Jacob Bernoulli made a now obvious observation about probability over three-and-a-half centuries ago: If nothing is known about the outcome of a random event, all outcomes can be assumed to be equally probable. Bernoulli’s Principle of Insufficient Reason (PrOIR) is commonly used. Throw a fair die. There are six outcomes, one for each face of the cube. The chance of getting five pips showing on the roll of a die is therefore one sixth. If a million lottery tickets are sold and you buy one ticket, the chances of winning are one in a million. This reasoning is intuitively obvious.
If the Die Is Loaded
The assumption about the die is wrong if the die is loaded. But you don’t know that. You know nothing. So Bernoulli’s PrOIR provides the best model based on the known. If the lottery is fixed and you’re not in on the fix, your chances of winning will be less that one in a million. Maybe zero. But you don’t know the game is fixed. You know and assume nothing. Under the circumstances, equal probability is the best assumption you can make.
In analysis of fine-tuning, No Free Lunch Theorems, and conservation of information, Bernoulli’s PrOIR is foundational. In thermodynamics, uniform distributions correspond to maximum entropy. In the absence of air currents or thermal gradients, the temperature is the same in the middle of the room as it is in the corners.
Those who disagree with Bernoulli’s PrOIR consistently misapply the principle. They don’t appreciate the definition of “knowing nothing.” The concept of “knowing nothing” can be tricky. The sentences “knowing nothing means knowing something” and “knowing nothing means knowing nothing” are both curious puns.
Strange Ideas in Economics
The most visible opposition of Bernoulli’s PrIOR comes from the economist John Maynard Keynes who is most famous for some strange ideas in Keynesian economics. Keynes’ problems with Bernoulli’s assumption are discussed in his book A Treatise on Probability. Two of his objections, Bertand’s Paradox and the distribution of reciprocals, are soundly debunked in Introduction to Evolutionary Informatics by Ewert, Dembski and me.
A third argument made by Keynes stems from the sort of data economists would deal with. Here is the example: Consider presenting a man who is either from Great Britain or France. You know nothing about the selection process. Bernoulli then says the chances of the man being French is one half. Consider a second situation where locations are finer grained. You are told the visitor is either from Scotland, Wales, or France. Is the chance the man is French now one third? Since both Scotland and Wales are part of Great Britain, what does this say about the first answer where the chance of being French is one half? Is this a case where Bernoulli’s PrOIR breaks down?
No. In reaching this contradiction, Keynes knew something. He did not know “nothing” as required by Bernoulli’s PrOIR.
Read the rest at Mind Matters News, published by Discovery Institute’s Walter Bradley Center for Natural and Artificial Intelligence.