A.I. On Bayes Theorem Stressing David Hume's Contribution!

"Is Bayes theorem a legitimate tool to evaluate miracles in the Bible?" 

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Yes, Bayes’ Theorem is a legitimate mathematical and philosophical tool for evaluating biblical miracles. However, because the theorem requires subjective inputs—like the initial probability of a miracle occurring and the reliability of ancient texts—scholars debate how definitively it can prove or disprove specific events. [1, 2, 3]

The Bayesian Formula:

Bayes’ theorem calculates the probability of a hypothesis (H) being true given certain evidence (E).

The Components:

Evaluating miracles using this theorem involves three key factors:

[1] Prior Probability P(M):

Your initial assessment of the likelihood of a miracle occurring before considering the specific evidence. Skeptics often argue this is near zero, while proponents argue it is higher based on the existence of an active, intervening Creator. [1, 3, 4]

Likelihood P(T|M):

The probability that we would have the existing historical records and testimony if the miracle actually happened. [1]

False Positive Rate \(P(T\vert{}\sim M)\):

The probability of getting that same testimony or record even if the miracle did not happen (e.g., due to human error, legend, or deception). [1]

The Debate in Scholarship:

While the method itself is logically sound, applying it to the Bible sparks intense scholarly disagreement: [1, 2]

The Skeptic's View:

Philosopher David Hume argued that the prior probability of a miracle is infinitely low because it violates established laws of nature. Consequently, the likelihood of human testimony being mistaken will almost always outweigh the probability of the miracle itself, making it irrational to believe. [*See Below]

Anselm "Faith Seeking Confirmation"

[First published 5/18/20] I think Anselm's dictum "faith seeking understanding" is to be understood in the history of theology and philosophy to be equivalent to "Faith Seeking Confirmation." If that's how it's historically used then that's what it means. Below is an updated edit from chapter 2 of my my book, Unapologetic: Why Philosophy of Religion Must End.

There is a common theme among St. Anselm's work and the work of other obfuscationist theologians and philosophers that needs to be highlighted. It’s called faith seeking confirmation. We see this in Anselm with regard to his new atonement theory and his ontological argument.

Anselm therefore is exhibit “A” in defense of what atheist philosopher Stephen Law said: “Anything based on faith, no matter how ludicrous, can be made to be consistent with the available evidence, given a little patience and ingenuity.”1 If I could pick one sentence, one aphorism, one proverb that highlights the main reason philosophy of religion (PoR) must end, it’s Law’s. I’ll call it Law’s law of faith.

We Already Use Bayes' Formula

Via John Earman.

I guarantee you that everyone uses Bayes' formula whenever considering new information. We ask ourselves implicitly and/or explicitly what the Pr (probability) is of a new H (hypothesis) given E (evidence) and our K (background knowledge).

That is, when considering a new hypothesis we fact-check it based on the evidence plus our existing background knowledge.  

Of course we do this imperfectly, sometimes woefully so, but we try.

The only question left is when and where should we use the math.

“He’s Got the Whole World in His Hands”

Oh please, give me a break: It's ridiculous bragging!  

For a long time I have suggested that people should carefully study—and absorb the truth about—World War I. In John Loftus’ Second Edition of God and Horrendous Suffering, he included my essay, The Problem of World War I, in which I present the case that this brutal conflict should have put an end to belief in a good, caring, all-powerful god. World War II reenforces this conclusion, as Darrell Ray has pointed out in his book, The God Virus: How Religion Infects Our Lives and Cultures: 

“It took two world wars for Europeans to realize that the prayers of millions of people were not answered. It doesn’t take much intelligence to see that god isn’t working too well when 92 million people died in two world wars, or to see the complicity and cooperation of the Pope, Lutheran clergy and Christians with Hitler during WWII.” (page 75)

Bayes Theorem and Miracles: Three Cases Studies:

Philosopher Michael P. Levine writes a bit eccentrically about miracles but he also has some incisive things to say. For people who think David Hume’s arguments against miracles are obsolete and should be ignored, Levine says, “The locus classicus for modern and contemporary philosophical discussion of miracles is Chapter X (‘Of Miracles’) of David Hume’s Enquiries.”[1] [“Locus classicus is a Latin term used in literature, law, and other academic fields. It refers to the original or most authoritative passage or source for a particular word, phrase, concept, or idea. In other words, it is the classical or standard example that is most often cited as the definitive reference for a subject.”] So he proceeds to analyze what Hume wrote.

