (364/365) × (363/365).

As we expand the group, the calculations continue in a similar manner. The more people in the room, the greater the chance of finding at least two people sharing a birthday.

Consider a group of four people. The probability that four people have different birthdays is (364 × 363 × 362)/(365 × 365 × 365). To find the probability that at least two of the four share a birthday, we subtract this number from 1. Thus, the odds of having at least two people with the same birthday in a group of four are about 1.6%.

As the number of people in the room increases, the probability of at least two sharing a birthday grows:

With 5 people, the probability is 2.7%.

With 10 people, the probability is 11.7%.

With 16 people, the probability is 28.1%.

With 23 people, the probability is 50.5%.

With 32 people, the probability is 75.4%.

With 40 people, the probability is 89.2%.

THE PARADOX UNVEILED: IT’S NOT JUST ABOUT BIRTHDAYS

You might be wondering why we need just 23 people to reach a 50% chance of finding shared birthdays. This can be explained by how many possible pairs can be made in a group. In a group of 23, there are 253 unique pairs. Each of these pairs has a 1/365 chance of sharing a birthday, and all these possibilities add up. This is what makes the birthday problem so counterintuitive. Basically, when a large group is analysed, there are so many potential pairings that it becomes statistically likely for coincidental matches to occur.

This is a perfect demonstration of the concept of multiple comparisons and an example of the so-called ‘Multiple Comparisons Fallacy’.

The same reasoning applies to balls being randomly dropped into open boxes. Assume there is an equal chance that a ball will drop into any of the individual boxes, and there are 365 such boxes, into which 23 balls are randomly dropped. There is an an equal chance, we assume, that a ball will drop into any specific box. Now, there is just over a 50% chance in this scenario that there will be at least two balls in at least one of the boxes. Randomness produces more aggregation than intuition leads us to expect.

YOUR PERSONAL BIRTHDAY CHANCES: WHERE DO YOU STAND?

The reason for the paradox is that the question is not asking about the chance that someone shares your particular birthday or any particular birthday. It is asking whether any two people share any birthday.

While the birthday problem shows the increased likelihood of shared birthdays in a group, the chance that someone shares your birthday specifically is a different question.

In a group of 23 people, including yourself, the probability that at least one person shares your birthday is much lower than 50%—it’s about 6%. This is because there are only 22 potential pairings that include you.

Even in a group of 366 people, the probability that someone shares your specific birthday is only around 63%.

CONCLUSION: THE MAGIC OF PROBABILITY AND THE BIRTHDAY PARADOX

The Birthday Paradox reveals an intriguing counterintuitive fact about probability: a group of just 23 people has a greater than 50% chance of including at least two people who share the same birthday. It sheds light on the intricacies of probability by demonstrating how many opportunities there are for matches to occur, even in seemingly small groups. For example, if you can find out the birthdays of the 22 players at the start of a football game, and the referee, more than half of the time two of them will share a birthday.

This fascinating concept has applications way beyond birthdays. It’s also very important for the safety and performance of computer systems and online security. This idea helps specialists prevent and deal with issues that occur when data unexpectedly overlaps. Understanding the paradox is crucial, therefore, for those who design and secure computer systems, helping them to make these systems more reliable and efficient.

Nevertheless, it’s in the social setting of parties where the paradox becomes a delightful surprise. Next time you’re among friends or at any casual meet-up, consider introducing this paradox; you might just bring to life the unexpected magic of probability!

William Shakespeare

The Timeless Bayesian

When Should We Trust a Loved One? Exploring a Shakespearean Tragedy

OTHELLO: THE BACKGROUND

Created by William Shakespeare, ‘Othello’ is a play centred around four main characters: Othello, a general in the Venetian army; his devoted wife, Desdemona; his trusted lieutenant, Cassio; and his manipulative ensign, Iago. Iago’s plan forms the central conflict of the play. Driven by jealousy and a large helping of evil, Iago seeks to convince Othello that Desdemona is conducting a secret affair with Cassio. His strategy hinges on a treasured keepsake, a precious handkerchief which Desdemona received as a gift from Othello. Iago conspires successfully to plant this keepsake in Cassio’s lodgings so that Othello will later find it.

UNDERSTANDING OTHELLO’S MINDSET

Othello’s reaction to this discovery can potentially take different paths, depending on his character and mindset. If Othello refuses to entertain any possibility that Desdemona is being unfaithful to him, then no amount of evidence could ever change that belief.

On the other hand, Othello might accept that there is a possibility, however small, that Desdemona is being unfaithful to him. This would mean that there might be some level of evidence, however overwhelming it may need to be, that could undermine his faith in Desdemona’s loyalty.

There is, however, another path that Othello could take, which is to evaluate the circumstances objectively and analytically, weighing the evidence. But this balanced approach also has its pitfalls. A very simple starting assumption that he could make would be to assume that the likelihood of her guilt is equal to the likelihood of her innocence. That would mean assigning an implicit 50% chance that Desdemona had been unfaithful. This is known as the ‘Prior Indifference Fallacy’. If the prior probability is 50%, this needs to be established by a process better than simply assuming that because there are two possibilities (guilty or innocent), we can ascribe automatic equal weight to each. If Othello falls into this trap, any evidence against Desdemona starts to become very damning.

THE LOGICAL CONTRADICTION APPROACH

An alternative approach would be to seek evidence that directly contradicts the hypothesis of Desdemona’s guilt. If Othello could find proof that logically undermines the idea of her infidelity, he would have a solid base to stand on. However, there is no such clear-cut evidence, leading Othello deeper into a mindset of anger and suspicion.

BAYES’ THEOREM TO THE RESCUE

Othello might seek a strategy that allows him to combine his subjective belief with the new evidence to form a rational judgement. This is where Bayes’ theorem comes in. Bayes’ theorem allows, as we have seen in previous chapters, for the updating of probabilities based on observed evidence. The theorem can be expressed in the following formula:

Updated probability = ab/[ab + c (1 − a)]

In this formula, a is the prior probability, representing the likelihood that a hypothesis is true before encountering new evidence. b is the conditional probability, describing the likelihood of observing the new evidence if the hypothesis is true. And finally, c is the probability of observing the new evidence if the hypothesis is false. In this case, the evidence is the keepsake in Cassio’s lodgings, and the hypothesis is that Desdemona is being unfaithful to Othello.

