Bayes’ Theorem, With No Maths
EVENT 1: A crazy driver hits your car and whizzes past. You get out and see your car is badly dented. The driver has disappeared. You're fuming, but there's nothing you can do. You tell yourself, at least you're alive. It could have been worse.
EVENT 2: You drive on. You notice that driver in front of a hospital. He's standing in front of an ambulance. The driver speeds into the hospital with a patient.
EVENT 3: Your prayers are now with the patient. Your anger has evaporated.
Before Event 1, you had a clean information set. You thought what anybody would have thought based on universal experience: the driver must be careless and reckless. Before Event 2, you're happy you survived. Event 3 is based on what you learned from Event 2. Your outlook towards the driver has changed. Who knows if that patient is still alive?
As new sets of information appear, we adapt our perception. In probability science, this is known as the Bayes' theorem.
British mathematician Thomas Bayes (1702-1761) proposed that our perception towards the world is based on "prior probabilities". This is the probability set we start with. As we confront new experiences, the prior probability sets also change – "conditional" to what we experience. And with it, our perception changes.
We look at the world based on existing beliefs. When we confront overwhelming evidence that shatters our beliefs, Bayesian rationalists argue that our perception towards the world also changes.
Bayes didn't live to see the impact of his thoughts. His friend Richard Price published his work in 1763. Years later, it caught the attention of French mathematician Simone Pierre Laplace. Laplace introduced "conditional probabilities" and formalised the theorem in the equational form we see today.
Remember the 2016 T20 World Cup final between England and West Indies? Let's replay the last over. You have no "prior" knowledge of how the match ended.
England made 155/9. A competitive score. At the beginning of the 20th over, West Indies were 137/6. Their star batsman Marlon Samuels was at the non-striker's end. A relatively unknown Carlos Brathwaite would face England's star all-rounder Ben Stokes.
David "Bumble" Lloyd of England and Ian Bishop of West Indies were in the commentary box. Surely, West Indies stood no chance against Stokes. Brathwaite sent the first ball for a six off long leg. Bumble was ecstatic; Bishop, silent. West Indies now needed 13 from five balls.
The next ball goes for a six off midwicket. Seven needed off four. Are you changing opinions "conditional" to new probability sets?
Third ball: six! Scores are level. Three balls remain. Minutes ago, many thought the game was sealed for England. Bumble is shouting at the top of his lungs. The rest is history – Brathwaite sends the fourth ball for a six off long-on. From nowhere, Bishop screams, "Carlos Brathwaite! Remember the name!" What Bishop hadn't dreamt of, was now reality.
As probability sets changed, alliance tilted from England to the West Indies. This is the crux of Bayes' theorem stripped of all mathematical vocabulary.
According to Bayesian logic, we change our perception as we confront overwhelming evidence. The first story supports this. In the second, even after watching a wonderful match, some people will still support England. This tells us Maths has its limitations. People don't always respond to mathematical predictions. Once we know the limits, we appreciate the beauty and the beast of a discipline.