Showing posts with label Probability. Show all posts
Showing posts with label Probability. Show all posts

Wednesday, 4 February 2026

Answered Prayer & Probability

 

A lot of people don't believe that God answers prayers. I will show you why we can be reasonably sure He does. God answering prayers is basically this; a Christian prays for x, x happens, therefore x happened because God answered the prayer. That happens a lot, but this apparent pattern is complicated by two common exceptions: times when a Christian prays for x, and x does not occur; and times when x occurs without anyone praying for it at all.

It is easy to understand why both exceptions occur, though; when a Christian prays for x, and x does not occur, we ought to conclude that x was the wrong prayer or not aligned with God's will. And x occurring without anyone praying for it at all does not tell us anything much about whether God answers prayers, any more than a fair coin landing heads tells us whether coins in general are biased.

A more appropriate way to affirm that God answers prayers is to look at what is being prayed for, and ask what the likelihood of the event is anyway without the prayer. For example, suppose Jack has a deck of cards, and prays that, after shuffling them, he will draw the king of diamonds. Since there are 52 cards in the deck, there is a 1 in 52 probability of this happening by chance alone. If 52 independent people were each to pray that they would draw the king of diamonds from their shuffled deck, probability suggests that, on average, one of them would succeed and might therefore conclude that their prayer had been answered.

What we are therefore looking for, in order to justify belief in answered prayer, is to consider events that occur where there is an astronomically low probability that it would happen by chance, but which are specifically and unambiguously prayed for in advance. This requirement precludes cases in which low-probability events are later interpreted as answers to prayer simply because they happened to occur, and excludes low-probability events that do in fact occur by chance, but lack any prior, specific prayer corresponding to them.

At this point, the sheer weight of Christian testimony should be ample evidence that God answers prayers - the kind of prayers which significantly undermine objections such as 'it is only anecdotal testimony' or 'there is no medical verification'. I have been instantly healed a few times immediately after prayer - from a pulled muscle in my leg, and from a chronic tooth pain - and I have witnessed a blind person given their sight back immediately after prayer, a severely deformed leg twist around, grow and be restored immediately after prayer, and a lady crippled and confined to a wheelchair all her adult life stand up and walk immediately after prayer. And they are merely a few experiential drops in a sea of miraculous testimony amassed worldwide.

The likelihood of any one of those events happening by chance is astronomically low, so the likelihood of any one of those events happening by chance immediately after prayer is even lower (for obvious reasons). Therefore, the most likely explanation is that these immediate healings were answers to prayer - especially when these events are considered cumulatively. If the likelihood of an instant healing is almost zero and it happens straight after prayer, the best explanation is the prayer caused it to happen. If the same thing repeats five times in just my experience, the best explanation being answered prayer becomes cumulatively stronger. When this pattern is multiplied across the experiences of Christians more broadly, answers to prayer become so much more plausible that it would be difficult to dismiss without adopting an unjustified scepticism.

 

EDIT TO ADD:

A friend asked about why miracles aren’t more attested to in terms of medical statistics.

My comment: Let me start with a question for you; even if we simply focus on the prayer examples I outlined - to keep it concrete and less abstract – I wonder why you focused not on them but on the proposition that if prayer worked like a predictable medical intervention that it would yield statistical differences in population health data. Please don’t misunderstand, I understand the appetite to broaden it to a wider empirical investigation, but if one is faced with gold standard evidence with 5 evident miracles, then a lack of consistent, predictable formal medical outcomes is not adequate to undermine it by itself.

I agree with your point that prayer and belief in God can enhance outcomes through psychological and community engagement – and given the truth of Christianity, one could reasonably expect that any behaviours that align with His truth can be expected to enhance well-being and utility.  But most Christians understand that prayer is about relationship with God Himself, and is therefore unlikely to be friendly to statistical analyses when treated too mechanically. God is not a God who likes to be tested as though He has something to prove – especially if one doesn’t approach Him with humility (not saying you lack humility). Once one is in a relationship with Him, and gets to experience the power of His love, grace and what He can do for us, those kinds of empirical probing can only fail to enchant. A bit like if a bodybuilder has shown us he can bench press 200kg, investigations about whether he can curl two 5kg pink dumbbells seem quite remiss.

Incidentally, I don’t think you’re being closed-minded or churlish at all - you’re attempting to apply a consistent standard of evidence, which is the very bedrock of empirical investigation. And it remains an interesting question about how some truthful phenomena manifest primarily through population-level statistics and how some operate more at the level of cumulative testimony. But as I’ve argued in other articles, the cumulative testimony in favour of Christianity is one of the strongest pieces of evidence for its truth. 