APPLYING BAYES’ THEOREM TO OTHELLO’S DILEMMA

Now, before he discovers the keepsake (new evidence), suppose Othello perceives a 4% chance of Desdemona’s infidelity (a = 0.04). This represents his prior belief, based on his understanding of Desdemona’s character and their relationship. Of course, he is not literally assigning percentages, but he is doing so implicitly, and here we are simply making these explicit to show what might be happening within a Bayesian framework.

Next, consider the probability of finding the keepsake in Cassio’s room if Desdemona is indeed having an affair. Let’s assume that Othello considers there is a 50% chance of this being the case (b = 0.5).

Finally, what is the chance of finding the keepsake in Cassio’s room if Desdemona is innocent? This would in Othello’s mind require an unlikely series of events, such as the handkerchief being stolen or misplaced, and then ending up in Cassio’s possession. Let’s say he assigns this a low probability of just 5% (c = 0.05).

BAYESIAN PROBABILITIES: WEIGHING THE EVIDENCE

Feeding these values into Bayes’ equation, we can calculate the updated (or posterior) probability of Desdemona’s guilt in Othello’s eyes, given the discovery of the keepsake. The resulting probability comes out to be 0.294 or 29.4%. This suggests that, after considering the new evidence, Othello might reasonably believe that there is nearly a 30% chance that Desdemona is being unfaithful.

IAGO’S MANIPULATION OF PROBABILITIES

This 30% likelihood might not be high enough for Iago’s deceitful purposes. To enhance his plot, Iago needs to convince Othello to revise his estimate of c downwards, arguing that the keepsake’s presence in Cassio’s room is a near-certain indication of guilt. If Othello lowers his estimate of c from 0.05 to 0.01, the revised Bayesian probability shoots up to 67.6%. This change dramatically amplifies the perceived impact of the evidence, making Desdemona’s guilt appear significantly more probable.

DESDEMONA’S DEFENCE STRATEGY

On the other hand, Desdemona’s strategy for defending herself could be to challenge Othello’s assumption about b. She could argue that it would be illogical for her to risk the discovery of the keepsake if she were truly having an affair with Cassio. By reducing Othello’s estimate of b, she can turn the tables and make the presence of the keepsake testimony to her innocence rather than guilt.

CONCLUSION: THE TIMELESS BAYESIAN

Shakespeare’s ‘Othello’ was written about a century before Thomas Bayes was born. Yet the complex interplay of trust, deception, and evidence in the tragedy presents a classic case study in Bayesian reasoning.

Shakespeare was inherently Bayesian in his thinking. The tragedy of the play is that Othello was not!

Bayes and the Reliability of Evidence

Exploring Bayes’ Theorem Through a Story

A version of this article appears in Twisted Logic: Puzzles, Paradoxes, and Big Questions. By Leighton Vaughan Williams. Chapman & Hall/CRC Press. 2024.

UNDERSTANDING EVIDENCE THROUGH A STORY

Imagine a situation where your friend, known for her upstanding character, is accused of vandalising a shop window. The only evidence against her is a police officer’s identification. You know her well and find it hard to believe she would commit such an act.

SETTING THE STAGE FOR BAYESIAN ANALYSIS

In Bayesian terms, the ‘condition’ is your friend being accused, while the ‘hypothesis’ is that she’s guilty. To apply Bayes’ theorem, we consider three probabilities:

Prior Probability: Based on your knowledge of your friend, you might initially think there’s a low chance of her guilt, say 5%. This is your ‘prior’—your belief before considering the new evidence.

Likelihood of Evidence If Guilty: Consider how reliable the officer’s identification is. If your friend were guilty, what’s the chance that the officer would identify her correctly? Let’s estimate this at 80%.

Likelihood of Evidence If Innocent: What’s the chance of the officer mistakenly identifying your friend if she’s innocent? Factors like similar appearances or biases could play a role. Let’s estimate this at 15%.

THE ITERATIVE NATURE OF BAYESIAN UPDATING

Bayes’ theorem allows for continual updates. If new evidence arises, you can recalculate, using your updated belief as the new ‘prior’. This process offers a dynamic way to assess situations as they evolve.

WHEN EVIDENCE DOESN’T ADD UP

In cases where evidence is equally likely whether the hypothesis is true or false, it doesn’t change our belief. It’s crucial to evaluate the quality of evidence, not just its existence.

CHALLENGES IN ASSIGNING PROBABILITIES

While assigning precise probabilities to real-life situations can be challenging, the exercise is invaluable. It forces us to think critically and systematically about our beliefs and how new information affects them.

The Unfolding Story

Now let’s consider the story in a little more detail. You’ve received a phone call from your local police station. An officer tells you that your friend, someone you’ve known for years, is currently assisting the police in their investigation into a case of vandalism. The crime in question involves a shop window that was smashed on a quiet street, close to where she resides. Furthermore, the incident took place at noon that day, which happens to be her day off work.

You had heard about the incident, but had no reason to believe your friend was involved. After all, she’s not a person known for reckless or unlawful behaviour.

However, this is where the narrative takes a twist. Your friend comes to the phone and tells you that she’s been charged with the crime. The accusation primarily stems from the assertion of a police officer who has positively identified her as the offender. There’s no other evidence, such as CCTV footage or eyewitness testimonies, to substantiate the officer’s claim.

She vehemently maintains her innocence, insisting it’s a case of mistaken identity.

The Challenge

Now, as a follower of Bayes as well as being a close friend, you find yourself in a position where you need to evaluate the probability that she has committed the crime before deciding how to advise her. This challenge leads us to the central theme of this section—the application of Bayes’ theorem to real-life situations.

Before we proceed, let’s clarify our terms. The ‘condition’ in this context is that your friend has been accused of causing the criminal damage. The ‘hypothesis’ we aim to assess is the probability that she is indeed guilty.

Bayes’ Theorem and Its Application

So, how does Bayes’ theorem help us answer this question? Well, Bayes’ theorem is a formula that describes how to update the probabilities of hypotheses being true when given new evidence. It follows the logic of probability theory, adjusting initial beliefs based on the weight of evidence.

