Can Bayesianism Quantify True Belief?

The first part of Evelyn Waugh’s masterpiece Brideshead Revisited (1945) centers around the relationship between protagonist Charles Ryder and his friend Sebastian Flyte, whom he met at Oxford. Sebastian rebels against his rigid Catholic upbringing by living a life of debauchery. Charles begins the following dialogue by asking Sebastian about the faith:

“I suppose they try and make you believe an awful lot of nonsense?”
“Is it nonsense? I wish it were. It sometimes sounds terribly sensible to me.”
“But my dear Sebastian, you can’t seriously believe it all.”
“Can’t I?”
“I mean about Christmas and the star and the three kings and the ox and the ass.”
“Oh yes, I believe that. It’s a lovely idea.”
“But you can’t believe things because they’re a lovely idea.”
“But I do. That’s how I believe.”

This dialogue reveals that Sebastian is childish, which is a central theme of the novel. We may ask, however, whether his view of faith, which seems to make no use of reason, constitutes bad theology. More generally, we may ask, How should we determine what we believe? For some, it seems obvious that we ought to rely on logic and evidence and should not, in any context, accept ideas merely because they are lovely. Many philosophers, mathematicians, and other thinkers have contributed to the tradition of this evidence-based approach. Perhaps the greatest modern school of thought in this tradition is called Bayesianism, which incorporates ideas from many brilliant thinkers into a coherent system to describe how we know things. St. John Henry Newman, whom Pope Leo XIV recently named a doctor of the Church, has a different — and, I will argue, broader — epistemology, because it can incorporate the best aspects of Bayesianism as well as crucial aspects of human nature that Bayesianism misses.

It is natural to desire certainty in our beliefs. One way to obtain it is through deductive logic, in which some premises are assumed and others are proven. The classic example is:

All men are mortal.
Socrates is a man.
Therefore, Socrates is mortal.

In this argument, the first two sentences, called postulates, are not proven but are assumed to be true. If they are, then the third sentence is certainly true. This method of reasoning has the benefit of giving mathematical surety, but it has the obvious limitation of requiring postulates that are unproven. Sometimes the postulates can be deductively proven, but these proofs require additional unproven postulates.

In the past, certain logicians, most notably Bertrand Russell and Gottlob Frege, hoped we could approach certainty by having obvious postulates and proving everything else with deductive reasoning. The evidence for this system would be strengthened if there were very few postulates, and the system could show it did not contain contradictions. However, as mathematician Kurt Gödel demonstrated in his incompleteness theorem, you cannot use a logical system to prove its own consistency, and there will always be true statements outside the postulates that cannot be proven, and false statements that cannot be proven false. This theorem significantly weakened the hope that we might be able even to come close to a practical system of knowledge that is strictly logical.

Another method of obtaining postulates that can be used as starting points for knowledge is by looking at data. An epistemology called frequentism hypothesizes that if you repeatedly record an event, you can find the frequency at which that event will occur. For instance, if you say there is a 50 percent chance of getting heads when you flip a coin, what you mean is that if you flip it 1,000,000 times, the coin will likely land on heads very close to 500,000 times. The frequentist would argue that you can obtain probabilities on claims by collecting large amounts of evidence. A common challenge to frequentism is that most events cannot be identically repeated. It is easy to use frequentist statistics to determine the probability that an unbiased coin has a 50 percent chance of landing on heads, or that a biased coin might have an 80 percent probability of landing on heads, but the same method cannot be used to determine, say, the probability of a specific candidate beating another candidate in an election. The two candidates run against each other only once, so it is not possible to obtain a large set of repeated measurements. The fact that frequentism cannot accurately predict one-off events limits its usefulness as a general system to understand knowledge.