He tells us “there are various versions of Bayes’s theorem,” not just one, depending on the hypothesis to be resolved. With miracles they are many. He highlights John Earman’s formulation:

“The reader is invited to think of H as a hypothesis at issue; K as the background knowledge; and E as the additional evidence. Pr(H/E&K) is called the posterior probability of H. [i.e., the Probability of the Hypothesis divided by the background Knowledge]. Whereas Pr(H/K) and Pr(E/H&K) are respectively called the prior probability of H and the (posterior) likelihood of E.”[2]

Labeling this as “Hume’s argument” Levine adds: 

On the interpretation of Hume’s argument given above, a Bayesian analysis sheds no light whatsoever on the structure or substance of the argument, and can do nothing by way of either supporting or refuting the argument. Indeed, any Bayesian analysis of the question of justified belief in miracles must be otiose until the difficult and essential questions concerning “evidence” in relation to an allegedly miraculous occurrence are resolved - at which point any Bayesian analysis will add little except the technical complexity of a formal apparatus that may or may not “clarify” the structure of Hume’s argument. The balancing of probabilities is of no use until it is decided what goes into the balance - that is, what constitutes the evidence.[3]

Let's Highlight the Evidence (E). This is the crucial factor regardless of one’s background knowledge (K) or priors. Regarding a Christian Bayesian analysis, Levine comments, “The idea that the Bible provides historically acceptable grounds for believing in miracles reflects a wish-fulfilling rendering of what constitutes evidence.”[4] Everything hangs on the quantity and the quality of the evidence. It’s the indispensable input, the essential element, and the non-negotiable necessity.

On Miracles and the Importance of Contexts, Perspectives, Priors, Background Information, and Bayes Theorem

--Norman Geisler, a Christian apologist, is on record as saying:

The mere fact of the resurrection cannot be used to establish the truth that there is a God. For the resurrection cannot even be a miracle unless there already is a God. . . . The real problem for the Christian apologist is to find some way apart from the mere facts themselves to establish the justifiability of interpreting the facts in a theistic way. . . . No fact, event, or series thereof within an overall framework which derives all of its meaning from the framework can be determinative of the framework which bestows that meaning on it. For no fact or set of facts can of and by themselves, apart from any meaning or interpretation given to them, establish which of the alternative viewpoints should be taken on the fact(s). [Norman Geisler, Christian Apologetics (Grand Rapids, MI: Baker House, 1976), see pp. 94–98.]

Geisler's perspective stands in stark contrast to other ways of knowing whether Jesus arose from the dead. Geisler was endorsing Classical apologetics. Evidentialist apologetics by contrast, basically rejects it, since it seeks first and foremost the evidence.

--I. Howard Marshal. According to Christian scholar Marshall in his book, I Believe in the Historical Jesus (Eerdmans, 1977) “many historians—the great majority in fact—would say that miracles fall outside their orbit as historians. For to accept the miraculous as a possibility in history is to admit an irrational element which cannot be included under the ordinary laws of history. The result is that the historian believes himself justified in writing a ‘history’ of Jesus in which the miraculous and supernatural do not appear in historical statements. The ‘historical’ Jesus is an ordinary man. To some historians he is that and no more. To others, however, the possibility is open that he was more than an ordinary man—but this possibility lies beyond the reach of historical study as such.” (p. 59).

--William Lane Craig is on record as saying:

"J.L. Mackie's critique of miracles is particularly shockingly superficial."