To apply Bayes’ theorem, we need to estimate three crucial probabilities:

Prior probability (‘a’)

The prior probability refers to the initial assessment of the hypothesis being true, independent of the new evidence. In this scenario, it equates to the likelihood you assign to your friend being guilty before you hear the evidence.

Considering you’ve known her for years and her involvement in such an act is uncharacteristic, you might deem this probability low. After a thoughtful consideration of your friend’s past actions and character, allowing for the fact that she was off work on that day and in the neighbourhood, let’s say you assign a 5% chance (0.05) to her being guilty.

Assigning this prior probability requires an honest evaluation of your initial beliefs, unaffected by the newly received information.

Conditional probability of evidence given hypothesis is true (‘b’)

Next, you need to estimate the likelihood that the new evidence (officer’s identification) would have arisen if your friend were indeed guilty.

This estimate might be guided by factors such as the officer’s reliability, credibility, and proximity to the crime scene. For the sake of argument, let’s estimate this probability to be 80% (0.8).

Conditional probability of evidence given hypothesis is false (‘c’)

The third estimate involves figuring out the probability that the new evidence would surface if your friend is innocent. This entails gauging the chance that the officer identifies your friend as the offender when she isn’t guilty.

The probability could be influenced by several factors—perhaps the officer saw someone of similar age and appearance, jumped to conclusions, or has other motivations. For the purposes of our discussion, let’s estimate this probability to be 15% (0.15).

Probabilities Adding Up

An interesting point to note is that the sum of probabilities ‘b’ and ‘c’ doesn’t necessarily have to equal 1. Just for example, the police officer might have a reason to identify your friend either way (whether she’s guilty or innocent), in which case the sum of ‘b’ and ‘c’ could exceed 1. Alternatively, the officer may be reluctant to positively identify a suspect in any circumstance unless he is absolutely certain; in which case b plus c may well sum to rather less than 1. In this particular narrative, b plus c add up to 0.95.

Calculation and Interpretation

With these estimates in hand, we can now apply Bayes’ theorem, which calculates the posterior probability (the updated probability of the hypothesis being true after considering new evidence) using the formula: ab/[ab + c (1 − a)].

In our case, substituting the values results in a posterior probability of around 21.9%. What does this mean? Despite the officer’s confident identification (a seemingly strong piece of evidence), there’s only a 21.9% probability that your friend is guilty given the current information.

This result may seem counterintuitive. However, this discrepancy arises from our understanding of prior probability and the weight we assign to the new evidence. We must remember that the officer’s identification is only one piece of the puzzle, and its strength as evidence is balanced against the prior probability and the potential for a false identification.

Updating the Probability

The beauty of Bayes’ theorem lies in its iterative nature. Let’s suppose that another piece of evidence emerges—say, a second witness identifies your friend as the culprit. You can reapply Bayes’ theorem, using the posterior probability from the previous calculation as the new prior probability. This iterative process allows you to incorporate additional pieces of evidence, each of which updates the probability you assign to your friend’s guilt or innocence.

Cases Where Evidence Adds No Value

Consider a situation where ‘b’ equals 1 and ‘c’ also equals 1. This would imply that the officer would identify your friend as guilty whether she was or not. In such cases, the identification fails to update the prior probability, and the posterior probability remains the same as the initial prior probability.

The Imperfections of Assigning Probabilities

Now, it’s worth recognising the potential difficulty in assigning precise probabilities to real-life situations. After all, our scenario involves complex human behaviour and a unique event.

However, our inability to determine precise probabilities shouldn’t lead us to dismiss the process. In fact, this process of estimation is what we’re doing implicitly when we evaluate situations in our everyday lives.

While the results might not be perfect, Bayes’ theorem provides a systematic approach to updating our beliefs in the face of new evidence.

CONCLUSION: BAYESIAN REASONING IN REAL LIFE

Bayes’ theorem provides a structured approach to incorporating new evidence into our beliefs. It’s a tool that enhances our decision-making, offering a mathematical framework to navigate uncertainties, from everyday dilemmas to complex legal and medical decisions.

As we grapple with uncertainty, the application of Bayes’ theorem allows us to transition from ignorance to knowledge, systematically and rationally. Thus, whether we’re faced with a shattered shop window or any other challenging situation, we have a powerful tool to help us navigate our path towards truth.

When Should We Trust the Jury?

Exploring a Courtroom Tragedy

A version of this article appears in Twisted Logic: Puzzles, Paradoxes, and Big Questions. By Leighton Vaughan Williams. Chapman and Hall/CRC Press, 2024.

THE CONVICTION

In the final weeks of the 20th century, a lawyer named Sally Clark was convicted of the murder of her two infant sons. Despite being a woman of good standing with no history of violent behaviour, Clark was swept up in a whirlwind of accusations, trials, and appeals that would besmirch the criminal justice system and cost her dearly.

THE INVESTIGATION AND TRIAL—BUILDING A CASE ON UNCERTAINTY

The deaths of Clark’s two children were initially assumed to be tragic instances of Sudden Infant Death Syndrome (SIDS), a cause of infant mortality that was not well understood even by medical experts. However, the authorities became suspicious of the coincidental deaths, leading to Clark’s eventual trial. As the investigation evolved, it subsequently transpired that numerous pieces of evidence helpful to the defence were withheld from them.

STATISTICAL EVIDENCE—THE MISINTERPRETATION

The prosecution presented a piece of seemingly damning statistical evidence during Clark’s trial. One of their witnesses, a paediatrician, asserted that the probability of two infants from the same family dying from SIDS was incredibly low—approximately 1 in 73 million. He compared the odds to winning a bet on a longshot in the iconic Grand National horse race four years in a row.

THE PROSECUTOR’S FALLACY—THE DANGEROUS CONFLATION OF PROBABILITIES

The flaws in the statistical argument presented at the trial were both substantial and consequential. The paediatrician had mistakenly assumed that the deaths of Clark’s children were unrelated, or ‘independent’ events. This assumption neglects the potential for an underlying familial or genetic factor that might contribute to SIDS.