Bayesianism responds to the limitation of deductive logic and frequentist statistics. First developed by 18th-century British statistician and Presbyterian minister Thomas Bayes, it can be characterized by three premises. First, individuals have subjective degrees of belief. Rather than trying to base a system on objective statements about the world that can never have the desired certainty, Bayesianism rests on the subjective beliefs people hold for whatever reason. Second, individuals act on their beliefs. This premise prevents people from being dishonest about their level of belief. For instance, a political pundit who confidently declares that his preferred candidate has a 99.9 percent chance of winning an election but refuses to make an even bet in favor of his candidate almost certainly does not have as much confidence as he claims. Betting odds, because of their simplicity, are a commonly employed tool in Bayesianism for quantifying true belief. (People do not often bet on false beliefs merely because they want them to be true.) Third, individuals’ degree of belief can and should be changed by new evidence.

Bayes’ theorem, a rule for inverting conditional probabilities, is a mathematical way to update a prior belief. For instance, a doctor might believe, based on a gut feeling from his discussion with a patient, that there is a 20 percent chance the patient has cancer. After a certain test comes in positive, he might use Bayes’ theorem to update this probability to 56 percent. Even though the doctor’s initial assessment of 20 percent was based on an instinctual analysis, he used mathematical/logical rules to transform this into an updated probability of 56 percent. Thus, his final probability consists of a synthesis of both abstract logical reasoning and subjective instinctual judgment.

This synthesis is a major strength of Bayesianism. It allows beliefs to be improved with deductive certainty without the need for absolute postulates or repetition of events. Sound arguments, however, suggest it has a limited scope and, therefore, is not a complete theory of human knowledge. To be able to compare various forms of evidence, Bayesianism requires the quantification of probability. In certain situations, it is difficult to quantify such probabilities.

The Bayesian method of assuming that a person’s decisions reveal his degree of confidence can be applied to many situations, but not all. For instance, it can be difficult to quantify the degree of a person’s certainty that his car’s brakes will not fail. A Bayesian could look at how this certainty affects the driver’s actions. For instance, perhaps he declined to purchase an expensive safety feature and interprets this as a type of bet. If he wins the bet, he saves the cost of the safety feature; if he loses the bet, he loses his life. But to recover a numerical probability from this bet, the driver must compare the value of his life to the cost of the safety feature. Without this probability, Bayes’ theorem cannot be used to determine how additional evidence should update the probability, indicating that this situation is beyond the scope of Bayesianism. Thus, Bayesianism, properly applied, can be an extremely powerful tool of analysis, but it does not allow for a single method of human judgments across all domains.

Some have tried to develop an epistemology in which Bayesianism and the Christian view of faith are reconciled. For instance, formerly Protestant YouTuber Cameron Bertuzzi applied a Bayesian analysis in his decision to convert to the Catholic faith. Using the described method, he determined that he had a 93.8 percent certainty of the truth of the papacy. Due to this calculation, as well as other motivations, he accepted the truth of the papacy on faith. I will refer to the method of first using a Bayesian analysis to determine that Christianity is likely but not certain and then relying on faith to get certainty as Christian Bayesianism. This view of faith allows for Bayesianism with its extensive explanatory power to be married to the Christian understanding of the need for faith. However, we may ask whether it is the view of faith contained in Scripture or Tradition.

In Newman’s epistemology, which he developed in a number of works, including An Essay on the Development of Christian Doctrine, An Essay in Aid of a Grammar of Assent, and his Oxford sermons, he chooses to look at how ordinary people actually reach certitude, rather than developing a system that some intellectuals think they ought to use. Newman makes this choice not because he believes an idea is correct if the majority accept it but because he believes humans have a determinate nature. If a human being’s nature can be perfected, it will still be a human nature. Thus, the way to understand the perfect mode of human knowledge involves understanding how human knowing actually works. This understanding requires a descriptive account of human knowledge, not a prescriptive theoretical account of the subject.