Craig said this in his paper, The Resurrection of Theism, Endnote # 19. Craig was commenting on the late J.L. Mackie's book, The Miracle of Theism. Craig was prompted to say this while reading Alvin Plantinga’s critique of it titled, Is Theism Really a Miracle?, in Faith and Philosophy, [April 1986].

--J.L. Mackie by contrast argued against the belief in miracles. Is he really "shockingly superficial"? Let me quote from him since it's so good:

Every Human Is Born Into a Vast Bubble of Ignorance

Religion seems to benefit the most from this reality

The majority of humans seem unaware of their ignorance, and simply want to thrive as best they can in the world, have a good time, be entertained. But then there are some who are also born with curiosity. They want to know and learn, to find out as much as they can about our world and the cosmos. Religions in general are nervous about curious people. I’ve mentioned quite a few times what an elderly Catholic woman once told me—and it is worth repeating: “The priests told us never to think about what we learned in catechism."

Why Didn't David Hume Respond Upon Knowing Bayes Theorem?

I'm currently trying to resurrect David Hume from the dead. We've established that David Hume knew of Bayes Theorem and had two chances to edit his chapter "Of Miracles" in response, but didn't. This seems to be "a deliberate philosophical position," per William Vanderburgh.

As an author I've published a great many blog posts, papers, and books about a whole lot of issues; a small encyclopedia of them, if you will. The more that one publishes the more of a context informed readers have for understanding an author on a separate issue. So let's consider the relevant books of David Hume, apart from his History of England which was a massively influential six-volume historical work that thrust him into fame. Let's see if there's a reason he didn't respond to Bayes Theorem. (In what follows A.I. insisted that I let her help a bit.)

A Treatise of Human Nature was published in London in multiple volumes between 1739 and 1740. Together it was Hume’s most comprehensive philosophical work. It breaks down the origins of human knowledge, ideas, cause and effect, personal identity, the passions, and morality.

An Enquiry Concerning Human Understanding (1748). A more accessible, revised version of the first book of the Treatise. It specifically addresses skepticism, our understanding of cause and effect, free will, and the critique of miracles. Thomas Bayes’s work on probability appeared posthumously in 1763. His friend Richard Price discovered the manuscript and published it posthumously in 1767 along with his explanation of Bayesian methods for assessing miracles. We know Hume read Price's paper and liked it. But Hume neither addressed Bayesian arguments nor revised his account of miracles for the 1768 and 1777 editions of his Enquiry.

An Enquiry Concerning the Principles of Morals (1751): Hume’s dedicated work on ethics. By relying on empirical observations of human nature rather than abstract religious or rationalist dogmas, Hume presents morality as a practical, natural tool designed by humans to foster social harmony and mutual well-being.

The Natural History of Religion (1757). Hume treats religious history as a natural, secular phenomenon, fundamentally undermining the belief that religion is inherently rooted in reason or divine revelation.

Dialogues Concerning Natural Religion (1779). His last book published posthumanously. It explores the psychological and sociological origins of religious belief, arguing that religion stems not from rational contemplation or innate knowledge of God, but from the emotional drives of human nature, primarily fear, anxiety, and the desire for security.

An Excerpt from My 2019 Book, "The Case against Miracles"

David Hume’s arguments against miracles succeed despite the fact he didn’t use Bayes’ Theorem. William Vanderburg briefly addresses this issue in his article for Hume Studies.[1] He shows Hume was aware of Bayes’ Theorem but didn’t think he needed it to make his argument. 