Moreover, the paediatrician’s argument represents a common misinterpretation of probability known as the ‘Prosecutor’s Fallacy’. This fallacy involves conflating the probability of observing specific evidence if a hypothesis is true, with the probability that the hypothesis is true given that evidence. These are two very different things but easy for a jury of laymen to confuse.

THE PROSECUTOR’S FALLACY EXPLAINED

This fallacy arises from confusing two different probabilities:

The probability of observing specific evidence (in this case, two SIDS deaths) if a hypothesis (Clark’s guilt) is true.

The probability that the hypothesis is true given the observed evidence.

THE NEED FOR COMPARATIVE LIKELIHOOD ASSESSMENT

The Royal Statistical Society emphasised the need to compare the likelihood of the deaths under each hypothesis—SIDS or murder. The rarity of two SIDS deaths alone doesn’t provide sufficient grounds for a murder conviction.

PRIOR PROBABILITY—UNDERSTANDING THE LIKELIHOOD OF GUILT BEFORE THE EVIDENCE

Prior probability—a concept integral to understanding the Prosecutor’s Fallacy—is often overlooked in court proceedings. This term refers to the probability of a hypothesis (in this case, that Sally Clark is a child killer) being true before any evidence is presented.

Given that she had no history of violence or harm towards her children, or anyone else, or any indication of such a tendency, the prior probability of her being a murderer would be extremely low. In fact, the occurrence of two cases of SIDS in a single family is much more common than a mother murdering her two children.

The jury should weigh up the relative likelihood of the two competing explanations for the deaths. Which is more likely? Double infant murder by a mother or double SIDS?

More generally, it is likely in any large enough population that one or more cases of something highly improbable will occur in any particular case.

In a letter from the President of the Royal Statistical Society to the Lord Chancellor, Professor Peter Green explained the issue succinctly:

The jury needs to weigh up two competing explanations for the babies’ deaths: SIDS or murder. The fact that two deaths by SIDS is quite unlikely is, taken alone, of little value. Two deaths by murder may well be even more unlikely. What matters is the relative likelihood of the deaths under each explanation, not just how unlikely they are under one explanation.

Put another way, before considering the evidence, the prior probability of Clark being a murderer, given her background and lack of violent history, was extremely low. The probability of two SIDS deaths in one family, while rare, was still much higher than the likelihood of the mother murdering her two children.

The jury should weigh up the relative likelihood of the two competing explanations for the deaths. Which is more likely? Double infant murder by a mother or double SIDS?

More generally, it is likely in any large enough population that one or more cases of something highly improbable will occur in any particular case.

In a letter from the President of the Royal Statistical Society to the Lord Chancellor, Professor Peter Green explained the issue succinctly:

The jury needs to weigh up two competing explanations for the babies’ deaths: SIDS or murder. The fact that two deaths by SIDS is quite unlikely is, taken alone, of little value. Two deaths by murder may well be even more unlikely. What matters is the relative likelihood of the deaths under each explanation, not just how unlikely they are under one explanation.

Put another way, before considering the evidence, the prior probability of Clark being a murderer, given her background and lack of violent history, was extremely low. The probability of two SIDS deaths in one family, while rare, was still much higher than the likelihood of the mother murdering her two children.

THE NEED FOR COMPARATIVE LIKELIHOOD ASSESSMENT

The Royal Statistical Society emphasised the need to compare the likelihood of the deaths under each hypothesis—SIDS or murder. The rarity of two SIDS deaths alone doesn’t provide sufficient grounds for a murder conviction.

The Fictional Case of Lottie Jones

To illustrate the Prosecutor’s Fallacy, consider the fictional case of Lottie Jones, charged with winning the lottery by cheating. The fallacy occurs when the expert witness equates the low probability of winning the lottery (1 in 45 million) with the probability that a lottery win was achieved unfairly.

As in the Sally Clark case, the prosecution witness in this fictional parody commits the classic ‘Prosecutor’s Fallacy’. He assumes that the probability Lottie is innocent of cheating, given that she won the Lottery, is the same thing as the probability of her winning the Lottery if she is innocent of cheating. The former probability is astronomically higher than the latter unless we have some other indication that Lottie has cheated to win the Lottery. It is a clear example of how it is likely, in any large enough population, that things will happen that are improbable in any particular case. In other words, the 1 in 45 million represents the probability that a Lottery entry at random will win the jackpot, not the probability that a player who has won did so fairly!

Lottie just got very, very lucky just as Sally Clark got very, very unlucky.

THE AFTERMATH—TRAGEDY AND LESSONS LEARNED

Following her acquittal in 2003, Sally Clark never recovered from her ordeal and sadly died just a few years later. Her story stands as testament to the potential for disastrous consequences when statistics are misunderstood or misrepresented.

O.J. SIMPSON—AN ALTERNATE SCENARIO

Even in high-profile cases, such as American former actor and NFL football star O.J. Simpson’s murder trial in the 1990s, this same misinterpretation of statistics is prevalent. Simpson’s defence team argued that it was unlikely Simpson killed his wife because only a small percentage of spousal abuse cases result in the spouse’s death. This argument, though statistically accurate, overlooks the relevant information—the fact that about 1 in 3 murdered women were killed by a spouse or partner. This represents a very clear case of the misuse of the Inverse or Prosecutor’s Fallacy in argumentation before a jury.

CONCLUSION: THE IMPORTANCE OF STATISTICAL LITERACY

The importance of statistics in our justice system cannot be overstated. We must recognise the potential for misinterpretation and the potentially devastating results. A concerted effort to promote statistical literacy, particularly within our legal systems, can hopefully go a long way in preventing future miscarriages of justice.

When Should We Believe the Diagnosis?

Exploring the World of False Positives

A version of this article appears in Twisted Logic: Puzzles, Paradoxes, and Big Questions. By Leighton Vaughan Williams. Chapman & Hall/CRC Press. 2024.

THE FLU TEST SCENARIO: SETTING THE STAGE

Imagine this scenario: you twist your knee in a skateboarding mishap and decide to visit your doctor to have it looked at, just to be on the safe side. At the surgery, they run a routine test for a flu virus on all their patients, based on the estimate that about 1 out of every 100 patients visiting them will have the virus. This flu test is known to be pretty accurate—it gets the diagnosis right 99 out of 100 times. In other words, it correctly identifies 99% of people who are sick as sick, and equally importantly, it correctly clears 99% of those who don’t have the flu virus.

