Digression 2: Was the Heretical Formula Meant to Measure 'Miracles'? — The Strange Fate of Reverend Thomas Bayes and His Friend

A Century-Defining Discovery That Remained Unpublished During His Lifetime

Given that he is the father of a theory with such a profound impact on the world, one might imagine that Thomas Bayes (c. 1701–1761) was a world-renowned mathematical genius who left his mark on history. However, the facts of history defy such predictions (or assumptions). He was neither a professional mathematician nor a prominent university scholar; he was a Presbyterian minister who looked after a small church near London, England, in the 18th century.

Even more surprisingly, Bayes himself never published this groundbreaking theory, known as 'Bayes' Theorem,' during his lifetime. It is believed that for him, mathematics was merely a personal pursuit.

After he passed away quietly in 1761, his friend Richard Price, who was also a minister, was sorting through his belongings when he discovered an incredibly innovative mathematical formula among the old notebooks left behind. Price, intuiting that 'this might be an extraordinary discovery,' organized and added to the notes before sending them as a paper to the Royal Society in London. As a result, the theory finally saw the light of day several years after Bayes' death.

It is said that if his friend Price had overlooked the value of these notes and thrown them into the fireplace, the arrival of today's AI society might have been delayed by decades, or perhaps even more than a century.

Why Did a Minister Create a 'Heretical Formula'?

The orthodox statistics of the time (frequentism) was dominated by forward-looking inference, which aimed to 'predict results from causes.'

For example, if the clear cause is known—that 'a die has six sides'—the reasoning is to derive that 'the probability of rolling a one (the result) is one-sixth.' However, the formula Bayes left in his notebook was an approach known as 'inverse probability,' which is the exact opposite.

It was an approach that inferred causes from results—'calculating the probability of a hidden cause (truth) from the incomplete results (data) before one's eyes'—which was considered extremely heretical at the time.

So, why did a church minister feel the need to devise such a formula for reverse calculation?

In 18th-century England, the philosopher David Hume was causing a great stir in the religious world by arguing that 'miracles are contrary to the laws of nature, and therefore no testimony is sufficient to make them probable.'

The manuscript Bayes left behind dealt precisely with the problem of 'how to update the probability of an event occurring based on observational data,' which overlapped with the core of Hume's 'argument against miracles.' In other words, it is highly likely that this was a mathematical approach to the question: 'How should we update our beliefs if testimony or observational data accumulates in support of an event with an extremely low probability, such as a miracle?'

However, Bayes himself was extremely cautious and did not develop any religious arguments using terms like 'God' or 'miracles' in his manuscript. It was his friend Price, who discovered the notes, who explicitly applied this formula to counter Hume and engage in religious debate. The anecdote that 'Bayes tried to prove the existence of God' is now considered by historical research to be a misunderstanding of history born from confusing Bayes' work with Price's achievements.

The Mechanism of Bayesian Thinking
— 'Abduction' and 'Bayesian Updating'

The concept of Bayesian statistics is intuitively expressed by the phrase 'Posterior Probability ∝ New Fact × Prior Probability.' This is more than just a calculation; it is a mathematical model of the process by which we should rationally update our perceptions and inferences when we acquire new experiences or data. This process can be understood as a continuous movement of the following two steps.

Abduction (Setting the Prior Probability): A concept proposed by Charles Peirce, referring to the intuitive leap of forming a hypothesis from a surprising fact before one's eyes, thinking, 'If there were a law (hypothesis) like X, could this fact be explained well?' In Bayesian statistics, this corresponds to the act of intuitively setting the 'prior probability' (the initial estimate), which serves as the starting point for verification.

Bayesian Updating (Correction to Posterior Probability): A process of flexibly updating one's estimate by calculating how likely it is (posterior probability) each time new facts or evidence (likelihood) are obtained, relative to the initial estimate (prior probability) formed through abduction.

This structure overlaps perfectly with the 'editing process' of intuitively forming a hypothesis about a 'true theme' from chaotic events and increasing or fine-tuning one's confidence while checking it against facts. Furthermore, the 'Bayesian brain hypothesis' in neuroscience shares this same structure. It is believed that the brain does not merely view the outside world passively, but constantly forms predictions (hypotheses) first, instantly calculates the discrepancy with input data from sensory organs, and continues to correct those predictions (Bayesian updating) to construct the world.

In an uncertain world where an absolute correct answer (objective probability) is unknown, humans first establish subjective beliefs or assumptions (prior probabilities) that 'it must be this way.' Then, by comparing them with real-world data (new facts), they gradually approach the truth.

The fact that an approach born from the personal pursuit of an anonymous minister and the faith of his friend has become a master key that drives cutting-edge artificial intelligence and unravels the mechanisms of the human brain after 250 long years. When you learn of this strange fate, which could be called an irony of history, perhaps Bayesian statistics, which might look like dry and cold formulas, will appear to you as a very human-centric philosophy filled with romance.

*For reference----------------------------------

In modern statistics, the basic formula for Bayes' theorem is as follows:
P(A|B) = P(B|A) P(A) / P(B)

Each symbol that makes up the formula has the following meaning.

P(A|B) (Posterior probability): The probability that event A occurs given that event B has occurred. It represents the 'probability of A updated after obtaining new information (evidence B),' which is what we want to find using Bayes' theorem.

P(B|A) (Likelihood): The probability that event B occurs given that event A is occurring. It represents 'how likely result B is to occur if A is the cause.'

P(A) (Prior probability): The probability that event A occurs before obtaining new information (event B). This is the 'original guess' based on past experience or knowledge.

P(B) (Marginal likelihood/Evidence): The probability that event B itself occurs, regardless of event A. It is the total probability of event B, summing all possibilities.

What Bayes' Theorem Means
The most important point of this theorem is that it mathematically demonstrates 'how one should update their previous guess (P(A)) to a new guess (P(A|B)) when new data (B) is obtained.'