Earman chastises Hume for being unaware of Bayesianism and of mathematical probability generally. This is unfair on two counts. First, Bayes’s work on probability was not widely known in 1748 when Hume published the first edition of the Enquiry. Richard Price arranged the posthumous publication of Bayes’s essay only in 1763, and it remained obscure even after its publication; Price’s paper applying Bayesian methods to the evidence for miracles appeared in 1767. We know Hume read and admired that paper, but he neither addressed Bayesian arguments nor revised his account of miracles for the 1768 and 1777 editions of the Enquiry. This suggests that Hume ultimately did not view Bayes’s work as relevant to the argument against miracles. Second, Hume’s discussion of the probability of chances “shows without controversy that he was familiar with the basic concepts of probability based on the calculus of chances.” Given Hume’s familiarity with Pascalian probability in general, and his acquaintance (through Price) with Bayesian ideas, his non-numerical treatment of the evidential probability of miracles must be seen as a deliberate philosophical position, not as a result of negligence or ignorance.[2]

         What Hume is aiming at is seen in his general maxim: “Therefore we may establish it as a maxim, that no human testimony can have such force as to prove a miracle, and make it a just foundation for any such system of religion.” His twofold contention is not only that mere testimonial evidence for a miracle is insufficient for believing in a miracle, but also that miracles cannot be an adequate foundation for a religion. Hume is undercutting miracles and any religion born of miracles in one fell swoop. Going for the jugular vein of miracles does all the work, for it also deals a death blow to the religion of any miracle working god.

If miracles are the foundation for a religion, then an apologist for that religion cannot bring up a miracle working god to establish his supposed miracles. For miracles are supposed to be the basis for the religion and its miracle working god. One cannot reason backwards from the existence of their god to the miracle testimonies in the Bible. For the issue is whether the god of these miracles exists in the first place.

Don McIntosh Responds: "Reaffirming Belief in Miracles: A Brief Rebuttal to Loftus"

ABSTRACT:

In a hard-hitting and comprehensive reply to a previous paper by Don McIntosh defending the rationality of belief in miracles, John Loftus argued that McIntosh’s defenses are ultimately self-defeating: his denial of laws of nature in particular, coupled with his religious definition of miracles as “signs and wonders,” would leave no way to distinguish miracles from very rare but natural occurrences. In turn that would leave no way to discern true from false miracle claims, including the ancient and unverifiable claims in the Bible made by credulous and superstitious people. In this brief rebuttal McIntosh pushes back against Loftus’ arguments affirming inductive reasoning and the laws of nature/science, as well as his arguments against the divine creation of life, predictive prophecy, and miracles in the Gospels. McIntosh concludes that in light of the relevant information available, belief in a miracle-working God is rationally justified.

You can find it at The Secular Web. I don't care that much to respond. Perhaps my readers would want to do that?

On Resurrecting David Hume From the Dead!

I chose this title due to the double entendre. Hume, of course, might roll over in his grave if he saw it expressed in terms of a miracle. ;-) Here's what I wrote in my book, describing the uphill problem I have in defending David Hume on miracles in The Case against Miracles, p. 552].

Since Christians are convinced Hume’s arguments against miracles have been “refuted” (per William Lane Craig in a personal conversation), some atheists are trying to reach out to Christians by avoiding Hume entirely. Benjamin Blake Speed Watkins, who Hosts and Produces Real Atheology – A Philosophy of Religion Podcast, is one of them. He said, “I see arguments against miracles as separate from Hume. We can have substantive discussion without ever appealing to him.” Now this can be done, I don’t doubt. But I don’t see the need to do it. I also think we should give credit where credit is due. I can’t imagine ignoring David Hume, the philosopher who originally put the issue of miracles up for discussion, like no other had done before. That would be an injustice. I understand that Christians identify the best arguments against miracles with Hume’s arguments. I also understand they think Hume’s arguments fail. So I’m not oblivious they conclude that with Hume’s demise so also goes the case against miracles. Just the same, it makes sense to defend Hume if he can be defended, and I think he can. When we do so successfully it doubly undercuts Christian apologetics. For not only do we show Hume’s arguments defeat the believability of miracles. By showing Christians are wrong about Hume, it also shows how badly they argue their case on behalf of miracles.