Now, you take the test, and to your surprise, it comes back positive. What does this mean for you, exactly? You dropped in to have your knee looked at, and now it seems you have the flu.

To summarise the situation, imagine you’ve twisted your knee and, while at the doctor’s office, you’re given a routine flu test. The test is 99% accurate and is positive. But what are the actual chances that you have the flu? This scenario is perfect for exploring Bayes’ theorem and understanding false positives.

BREAKING DOWN THE INVERSE FALLACY

Here, we step into the tricky territory of probabilities, a place where common sense can often mislead us. So, what is the chance that you do have the virus?

The intuitive answer is 99%, as the test is 99% accurate. But is that right?

The information we are given relates to the probability of testing positive given that you have the virus. What we want to know, however, is the probability of having the virus given that you test positive. This is a crucial difference.

Common intuition conflates these two probabilities, but they are very different. If the test is 99% accurate, this means that 99% of those with the virus test positive. But this is NOT the same thing as saying that 99% of patients who test positive have the virus. This is an example of the ‘Inverse Fallacy’ or ‘Prosecutor’s Fallacy’. In fact, those two probabilities can diverge markedly.

To summarise, common sense might suggest a 99% chance of having the flu, aligning with the test’s accuracy. However, this confuses the probability of testing positive when having the flu with the probability of having the flu when testing positive—a common mistake known as the ‘Inverse Fallacy’.

So what is the probability you have the virus if you test positive, given that the test is 99% accurate? To answer this, we can use Bayes’ theorem.

APPLYING BAYES’ THEOREM

Bayes’ theorem, as we have seen, uses three values:

Your initial chance of having the flu before taking the test, which in our scenario was estimated to be 1 out of 100 or 0.01.

The likelihood of the test showing a positive result if you have the flu, which we know to be 99% or 0.99 based on the accuracy of the test.

The likelihood of the test showing a positive result if you don’t have the flu, which is 1% or 0.01, again based on the accuracy of the test.

When we plug these into Bayesian formula, we end up with a surprising result. If you test positive for the flu, despite the test being 99% accurate, there’s actually only a 50% chance that you really have it.

In other words, to find the real probability of having the flu, we consider:

Prior Probability: Your initial chance of having the flu is 1% (1 in 100).

True Positive Rate: The test correctly identifies the flu 99% of the time.

False Positive Rate: The test incorrectly indicates flu in healthy individuals 1% of the time.

The formula is expressed as follows:

ab/[ab + c (1 − a)]

where

a is the prior probability, i.e. 0.01,

b is 0.99.

c is 0.01.

Using Bayes’ theorem, we find a surprising result: even with a 99% accurate test, there’s only a 50% chance you have the flu after a positive result.

GRAPPLING WITH PROBABILITIES

The result can seem counterintuitive, and it’s worth taking a moment to understand why that is. The key is to remember that the flu is a relatively rare occurrence—only 1 in 100 patients have it. While the test may be 99% accurate, we have to take into account the relative rarity of the disease in those who are tested. The chance is just 1 in 100. The chance of having the flu before taking the test and the chance of the test making an error are both, therefore, 1 in 100. These two probabilities are the same, and so, when you test positive, the chance that you have the flu is actually just 1 in 2.

It is basically a competition between how rare the virus is and how rarely the test is wrong. In this case, there is a 1 in 100 chance that you have the virus before taking the test, and the test is wrong one time in 100. These two probabilities are equal, so the chance that you have the virus when testing positive is 1 in 2, despite the test being 99% accurate.

Put another way, the counterintuitive outcome arises because the flu is relatively rare (1 in 100), balancing against the test’s accuracy.

THE IMPLICATION OF SYMPTOMS AND PRIOR PROBABILITIES

This calculation changes if we add in some more information. Let’s say you were already feeling unwell with flu-like symptoms before the test. In this case, your doctor might think you’re more likely to have the flu than the average patient, and this would increase your ‘prior probability’. Consequently, a positive test in this context would be more indicative of actually having the flu, as it aligns with both the symptoms and the test result.

In this way, Bayes’ theorem incorporates both the statistical likelihood and real-world information. It’s a powerful tool to help us understand probabilities better and to make informed decisions. The bottom line, though, is that while a positive test result can be misinterpreted, it should, especially in conjunction with symptoms, be taken seriously.

The Role of Symptoms in Adjusting Probabilities

If you had flu-like symptoms before the test, this would increase your ‘prior probability’. Consequently, a positive test in this context would be more indicative of actually having the flu, as it aligns with both the symptoms and the test result.

CONCLUSION: THE BROAD APPLICATION OF BAYESIAN THINKING

While we’ve used the example of a flu test, the principles of Bayes’ theorem apply beyond the doctor’s door. From the courtroom to the boardroom, from deciding if an email is spam to weighing up the reliability of a rumour, we often need to update our beliefs in the face of new evidence. Remember, a single piece of evidence should always be weighed against the broader context and initial probabilities.

Lucy Letby: Victim of Flawed Statistics?

Exploring the Texas Sharpshooter Fallacy

Further discussion of the flawed use of statistics in the Courtroom is available in Twisted Logic: Puzzles, Paradoxes, and Big Questions, by Leighton Vaughan Williams. Chapman & Hall/CRC Press. 2024, and also in Probability, Choice, and Reason, by the same author and publisher,

The Texas Sharpshooter Fallacy and the Lucy Letby Case: A Statistical Illusion?

The Texas Sharpshooter Fallacy is a cognitive bias where patterns are imposed on random data after the fact, creating the illusion of meaningful correlation. In criminal cases, this fallacy can lead to wrongful convictions when evidence is selectively framed to confirm a pre-existing hypothesis while ignoring contradictory data. In the case of Lucy Letby, did this fallacy play a significant role in shaping the prosecution’s argument?

Breaking Down the Fallacy: The “Barn Wall” of Hospital Deaths

Imagine a barn wall riddled with bullet holes.

