On the opposite side of Faith (BR=1 or BR=0) and Certainty (FPR=0 or TPR=0) there is Prior Indifference: BR=0.5.
Prior Indifference is perfect ignorance: no clue at all about whether the hypothesis under investigation is true or false. Imagine an urn containing 100 balls, black and white, in unknown proportions. What is the probability of extracting a white ball? The immediate answer is: no idea, we just don’t know. This feeling of helplessness is what is known as Knightian uncertainty. We’d rather not answer the question but, if forced to, our thinking may be: there are 99 equiprobable proportions, ranging from 1 white/99 black to 99 white/1 black. Hence we take their average: 50%. Under the circumstances, it is clearly the best answer. It is the same answer we would give if we knew that the balls were 50% white and 50% black. But under Knightian uncertainty we don’t know the actual proportion – in fact we know that it is very unlikely to be 50/50. It is precisely such ignorance that motivates the answer.
Despite the equivalence, if we had to choose between betting on the extraction of a white ball from an urn with a known 50/50 proportion and an urn with an unknown proportion, we would prefer the former. This is known as Ellsberg paradox, or ambiguity aversion. We prefer known risk to unknown uncertainty. But Prior Indifference is the starting point in both cases.
So BR=0.5 does not necessarily mean that we know the prior probability of truth is 50%. It may simply mean that we know nothing at all – nothing that allows us to differentiate between true and false: Perfect Ignorance. Will my child be a football star? Was Desdemona unfaithful? Is the market price right? If our answer to these questions is: no idea, we are in the grip of the Prior Indifference (or Perfect Ignorance) Fallacy.
Why is it a fallacy? Because it is hardly ever true that we have no idea. Most times our priors already contain plenty of background evidence that is wrongly ignored: unknown knowns that somehow we fail to take into account. The father is not a Martian catapulted on Earth with no knowledge of football. He knows very well that talented players are rare. But hope and wishful thinking lead him to rely entirely on the coach’s opinion. Under Perfect Ignorance, he replaces support with accuracy, confidence and, ultimately, trust. Othello knows very well that Desdemona is a faithful spouse. But the evil Iago manages to gain his trust, undermine his sound prior and drive him to murder. A good company priced at a big discount to book value is likely to be an attractive investment. We should know it, and not heed Mr. Market’s opinion to the contrary. Correct priors guard us against Perfect Ignorance, keep us closer to the truth and prevent us from getting blinded by evidence.
Of course, correct priors are just a good starting point. They are neither a necessary nor a sufficient condition for convergence to the truth. Othello starts with the right prior, but then veers away from it and does a terrible job at gathering evidence. Conversely, a Turkish Muslim might start with a wrong-headed opinion about Arabs’ involvement in 9/11, but then discover and trust the truth.
Convergence to the truth only occurs as a result of a thorough tug of war between confirmative and disconfirmative evidence. We may start well by seeing that a stock is cheap, thus avoiding Mr. Market’s prior indifference. But whether or not the stock is a good investment is a hypothesis that we need to test against the available evidence, making sure that we gather it on both sides of the rope.