Richard Carrier recently said I'm wrong to try, via email:

I think Hume should be dealt with much more briefly as having said something useful but centuries obsolete and thus no longer worth distracting ourselves with. That just helps theists, because they can throw up sand and argue endlessly over his choice of sentences and wording, when we could just say “why are you still talking about that, that’s centuries obsolete?” and thus pull the rug out from under that entire tactic. Force them instead to deal directly with the modern math. They will have a far harder time coming up with convoluted rhetoric to evade simple math than to make hay out of some long dead provocateur's 18th century prose.

Then again:

I don’t see value in trying to fix or explain hundreds years old obsolete stuff, even when clever or historically important. That wastes a lot of energy to no point. It’s great if you are doing history of the 18th century or something. But in philosophy, the name “Hume” should have no value argumentatively. So I think it’s pointless to try and fix “him.” Forget him. We have the benefit of two hundred plus years of scientific, logical, and mathematical advances since. We should just formulate the correct argument and be done with it. His name never need be mentioned; and mentioning it is just a distraction that triggers the howling monkeys of Christian apologetics. Make them confront the correct argument, rather than waste time arguing over what Hume’s centuries-old selection of words meant. Which fits into my ideal principle: never give a Christian an opening to derail a conversation by arguing over irrelevant trivia instead of the actual point being made. Try to prevent rather than facilitate that. It’s hard. But it should be the best aim.

From the outset, just to be sure, I have argued against miracles without depending on Hume. I mentioned him twice in two papers with just one link each. I did so for The Secular Web. One is titled, Hail Mary: Was Virgin Mary Truly the Mother of God’s Son?, and another one is titled, Assessing McIntosh’s Sophisticated Defense of Miracles.

How Can Belief in a Smart, Loving God Survive a Thoughtful Reading of Genesis 1-11 ???

There are too many flawed, naïve ideas about an ancient god in these pages

In my article I posted here a few days ago, I described a few of the problems posed by the first few chapters of Genesis 1-11. The most horrifying of which is Yahweh’s genocide designed to kill off all human and animal life on the earth—well, the earth as it was understood at the time. Only Noah’s family, and the animals he had collected in the ark survived. This is extraordinarily bad behavior on the part of this god. How can modern devout folks brush it off, overlook it—or pretend it wasn’t so bad?

William Vanderburgh On "Hume’s 'Abject Failure' Vindicated"

[First published in Dec. 2018] William L. Vanderburgh defended Hume against John Earman in a very thorough article published in 2005 in Hume Studies, titled, "Of Miracles and Evidential Probability: Hume’s 'Abject Failure' Vindicated." [You can read the PDF right here.] Go read it, now! In it Vanderburgh shows David Hume probably knew of Bayes Theorem and never mentioned it for good reasons. 

I'm including a few of the important highlights below. I consider it an important contribution on Hume, Earman, and Bayes Theroem.

David Hume “Of Miracles” and Religions (#97-100) with Notes by John W. Loftus

David Hume: “A miracle may be accurately defined as a transgression of a law of nature by a particular volition of the Deity, or by the interposition of some invisible agent.”
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Hume’s target was the Christian religion with its miracles. He argued against the probability of human testimony on behalf of miracles focusing on the Christian religion with its miracle claims. The root word “relig” with its endings (-ion –ions -ious –ionist) occurs twenty times in this chapter].

Below are a few paragraphs from the end of Hume's chapter (#97-100) with notes.

Don McIntosh Responds: "Reaffirming Belief in Miracles: A Brief Rebuttal to Loftus"

From the title it's clear I didn't convince Don. Maybe I should have used Bayes Theorem? It should be up at "The Secular Web" by next week. He started with these kind words though:

It’s an honor and pleasure to be interacting with John Loftus again, this time on the question of miracles. A well known and widely published atheist philosopher, counter-apologist, and critic of Christianity (and miracle claims in particular), Loftus has answered my 2025 article defending the rationality of belief in miracles and concluded that my arguments are inadequate to justify such a belief. In the process Loftus has managed to strike a friendly and respectful tone even while hitting back with replies that are insightful, vigorous, brutally honest and rhetorically formidable. That’s just how it should be, I think, and I will do my best to answer in kind.