  • A skilled sharpshooter carefully aims at a pre-drawn target and hits the bullseye. This represents a genuine patterna case where evidence is gathered before forming a conclusion.
  • A Texas sharpshooter, on the other hand, fires randomly at the barn, then paints a target around the densest cluster of bullet holes, claiming accuracy. This is a false pattern, created by selectively highlighting data that supports a conclusion while ignoring data that doesn’t.

The key mistake in the Texas Sharpshooter Fallacy is that the pattern is imposed after the data is already collected, rather than discovered through an objective analysis of all relevant information.

How This Applies to the Lucy Letby Case

1. The “Barn Wall” = All Neonatal Unit Deaths

  • The neonatal unit at the Countess of Chester Hospital experienced multiple infant deaths and collapses over a specific period.
  • The prosecution focused only on the subset of deaths and collapses that occurred during Letby’s shifts, effectively painting a target only after identifying her as a suspect.
  • This ignores other infant deaths and medical complications that occurred during the same period when Letby was not present, much like ignoring other bullet holes on the barn wall.

2. Painting the Target Around Letby

  • The prosecution used a chart showing that Letby was present at all the deaths/collapses for which she was charged.
  • However, at least six other deaths during the same period were excluded from this analysis because Letby was not present for them.
  • This selective focus creates a misleading illusion:
    • If Letby had been present for those deaths, they likely would have been included in the charges.
    • Because she was absent, they were ignored, despite potentially having the same medical causes as the deaths attributed to her.

This is a classic case of defining a pattern after seeing the data, rather than objectively analysing all neonatal deaths to determine if there was truly an unusual pattern.

Why This Statistical Error Matters

The Texas Sharpshooter Fallacy distorts the perception of probability and causation. In Letby’s case, it led to several key statistical misunderstandings:

1. Random Clustering Happens Naturally

  • In any high-risk medical environment, adverse events will cluster randomly without intentional wrongdoing.
  • Letby worked many shifts, increasing the likelihood that she would be present during multiple tragedies by chance alone.
  • The prosecution failed to show whether other nurses, working similar hours, might also have appeared in clusters if all deaths had been analysed.

2. Base Rate Neglect: Ignoring the Expected Frequency of Nurse Presence

  • The prosecution claimed that Letby’s presence at so many incidents was statistically improbable.
  • But how often were other nurses present for multiple collapses?
    • If most nurses worked 40% of shifts, but Letby worked 60%, she would naturally be present for more deaths.
    • Without comparing her shift pattern to other nurses, the statistical claim that her presence was “too unlikely to be coincidence” is unsubstantiated.

3. Confirmation Bias: Interpreting Evidence Through a Guilt-Focused Lens

  • Once Letby was identified as a suspect, investigators re-examined medical cases only from shifts she worked, looking for signs of wrongdoing.
  • This ignores cases with similar medical outcomes that occurred when she was not present.
  • If the same unexplained symptoms or medical complications were found in cases where Letby wasn’t working, the argument that she deliberately caused harm would weaken significantly.

4. The Prosecutor’s Fallacy: Misinterpreting Probability

  • The jury was told that the probability of Letby being present for all these deaths by chance was “1 in 3.5 million”.
  • This misleading argument makes two major mistakes:
    1. It assumes each death is an independent random eventwhen clusters happen naturally due to factors like seasonal infections, staffing levels, and equipment failures.
    2. It ignores alternative explanations, including poor hospital conditions and misdiagnosed medical complications, which might have been responsible for many of the deaths.

Expert Criticism: The Fallacy in Action

Several statisticians and medical experts have questioned the statistical reasoning behind Letby’s conviction:

  • Dr. Richard Gill (Former Chair of Mathematical Statistics, Leiden University): Argued that the prosecution’s statistical argument was a “classic Texas Sharpshooter” mistake, cherry-picking data and excluding deaths where Letby wasn’t present.
  • Prof. Jane Hutton (Professor of Statistics, Warwick University): Emphasised that all neonatal deaths should be analysed, not just a subset supporting the prosecution’s narrative.
  • Medical Experts: Pointed out that the hospital’s mortality rate remained high even after Letby was removed from duty, suggesting systemic failures rather than the actions of a single nurse.

The Danger of the Texas Sharpshooter Fallacy in Criminal Justice

The Letby case is a textbook example of why cherry-picked statistics can create false narratives in the courtroom.

  • Humans instinctively seek patterns, even in random data. When jurors see a chart where Letby’s name is the only one with multiple deaths, they may assume intent, even if the pattern is artificially constructed.
  • In ambiguous medical cases, statistical manipulation can override weak physical evidence and lead to wrongful convictions.
  • By focusing on Letby as a “bad actor”, the hospital avoids scrutiny over systemic failures in neonatal care, including understaffing, medical errors, and resource shortages.

The Bigger Picture: Does This Prove Letby’s Innocence?

The Texas Sharpshooter Fallacy does not prove Letby is innocentbut it does cast significant doubt on the prosecution’s statistical reasoning. When combined with:

  • Disputed medical evidence (e.g. air embolism diagnoses contradicted by experts).
  • No direct witnesses to wrongdoing.
  • A struggling hospital with a high infant mortality rate, even after Letby’s departure.

…it suggests that the “pattern” of Letby’s presence at deaths may have been artificially constructed rather than genuinely significant.

In Justice, as in Statistics, Correlation ≠ Causation

If the jury was swayed by a pattern that was painted after the fact, then Letby may have been convicted not on solid proof, but on a fallacy. This case serves as a cautionary tale: when statistics are weaponised in courtrooms, they must be scrutinised rigorously, because mistaking correlation for causation can cost an innocent person their life.

When Should We Believe the Eyewitness?

Bayes and the Taxi Problem

A Version of this article appears in Twisted Logic: Puzzles, Paradoxes, and Big Questions. By Leighton Vaughan Williams. Chapman & Hall/CRC Press. 2024.

THE BASICS OF THE TAXI PROBLEM

Let’s set the stage for our story. We’re in New Brighton, a city with a fleet of 1,000 taxis. Of these, 850 are blue and 150 are green. One day, a taxi is involved in an accident with a pedestrian and leaves the scene. We don’t know the colour of the taxi, and we don’t have any reason to believe that blue or green taxis are more likely to be involved in such incidents.

