The Quixotic Quest for Certainty

Once you eliminate the impossible, whatever remains, no matter how improbable, must be the truth. – Arthur Conan Doyle

Certum est, quia impossiblile. – Tertulliano

I recently noted that the more wondrous a miracle, the harder it is to show that it has indeed occurred (Krueger, 2022). This may seem trivial. What is not trivial is that David Hume and Thomas Bayes (or rather the Bayesians who followed him) came to different conclusions regarding the chances of a positive verdict on miracles. Hume set a standard, so high one gets the impression he sought to prove that miracles cannot occur. The Bayesians, by contrast, see a glimmer of hope that with a sufficient number of credible witnesses, the virtually impossible can creep toward a discernible likelihood (Earman, 2002).

The number of concurring witnesses, if their reports are independent of one another, is a crucial ingredient of the Bayesian probability nudge. As positive reports pile up, a discernible likelihood turns into a high probability and then into moral certainty. But a moral certainty is not a mathematical certainty. If the probability that the claim in question is true can approximate, we wonder if it can be 1. Hume argued that it cannot. Learning can support a strong belief, but it cannot deliver the sort of certainty one finds in mathematical or logical deductions.

I will show that Bayes, who, as Earman suspects, sought to refute – or at least shed doubt on – Hume’s skepticism ends up reaching the same conclusion as Hume. Bayes, it has been said, developed his probability calculus in hopes of finding an inductive proof of God. It’s been suggested that Bayes did not publish his work on probability during his lifetime because he came to think that his efforts had failed. Hume, it seems, won the argument.

The question on the table is whether there is a way for evidence or data to move us from a probability of less than 1 to a probability of 1. Can the chasm between uncertainty and certainty be crossed? Consider Bayes’s Theorem and its implications. In the formula below, H refers to ‘Hypothesis,’ or a particular belief, and D refers to ‘Data’ or evidence or observations.

The probability p(H|D) is the product of the prior probability of the hypothesis, p(H), and the ratio of the probability of the Data if H is true, p(D|H), over the overall probability of the data under all hypotheses, p(D). The latter can be written as p(D) = p(H) x p(D|H) + p(-H) x p(D|-H). To wit:

Bayes Theorem

Source: J. Krueger

The strength of belief after gathering data, p(H|D), is the product of the strength of belief before the data were gathered, p(H), and the degree to which the data favor this hypothesis H, p(D|H) / p(D).

Importantly, p(H) and the ratio p(D|H) / p(D) cannot vary independently. If they could, we might start with the strong belief in H being true (e.g., p(H) = .9) and find data that are twice as likely if H is true than if H is false (e.g., p(D|H) = .8 and p(D|-H) = .4). If this were possible, we would have .9 x 2 = 1.8 for p(H|D). However, 1.0 is the upper bound of probability. There is no numerical or conceptual room for a hyper-certainty beyond that which is certain.

As p(H), the prior strength of the belief, approaches 1 from below, the maximum ratio for the strength of the evidence in favor of H (i.e., p(D|H) / p(D)) will approach 1 from above. A ratio greater than 1 entails that p(H|D) > p(H). Is it possible that the ratio can take precisely that value which moves us from p(H) < 1 to p(H|D) = 1? The answer is that this is, in fact, possible, but the victory comes at a price. If the ratio = 1/p(H), the posterior probability of the hypothesis, p(H|D), = 1, and this is so regardless of the hypothesis’s prior probability. The only necessary condition for attaining belief certainty is that the data cannot occur if the belief is false, that is, p(D|-H) = 0, while at the same time there is a non-zero probability that the data occur if the hypothesis is true, p(D|H) > 0.

This is the Bayesian version of Hume’s skepticism of miracles. If a miracle has a very small prior probability, say p(H) = .01, and even if certain observational data will appear given that there is a miracle, say p(D|H) = .001, the posterior probability of the miracle given the data is p(H|D = 1 only if the data cannot be observed if the miracle hypothesis is false, p(D|-H) = 0.

In plain English, the only way to turn a probabilistic belief into a certainty is to have data we know are impossible if the belief is false. In Hume’s example, we need absolute certainty that the witnesses are not mistaken and are not lying in order to be certain about the truth of their miraculous claim. Certainty, in other words, cannot emerge from uncertainty. Certainty of belief requires certainty of data (or rather, certainty about the absence of data if the hypothesis is false) – and that we rarely have in this world.

Miracles and other extraordinary claims must remain improbable in order to maintain their mystique. Supportive data need not replicate; in fact, believers may wish that miracles not be reproducible. The barrier standing in the way of believing extraordinary claims is that the required condition of never making the observations of interest when non-miraculous forces are at work (lying or mistaken witnesses) requires proof of an impossibility.

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Digestifs

[1] After my last epistemological crisis, I went down to Hoca Camide’s Kappadokian Kave where I found him burning the candle at both ends. “Hoca,” I said, “I think it’s so unfair that scientists get canceled and abused when their findings don’t replicate, but the theologians rejoice when their miracles don’t replicate – for that’s what makes them miracles.” “Junior,” Hoca muttered, “You came all this way to tell me this?” “Well yes,” I said. “and I won’t do it again. Consider it a miracle.”

[2] I once asked Hoca Camide, who had studied with the sophists of Acragas, “Hoca, if it is true that if it’s possible, it is probable, and if it is true that if it is probable, it is certain, is it then true that if it is possible, it is certain?” “Perhaps,” Hoca snorted.