How do people usually determine their beliefs? Reason is often involved but almost never solely responsible. Other faculties, such as desire and intuition, contribute as well. In sermons 10, 11, and 12 of his celebrated Oxford sermons, Newman uses this fact to integrate faith and reason. Deductive reasoning is not meant to be a stand-alone faculty and, thus, cannot be completely divorced from the other faculties that help people reach conclusions. Rather, in human belief, in matters of faith as well as earthly matters, a person reaches a conclusion as a unified whole. This method is a part of human nature, and it makes sense to apply it to faith. Newman observes that this view of faith, unlike a strictly evidence-based approach, matches St. Paul’s description: “Faith is the substance of things hoped for” (Heb. 11:1).

The idea that hope is the motivation for faith might seem to confirm the worst criticisms nonbelievers lodge against Christians, such as that their faith is merely wish-fulfillment. They might argue that if human nature tells people to believe false conclusions merely because they are desirable, then faith is flawed, and Christianity should not be followed. For example, many can convince themselves, on very flimsy arguments, that beer or chocolate is healthy. There is something childish about this reasoning. Such people could see the truth more clearly if they considered the matter detached from their desires. Thus, the nonbeliever might argue that developing faith based on our loves would not be a good thing. However, the lover of chocolate or beer differs from the lover of God in the source and object of his love. In Sermon 10, Newman observes:

Love of the great Object of Faith, watchful attention to Him, readiness to believe Him near, easiness to believe Him interposing in human affairs, fear of the risk of slighting or missing what may really come from Him; these are feelings not natural to fallen man, and they come only of supernatural grace; and these are the feelings which make us think evidence sufficient, which falls short of a proof in itself. The natural man has no heart for the promises of the Gospel, and dissects its evidence without reverence, without hope, without suspense, without misgivings; and, while he analyzes that evidence perhaps more philosophically than another, and treats it more luminously, and sums up its result with the precision and propriety of a legal tribunal, he rests in it as an end, and neither attains the farther truths at which it points, nor inhales the spirit which it breathes.

The logical arguments for the probability of Christianity are valid and sound, but they are a poor substitute for faith, which can achieve what arguments can and more, because it includes the whole of the person in his ability to reach a conclusion. Faith is a fully human act, not an algorithmic assessment. It perfects the person in his entirety, including but not limited to the possession of correct beliefs.

We might ask whether the faith of a Christian Bayesian is insincere. I argue that in most cases it is not. Christians whose aim is to separate argument from love do have a sincere faith, because it is, paradoxically, based on love. For most people, love for God does not emerge fully formed; it begins with a love for a created thing. For instance, a musician might be drawn to a love for God through his love for the beauty of a melodic hymn, or an architect by her love for the beauty of a Gothic church. Extremely logical people can encounter God through their love for the strength of apologetical arguments. The proper use of deductive reason is a beautiful thing. There are many good arguments that suggest the probability of the Church’s claims, and they ought to be loved. Mathematicians, logicians, and apologists are right to love a God to whom the logical evidence (even in the absence of faith) points. Thus, for a Christian Bayesian, the love for objective, unmotivated reason can lead to a love for the Divine Creator, whose supernatural grace gives the Christian Bayesian a supra-logical confidence in the Church’s teaching.

Such people, however, are at risk of taking their arguments too seriously. If they start to believe they have earned their faith through their intellect, or sneer at those who come to the faith through a different love, their faith ceases to be genuine. They no longer love God through a love for logic and instead reduce God to a tool for their love for logic. Faith, we must remember, remains constantly a gift and never a prize.

Even though a strictly rationalist view might lead to the probability of the truth of Catholicism, Catholicism is not meant to be held as a proposition but lived as a life. As this life requires the fullness of our human nature, Catholicism must be chosen in a specifically human way, even though more rational means also point to it. There is a beautiful paradox in this understanding of faith: a lower form of reasoning that all humans use can draw our minds to the highest things that can be reasoned. Sebastian Flyte in Brideshead Revisited is not wrong in his understanding of the faith. His idea that we should choose Catholicism because it is a lovely idea seems wrong because it appears childish. But the childishness with which he expresses his faith, characterized by a simplicity that unifies his whole person, may well be the childlikeness Our Lord tells us is required to enter the Kingdom of Heaven.

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