An independent eyewitness now comes forward. She saw the accident and claims that the taxi was green. To verify the reliability of her account, investigators conduct a series of observation tests designed to recreate the conditions of the incident. These tests reveal that she is correct about the colour of a taxi in similar conditions 80% of the time.

So, what is the likelihood that the taxi involved was actually green?

INITIAL PROBABILITIES AND INTUITIVE ESTIMATES

Your first instinct might be to believe that the chance that the taxi was green is around 80%. This assumption is based on the witness’s track record of identifying the colour of a taxi accurately. However, this conclusion doesn’t consider other crucial information—the overall number of blue and green taxis in the city.

Given the total taxi population, only 15% of them are green (150 out of 1,000), while a substantial 85% are blue. Ignoring this ‘base rate’ of taxi colours leads to a common mistake known as the ‘Base Rate Fallacy’.

APPLYING BAYES’ THEOREM TO THE TAXI PROBLEM

Bayes’ theorem is a method that helps us adjust our initial estimates based on new evidence but allowing for this base rate of the total numbers of blue and green taxis. In this way, it offers a means of updating our initial estimates after taking account of some new evidence.

For our Taxi Problem, the new evidence is the witness statement. The witness says the taxi was green, and we know that there’s an 80% chance that she is correct if the taxi was indeed green (based on her observation test). But there’s also a 20% chance that she would mistakenly say the taxi was green if it were blue.

Bayes’ theorem helps us adjust initial beliefs with new evidence, considering the base rate. Here’s how it works in the Taxi Problem:

Prior Probability: Initially, there’s only a 15% chance (150 out of 1,000 taxis) that the taxi is green.

Conditional Probability of Green Taxi (If Witness Correct): The eyewitness is correct 80% of the time.

Conditional Probability of Green Taxi (If Witness Incorrect): There’s a 20% chance the eyewitness would mistakenly identify a blue taxi as green.

After applying Bayes’ theorem, the adjusted (or ‘posterior’) probability that the taxi is green is just 41%, using the formula: ab/[ab + c (1 − a)].

THE ROLE OF NEW EVIDENCE AND MULTIPLE WITNESSES

What happens if another eyewitness comes forward? Suppose this second witness also reports that the taxi was green and, after a similar set of tests, is found to be correct 90% of the time. Now we should recalculate the probabilities using the same principles of Bayes’ theorem but including the new evidence.

The updated ‘prior’ probability is no longer the original 15%, but the 41% we calculated after hearing from the first witness. After running the numbers again, using Bayes’ formula, the revised probability that the taxi was green increases to 86%.

INTERPRETING WITNESS TESTIMONIES WITH BAYES’ THEOREM

Let’s dive a bit deeper into the implications of these results. Here are some situations that may seem counterintuitive at first, but make sense when we apply Bayes’ theorem:

The 50-50 Witness: Suppose we have a witness who is only right half the time—in other words, they are as likely to be right as they are to be wrong. Our intuition tells us that such a witness is adding no useful information, and Bayes’ theorem agrees. The testimony of such a witness doesn’t change our prior estimate.

The Perfect Witness: Now, imagine a witness who is always right—they have a 100% accuracy rate in identifying the taxi colour. In this case, if they say the taxi was green, then it must have been green. Bayes’ theorem concurs with this conclusion.

The Always-Wrong Witness: What about a witness who always gets the colour wrong? In this case, if they say the taxi is green, then it must have been blue. Bayes’ theorem agrees. We can trust this witness by assuming the opposite of what they say is the truth.

In summary, a 50% accurate witness adds no value to our estimate. A 100% accurate witness’s testimony is definitive. An always-wrong witness inversely confirms the truth.

THE BASE RATE FALLACY AND ITS IMPLICATIONS

The Base Rate Fallacy occurs when we don’t give enough weight to ‘base rate’ information (like the overall number of blue and green taxis) when making probability judgments. This mistake can lead us to overvalue specific evidence (like a single eyewitness account) and undervalue more general information like the ratio of blue to green taxis. Even so, the eyewitness may still be correct.

Again, if someone loves talking about books, we might intuitively guess that they are more likely to work in a bookstore or library than as, say, a nurse. But there are many more nurses than there are librarians or bookstore employees, and many of them love books. So, taking account of the base rate, we may well conclude that it’s more likely that the book enthusiast is a nurse than a bookstore employee or librarian.

AVOIDING THE BASE RATE FALLACY

The Base Rate Fallacy leads us to ignore general information (like the ratio of blue to green taxis or nurses to librarians) in favour of specific evidence (an eyewitness account or specific bit of information). It’s essential to balance specific and general information to avoid skewed judgments.

THE UNVEILING OF THE TRUTH

In the case of the New Brighton Taxi Problem, the mystery was solved when CCTV footage surfaced. The taxi involved was revealed to be yellow, a twist no one expected. Not really—there are no yellow taxis in New Brighton. In fact, both eyewitnesses were correct and the taxi was green.

CONCLUSION: TRUTH AND TESTIMONY

While our story was hypothetical, the principles it illustrates are very real and applicable in a wide variety of situations and circumstances. Bayes’ theorem, base rates, and new evidence are all important parts of the detective’s toolkit.

The Wonderful World of Mr. Bayes

An Exploration in Probability

A version of this article appears in TWISTED LOGIC: Puzzles, Paradoxes, and Big Questions, by Leighton Vaughan Williams. Published by Chapman & Hall/CRC press. 2024.

When Should We Update Our Beliefs?

Imagine emerging from a cave for the first time and watching the sun rise. You have never witnessed this before, and in this thought experiment, you are unable to tell whether it’s a regular occurrence, an infrequent event, or a once-in-a-lifetime happening.

As each day passes, however, you observe the dawn again and again: you gradually grow to expect it. With each sunrise, you become more confident that this is a regular event. With this growing confidence, you forecast that the sun will rise again the next day.

This is an illustration of what so-called Bayesian reasoning is about. Bayes’ theorem is a tool that allows us to adjust our understanding of the world based on our observations over time. It represents a process of continuous learning and understanding, pushing us gradually nearer to the truth as we are exposed to more experiences and to more information.

That’s the essence of Bayesian reasoning: adjusting our beliefs based on new information.