His initial defense of miracles can be seen here: LINK

My one and only rebuttal can be seen here: LINK.

One Thing More: We Don't Need No Bayes Theorem!

Here's a paper I published at The Secular Web on the virgin birth of God's Son, titled Hail Mary: Was Virgin Mary Truly the Mother of God’s Son? In it I present a good case against such a miracle. Readers can judge for themselves without having to follow the math of Bayes. In fact, MANY MORE readers can follow my paper since they don't need to follow the math, while those who want it, can do it. Mic Drop!

Bayes Theorem with Regard to Vincent Torley

I previously mentioned knowing only one Christian who utilized Bayes and calculated a lower probability for the resurrection of Jesus than before. He went from a 60% probability to 50% to 25% probability. He still believes because after all, it's faith. All other Christians who use Bayes end with their probabilities well into the probability zone, some as high as 95%!

That's what I see. All but one Christian who utilized Bayes comes away with a high probability that Jesus arose from the dead, or that Christianity is true, or that God exists. On the surface Bayes only helps believers confirm their faith. Bayes ought to help believers re-think their faith, but it does no such thing. Granted, not much can change the diehard believer. But my point is that there is something wrong with an allegedly helpful theorem that doesn't force more believers to change their minds [edit: or the minds of non-believers for that matter]. If using Bayes won't help them [edit: or us] why bother?

The Christian believer I mentioned is Vincent Torley. You can read his cogitations on the matter at The Skeptical Zone. Here's his conclusion:

Since my estimate of the total probability of the various Type A skeptical explanations is less than 50%, and since the posterior probability of the Resurrection is much greater than that of the various Type B explanations, belief in the Resurrection is rational, from my perspective.

Based on the evidence, I estimate that there’s about a 60-65% 55-60% chance that Jesus rose from the dead. That means I accept that there’s a 35-40% 45-50% chance that my Christian faith is wrong. [Notice his lowering edits?]

However, I can understand why someone might rate the probabilities of hypotheses 3(a), 3(b) and 3(c) at 20% each, instead of 10%. For such a person, belief in the Resurrection would be irrational, since the total probability of the Type A skeptical hypotheses would exceed 50%.

Summing up: a strong case can be made for the reality of Jesus’ Resurrection. However, a responsible historian would not be justified in asserting that Jesus’ Resurrection is historically certain. As we’ve seen, such a conclusion depends, at the very least, on the claim that there is a significant likelihood that there exists a supernatural Being Who is capable of working miracles, which is something the historian cannot prove. In addition, estimates of the probabilities of rival hypotheses will vary from person to person, and there seems to be no way of deciding whose estimate is the most rational one.

Later, after reading through Michael Alter's book, Resurrection: A Critical Inquiry, Torley came to calculate the probability of the resurrection of Jesus to be 20-25%. LINK But here is the rub that doesn't feel too good. Torley was not prompted to change his mind due to any Bayesian analysis since Alter doesn't use it! Alter merely hammered home a comprehensive evidential case! Checkmate Bayesians!

There Is So Much Bad Theology, Rubbish, Goofy Folklore—Just in Genesis

It's a way to get Bible study off on the wrong foot

For a long time, there’s been a meme floating around Facebook, with this message:

“To most Christians, the Bible is like a software license. Nobody actually reads it. They just scroll of the bottom and click, I agree.” 

These days we are swamped with so many forms of entertainment, e.g., movies, TV, sports and hobbies, travel and vacations. In this situation, it’s a rare devout person who decides to read the Bible from cover-to-cover—maybe on the chapter-a-day plan. After all, a magnificent copy of the Bible is on display on the church altar. It must be worth reading, right? That adoration might continue if the devout reader embarks on this endeavor without critical thinking skills fully engaged.

