Will the sun rise tomorrow? Of course it will. But how can we be sure? There are two ways: Faith and Certainty.
In Bayes’ Theorem:
Faith is BR=1, a prior belief in the truth of the hypothesis, which requires no evidence and which no amount of evidence, however strong, will change. Faith implies PP=1: the hypothesis must be true, on the grounds of pure reason, irrespective of evidence.
Certainty is FPR=0, a posterior belief in the truth of the hypothesis, achieved through the accumulation of evidence. Certainty also implies PP=1: the hypothesis is true, irrespective of the initial Base Rate. But truth is not a prior necessity: it is the result of the lack of False Positives. As David Hume famously put it, there is no way we can know that the sun will rise tomorrow. We can only expect it, based on the fact that it has always done so and never appeared to do otherwise. Sunrise cannot be demonstrated a priori, or proven on the grounds of conclusive evidence. It can only be expected a posteriori, on the basis of an overwhelming accumulation of confirmative evidence. Therefore, no matter how large the amount of accumulated evidence, PP will tend to but never reach 1.
In practice, the accumulation of confirmative evidence can be as good as conclusive. While, strictly speaking, a Smoking Gun is a single piece of conclusive evidence incompatible with False Positives, the lack of False Positives in sunrise can be so overwhelming as to be a de facto Smoking Gun. In this sense, evidence proves that the sun must rise again tomorrow.
Will a giant parmesan cheese rise tomorrow? Of course not.
In this case Faith is BR=0, a prior belief in the falsity of the hypothesis. Faith implies PP=0: the hypothesis must be false. Likewise, Certainty is TPR=0, a posterior belief in the falsity of the hypothesis. Certainty also implies PP=0: the hypothesis is false, not as a prior necessity, but as a result of a lack of True Positives. There is no way to know that cheese will not show up tomorrow. We can only expect it, based on the fact that it has never done so and always appeared to do otherwise. Hence PP will tend to but never reach 0. There is a preponderance of disconfirmative evidence against cheeserise: a de facto Perfect Alibi.
So Faith and Certainty have the same effect: they both drive probability to one of the two boundaries of its spectrum. But whereas Faith requires no evidence, Certainty is entirely based on the accumulation of evidence to its conclusive end.
The distinction is largely inconsequential when evidence is hard and its accuracy can be properly measured through a controlled, replicable experiment. But soft evidence is in the eye of the beholder: its perceived accuracy coincides with the observer’s confidence in using it as a sign for evaluating the probability of the hypothesis. Therefore, it is essentially a matter of trust.
Most of what we know is not the result of direct experience, but of trusting the source of the evidence. That’s how we know that the Coliseum is in the centre of Rome, even if we have never been there. The accuracy we attach to soft evidence is ultimately our decision. We decide to trust the Rome Tourist Guide as to the Coliseum’s whereabouts, because we attach to it a very high TPR and a very small FPR – i.e. a very large Likelihood Ratio TPR/FPR. The guide has our full trust: we regard its indication as conclusive evidence. If the guide says it, it must be true. It is an entirely reasonable decision, based on attaching a zero probability to the chance that the guide’s authors may have made a mistake or lied.
We do the same with all evidence. To each piece we attach a Likelihood Ratio, which is ultimately based on trust. The evidence may come from myriad of sources: newspapers, TV, books, conversations; teachers, parents, friends; scientists, politicians, clerics. The trust we place on evidence is greatly influenced by its source. Smoking is bad for your health is more likely to be true if said by your doctor than by your mother; IBM shares are more likely to be a good investment if you hear it from Warren Buffett than from your uncle. Greed is more likely to be a capital sin if the Bible says so than if you read it on Hello! Magazine. In fact, you may go as far as giving the doctor, Warren and the Bible your full trust, irrespective of what you thought before. If they say it, it must be true: a Smoking Gun. As with the Rome guide book, it is your decision.
Trust does not require Certainty. As long as it is highly trusted, the source of evidence will have a large influence on our beliefs. Of course, trust may not be omnipotent: if Buffett says that elephants fly we won’t believe him (although I know a few who would give it a thought). This is because we attach a miniscule prior probability to flying elephants: whatever anyone says, we won’t believe it until we see one – whatever Warren says.
But as long as the source of evidence does not stray too wide from the confines of its credibility, prior indifference kicks in: unfiltered by our priors, evidence blinds us. As a result, our posterior probability is entirely determined by trust. Under prior indifference, what we believe depends on who said it.
It’s true: I heard it on TV.