THE BIRTH OF BAYESIAN THINKING

The Bayesian perspective on the world can be traced to the Reverend Thomas Bayes, an 18th-century clergyman, statistician, and philosopher. The Bayesian approach advocated predicting future events based on past experiences. His ideas were in a fundamental sense different from the prevailing philosophical ideas of his time, notably those of Enlightenment philosopher David Hume.

Hume argued that we should not justify our expectations about the future based on our experiences of the past, because there is no law stating that the future will always mirror the past. As such, we can never be certain about our knowledge derived from experience. For Hume, therefore, the fact that the sun had risen every day up to now was no guarantee that it would rise again tomorrow. In contrast, Bayes provided a tool for predicting the likelihood of such events based on past experiences and observations. His method can be applied consistently to the sciences, social sciences, and many aspects of our everyday lives.

Unlike the philosopher David Hume, who argued that past experiences don’t guarantee future outcomes, Bayes focused on how we can use past events to predict the likelihood of future ones. Bayes’ approach is not just academic; it’s a practical tool.

BAYES’ THEOREM: AN EVERYDAY TOOL FOR REFINING PREDICTIONS

So how does what is known as Bayes’ theorem help us in our everyday lives and beyond? As it turns out, it’s an important way of helping us to refine our belief of what is true and what is false. Let’s look more closely into this by breaking Bayes’ theorem down into its key components:

Establish a Prior Hypothesis: The starting point in Bayesian reasoning involves the establishment of an initial hypothesis, which may or may not be true. This hypothesis, also known as the ‘prior’ belief or ‘prior probability’ that you assign to this belief being true, is based on the information available to you. For instance, if you’re trying to predict whether it will rain tomorrow, you might estimate the initial likelihood (or ‘prior probability’) based on your personal observation of current weather patterns or conditions.

Observe New Evidence: Once you establish a prior probability, you’ll then need to consider updating this when any new information becomes available. In the weather example, evidence could be anything from new dark clouds gathering or else dispersing to a sudden rise or drop in temperature.

Assess to What Extent This New Evidence Is Consistent with Your Initial Hypothesis: Bayesian reasoning doesn’t stop at just gathering evidence. It also involves considering evidence that is consistent with, or inconsistent with, your initial hypothesis. For example, if there is an increase or decrease in wind speed, this might be considered additional evidence that you should take into account in estimating the probability of rain.

Let’s break down again how Bayes’ theorem helps us refine our beliefs:

Establishing a Starting Point (The Prior Hypothesis): Imagine you’re trying to predict if it will rain tomorrow. Your ‘prior hypothesis’ is your initial estimate, based on what you currently know about the weather conditions.

Incorporating New Information (New Evidence): Now, suppose you observe unexpected dark clouds gathering in the sky. This new information should logically influence your prediction about the weather.

Combining Old and New Insights (Assessing Consistency): Bayesian reasoning involves integrating the new evidence with your initial estimate. You assess whether the appearance of dark clouds increases the likelihood of rain tomorrow.

By applying Bayes’ theorem, you adjust your belief based on the new evidence. If dark clouds often lead to rain, you increase your belief that it will rain. If not, you adjust accordingly.

Visualising Bayes’ Theorem

Think of Bayes’ theorem as a formula that combines your initial estimate with new information to give you a better estimate.

Beyond Weather: The Broad Applications of Bayes’ Theorem

Bayesian reasoning isn’t just about predicting the weather. It’s used in medicine to interpret test results, in finance to assess investment risks, in sports for game strategies, and so on. It’s a tool that refines our understanding, helping us make more informed decisions.

HOW BAYES’ THEOREM ALLOWS US TO UPDATE OUR BELIEFS

In essence, Bayes’ theorem permits us to establish an initial hypothesis, and to enter any supportive and contradicting evidence into a formula which can be used to update our belief in the likelihood that the hypothesis is true.

Consider a scenario where we evaluate our initial hypothesis. For simplicity, we label the probability that this hypothesis is correct as ‘a’. This probability is our starting point, reflecting our initial estimate based on prior knowledge or assumptions before encountering new data.

Next, we introduce ‘b’, which represents the likelihood that some new evidence we come across is consistent with our initial hypothesis being true. This is a critical element of Bayesian updating.

Conversely, ‘c’ is used to denote the probability of observing the same new evidence but under the condition that our initial hypothesis is false. This estimate is equally essential because it helps us understand the significance of the evidence in the context of our hypothesis not being true.

With these definitions in place, Bayes’ Theorem provides a powerful formula: Revised (posterior) probability that our initial hypothesis is correct = ab/[ab + c(1-a)]

This formula is a mathematical tool that updates our initial belief ‘a’ in light of the new evidence.

The result is an updated (or ‘posterior’) probability that reflects a more informed stance on the initial hypothesis.

This process, termed Bayesian updating, is a methodical approach that enables us to refine our beliefs incrementally. As we gather more evidence, we iteratively apply this updating process, allowing our beliefs to evolve closer to reality with each new piece of information. This ongoing refinement is a cornerstone of the Bayesian approach, emphasising the importance of evidence in shaping our understanding and beliefs.

BAYES’ THEOREM: A POWERFUL TOOL

Bayes’ theorem offers us a weapon against biases in our intuition, which can often mislead us. For example, intuition can sometimes lead us to ignore previous evidence or to place too much weight on the most recent piece of information. Bayes’ theorem offers a roadmap that assists us in balancing the weight of previous and new evidence correctly. In this way, it provides a method for us to fine-tune our beliefs, leading us gradually closer to the truth as we gather and consider each new piece of evidence.

CONCLUSION: THE BAYESIAN BEACON

Bayes’ theorem is more than a mathematical concept; it’s a guide through the uncertain journey of life. It teaches us to be open to new information and to continually adjust our beliefs. From daily decisions like weather predictions to complex scientific theories, Bayes’ theorem is a bridge from uncertainty to better understanding, helping us navigate life’s puzzles with more confidence and precision.

It does so in a structured way, dealing with new evidence, guiding us gradually to more informed beliefs. It encourages us always to be open to new evidence and to adjust our beliefs and expectations accordingly. Bayes’ theorem is in this sense a master key to understanding the world around us.