A.I. On How Bayes Theroem is Best Used

I asked A.I. How Bayes Theorem is best used.

The answer:

Bayes’ Theorem is best applied in the following scenarios:

Medical and Legal Diagnostics: 

It clarifies confusing test results. For example, if a test is \(99\%\) accurate for a disease that only affects \(1\%\) of the population, a positive test actually means you are far more likely to be disease-free (high false-positive rate).

Machine Learning: 

It helps software automatically identify spam emails by checking the likelihood of certain words appearing, given that the email is spam.

Risk Assessment & Forecasting: 

Investment analysts use it to continually update the probability of market trends or a borrower's default risk as new quarterly data or economic conditions emerge.

Bayes Theorem On Trial

Only one Christian believer I know who used Bayes, went from 60% probability to 50% to 25% probability. He still believes because after all, it's faith. All other Christians who use Bayes end with their probabilities well into the probability zone, some as high as 95%!

That's what I see. All but one Christian who utilizes Bayes comes away with a high probability Jesus arose from the dead, or that Christianity is true, or that God exists.

On the surface Bayes only helps the believer confirm his or her faith. Bayes ought to help believers re-think their faith but it does no such thing.

Bayes On Trial: The Problem of Bayes in a Nutshell

In a nutshell what's wrong with using Bayes Theorem: Using it to determine the probabilities of miracles in the ancient pre-scientific world. The problem is the attempt to take miracles seriously. The biblical authors never thought anyone could prove they happened as recorded.

Bayes Theorem On Trial!

BAYES ON TRIAL! Imagine trying to convince someone of Einstein's equations, Isaac Newton's second law of motion (f=ma), or of biological evolution, by using Bayes Theorem. We never do that, nor do we use it on a great many other evidence based issues. The objective evidence is convincing all on it's own.
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A.I. on Bayes Theorem. I typed in these words: "Criticisms of bayes theorem theory"
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While Bayes’ Theorem is an undisputed mathematical formula, criticisms center on its application in Bayesian statistics and epistemology. The primary debates involve subjectivity in prior beliefs, methodological ambiguity, and computational/logical limitations.

1. The Subjectivity of Prior Probabilities (The Problem of the Priors). Bayes' theorem requires defining a "prior"—an initial assumption or belief about the probability of an event. Critics, particularly classical/frequentist statisticians (e.g., Fisher and Neyman), argue that because these initial beliefs are subjective, the resulting probability is also subjective, making it scientifically unreliable. If a prior is heavily biased or uninformed, it can take an excessive amount of new data to correct the final outcome.

2. Lack of Uniqueness (The Objective Bayes Dilemma). To counter subjectivity, "Objective Bayesians" use uninformative priors (representing ignorance). However, critics argue there is no single mathematically rigorous way to define a "true" state of ignorance. Depending on the mathematical method chosen, the resulting prior can inadvertently introduce hidden assumptions that distort the analysis.

3. Falsifiability and Flexibility. In scientific modeling, the sheer flexibility of Bayesian frameworks (with choices of priors, likelihoods, and utility functions) allows researchers to continuously adjust models until they achieve desired results. Critics argue this makes Bayesian theories susceptible to becoming unfalsifiable, or used to spin "just-so stories" to validate pre-existing conclusions rather than testing them objectively.

4. The Bayesian Trap and Pseudoscience. In applied settings, misinterpreting the odds or failing to seek alternative explanations can lead to over-diagnosis and confirmation bias. Practitioners can fall into the "Bayesian trap", where strong prior assumptions or repeated observations lead to a false sense of certainty, blinding the researcher to anomalies or the need for new experiments.

5. Logical Omnipotence and Unification. When applied to human reasoning or epistemology, critics point out the problem of logical omnipotence. Pure Bayesian theory assumes that a perfectly rational agent instantly assigns probabilities to all possible alternative hypotheses and knows all logical consequences of their beliefs, which is an impossible cognitive burden for actual human beings or finite computers.