Saturday, 3 January 2026

Beware Richard Carrier: Don’t Fall For His Deceptive Methods

 

Perhaps the best thing that’s ever happened to slippery grifter and unpleasant manipulator Richard Carrier on the Internet is my set of articles in which I try to help him reason better, be more honest, and become a kinder person (see hereherehere and here). But, alas, he got totally triggered and decided to write a horrendous dismissal of my critique of his terrible article on using Aristotle’s existence to undermine Jesus’ existence.

Now, I normally wouldn’t waste this sort of time on discredited people like Carrier – he’s the young earth creationist of the world of mythicist history. But as regulars will know, I do use the exposing of fraudulence and bad reasoning to showcase deeper psychological and philosophical tools at play in how masses of people can be manipulated, in the hope that it will help others see through the cheat. So, blog posts like these can serve as useful tools in aiding readers in critical evaluation and spotting bad arguments, even when they are - and perhaps especially when they are - presented with the outward appearance of sophistication.

Carrier’s modus operandi is a classic case in point – and while he’s not taken seriously by virtually all credible professionals in any of the subjects about which he writes extensively, he does seem to have assembled a small micro-cult following from a few thousand people who he’s duped into thinking he’s someone worth following because he’s onto things that even most experts have missed. Spoiler alert: he’s not; he’s a crank, and, from my personal experience, a deeply unpleasant one at that – which I’m going to expose, in a moment by showing how he manipulates the subject matter to sound like he’s offering sound refutations, but first, on the psychology of how his readers could possibly fall for such a blatantly untrustworthy front man.

I don’t care if Carrier reads this – he is almost certainly never going to change. But I do want his readers to read this, because they deserve better, and should want so much better for themselves, just as I want so much better for them. I’ll use Carrier’s own words from his article criticising me to show you every element of his deception, in the hope that it will help his readers see through him, and in the hope that it will give my readers a lesson more generally in how to see through people like Carrier.

Because the first thing that makes people like Carrier interesting for all the wrong reasons is how it can be the case that his readers can’t see through him, when it should be so much more obvious to them that he’s not to be trusted.

As a case in point, let me state right here that I’m going to submit this on the comments section of his blog post, but I’ll bet you don’t see it appear – because he won’t allow anything in the comments that shows him up at this level.

Before I get to Carrier’s errors of reasoning and mistaken arguments, I’ll first expose the primary psychological cheats he employs. 

1) The dismissive insult
Carrier’s tactic of insulting dismissals, the one he employs most often (“C.S. Lewis may have been the worst philosopher of the twentieth century”, “AI Is garbage“, “Jordan Peterson is a crank”, any expert who disagrees with him is a ‘buffoon’, ‘doofus’, ‘bozo’ etc ), is a deliberate strategy employed by many manipulators of his nature. Because Carrier’s worst fear is being exposed, his ‘go to’ strategy is to try to intimidate readers and critics into thinking disagreement = stupidity. It’s an attempt at psychological dominance and frame control, where his own followers are made to feel judged for even contemplating someone is onto him, and encouraged to dismiss not just all the experts who have exposed his work as wrong and his arguments as sub-standard, but be dissuaded from checking it out for themselves. That is what cult leaders do – it’s as old as the hills.

Of course, it can only work on people either too uniformed or too malleable to spot the cheat. I’ll give you an example which appears in literally the first thing he says in introducing his critique of me; he says “Today I make an example of an internet buffoon”. Now wait. Just stop and ask yourself several things about that. Any of his readers could look at my blog and find exactly who I am. I’m someone with over 1000 blog posts on the Philosophical Muser site to which he linked, and hundreds of articles of mine for other institutions. In that body of work you would find some of the most comprehensive and consistent in depth analyses of most subjects out there – there is no way anyone with half a brain could look at my material and write me off an as ‘Internet buffoon’. Moreover, Carrier knows I’m not an ‘Internet buffoon’, so why did he lie and say I am? In other words, why is the very first thing he said a lie? It’s easy, because unless he lies, he can’t get his readers on side, and by starting with a blatant lie he can lay the foundation down to have them fall for his nonsense reply to my article.