6. The "Bayesian Brain" Circularity. In cognitive science, the Bayesian brain hypothesis suggests that perception and learning are based on probabilistic inferences. Critics argue this approach can be ambiguous and methodologically circular—suggesting that because a system "looks" rational, it must be using Bayesian calculations, despite a lack of neurological evidence for such complex mathematics at the process level.

Richard Carrier Recommends Three More Books of Mine!

The category of books here is the Philosophy of Religion (POR). Below are summarized comments from him. The essay is linked below.

One) Unapologetic: Why Philosophy of Religion Must End (2016).

Unapologetic is an important book to engage with, whatever you think of its thesis in the end. And being the metaphilosophy of POR, it is an essential have for any POR shelf or library.

Two) God and Horrendous Suffering (2026).

What Loftus has assembled is a huge and excellent series of unique and well-sourced essays by numerous experts that supplements any discussion of the Argument from Evil (AfE). You won’t find the brilliant takes and studies and analyses in here anywhere else in one place. And it’s so good and far-reaching, it’s definitely a must-have for any library or list that you expect to cover the AfE well.

It opens with a professional journal debate on the AfE between John Loftus and Don McIntosh. And then chaper-by-chapter covers every conceivable digression in solid form.

Overall, every chapter has use in boning up on why the AfE is actually a powerful argument with no credible defense yet, all from different angles of methodology and subject. All of it quite good. As with any book I can’t have vetted every fact claim in it, but at my reading, it’s generally solid, and its approaches so diverse and unique that I think God and Horrendous Suffering is an essential appendix to any fullgoing knowledge of AfE. You will get valuable information and fresh perspective from every chapter.

Three) The Case against Miracles 2022).

It’s a subject I’ve long been a specialist in, and I can vouch for the fact that there is nothing better in its category.

It contains a lot of great analysis from a plethora of experts, all more than sufficient to make The Case Against Miracles an essential addition to any POR library.

Conclusion:

All three of these books get overlooked a lot, which is why I am bringing them back into view.

But as I don’t research this subject so thoroughly as to know all the best literature on every subject in so vast a heap of tinfoil as the POR, and I think PoR is a really annoying distraction anyway, I won’t be building a recommendations list in it. But if you are interested in building one...the Loftus volumes I just discussed today, plus the two items that are on my Philosophy List (Outsider Test for Faith, and Primer), make a really good—and essential—start.

LINK.

An Interview on Atheist Morality

Dear Dr. Loftus, 

I am a secondary school student currently researching the historical change in morality and its link to the rise in secularism for a literature project. I recently read your essay about atheist morality and found it very helpful for understanding secular ethical perspectives from a young Christians view. Thank you!

I was wondering if you might be willing to answer a few questions about your argument.

1. In your essay, you argue that morality can develop naturally through human evolution and social cooperation. Do you believe this completely rules out the possibility that morality could have a divine origin? Possibilities do not count! I have fully accounted for morality.  

On Assessing McIntosh's Sophisticated Defense of Miracles

Do you want to read a sophisticated defense of miracles and an equally sophisticated rebuttal?

The defense of miracles was written by Don McIntosh: LINK

The rebuttal was just published today, written by me! LINK.

Why Won't William Lane Craig Debate Me? (Plus Further Comments On Horrendous Suffering).

In this episode I started watching and I commented. Phil Harper was there!!! Then unexpectedly Derek Lambert sent me a link to be in the show. That was nice! I think it went well, starting early on with the question why William Lane Craig won't debate me. 

 

  

 

I had mentioned my novel approach to the problem of evil, calling it the minimal facts approach, LINK. Get my book on horrendous suffering, LINK. Derek Lambert's show is most excellent! Check out his other amazing videos!

Dr. Richard Carrier Highly Endorses My Book!

Thank you Richard Carrier for this enthusiastic endorsement of my book, "The Outsider Test For Faith"! Thanks also for accurately applying the outsider test to areas outside the philosophy of religion! LINK

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