Think about what that means if you’re one of his readers – he just lied to you, knowing that if you searched through my work you’d know he’s lying because you’d know I’m not an ‘Internet buffoon’ – I’m actually much more on top of these subjects than Carrier (which is why he refused to debate me live on video – see below). But it also shows something else telling; it shows that Carrier has contempt for his readers – he doesn’t care that you could find out the truth really easily – he’s just expecting you to swallow it without care because he has so little regard for you. And if you think about it, that’s exactly what you’d expect. He can’t really value his readers beyond your merely as serving as props for his arch manipulations, because he thinks most of you are so gullible you’ll just swallow it up hook, line and sinker. How can he respect people who he knows fall for his cunning deceptions, and thank him for it afterwards?

It reminds me a bit of the Nigerian prince scam, where virtually nobody falls for it, but the scammers still use the Nigerian prince method on the basis that if you're one of the few gullible people left in the world who hasn't been apprised of the Nigeria scam and are likely to fall for it, you will be of those for whom the uniformity of the 'Nigeria' email won't be alarming and prohibitive. With the simplicity of using Nigeria each time, the scammers save having to waste time with all the false positives, and they will continue to catch in their net the few gullible fish still in the sea. Carrier has contempt for his followers because the fact that you’ve become one of his followers has already showed him how gullible you are. As said, I want so much better for you.

By the time he’s moved on to his next line in the article, he’s already told a second big lie: saying my private messages to him reflect “someone painfully immature and emotionally irrational, with no interest in learning or understanding or engaging in any kind of intellectual discussion”. Well, yes, they do – but they reflect those things about Carrier, ironically – and I have screen shots of them to back that up if ever needed. I won’t labour on that point, but suffice to say, Carrier wasn’t willing to take up my offer of a live debate – where he knows there’s nowhere to hide and where the fraudster is forced to think on their feet, and can’t hide behind carefully manipulated, long-winded written texts – he showed a firmer desire to have a written debate, but when I left a voice message to explore this, he said, ‘I don’t listen to voice messages’ (which is ridiculous in this context, but expected from a narcissist like him) – and then when I sent him a written message giving some suggestions about what we could debate, he simply brushed it off with ‘tl;dr’. He’s an absolute joke of a man.

I mentioned that he’s a narcissist – which you can probably sense by his own words on his Facebook profile:

Yes, I'm the real Richard Carrier. Sadly, I'm much too busy to hang out here much. But I have to maintain a FaceBook page to prevent malicious hacks from pretending to be me.

What a scream that is – compounded by the fact that most narcissists have no idea how deep their own narcissism runs. Virtually no one outside his cult knows who Richard Carrier is, yet he has the audacity to assert “Yes, I'm the real Richard Carrier”, who’s “much too busy” to hang out with the likes of you lot, but I’d better have a page to “prevent people from pretending to be me”. Haha, I honestly can’t think of a single scenario in the world where someone would have the even a flicker of inclination to pretend to be Richard Carrier. Hilarious. 

2) Deflection through projection
Now, let me show you another tactic employed my manipulators like Carrier – he chooses terms of insult, like ‘childish’, to shift attention away from his deceptive methods and onto a manufactured image of his critic as the villain. It’s one of the oldest psychological tricks that insecure thinkers like him play. He wants you to think I’m childish so you don’t take my criticisms seriously - in fact, he uses this accusation that I’m childish EIGHTEEN times in the article, no less. Again, it’s another fat lie that shows utter contempt for you, his readers, because you can quite easily find out for yourself that I’m not a childish person. What I do have, as evidenced from highlights of what I sent Carrier in messenger over the past few months, is the measure of him, and a desire that he no longer gets away with his manipulations online. That is the opposite of childishness. Here are some examples of what I sent to him: 

I've looked at some of your writings today, and two of the primary things that stand out are this: 

1) You pretty much always conjure up in your head what you want the conclusion to be, and then use any squalid method possible to argue for that. 

2) Your body of work is similar to that of a young earth creationist, except with the signs reversed. You leave out so much of the vital material, you distort the arguments of your opponents, you argue dishonestly, and your operate on counterfactual peripheries in the hope of forging a niche reputation, where any expert in the field would easily show your ideas to be wrong, and you  produce material that relies on the ignorance and pliability of your audience, where your only recognition comes from people too intellectually lightweight to see through you. 

Your writings don’t come across that way – the most important thing to you is how others perceive you, and your asset or non-assent to truth is driven by how you think you can get more popular and regarded, which means claiming to hold positions that are niche enough to give you a certain type of following. You also claim to care what people who disagree with you think, but that is not backed up by your policy of limiting comments on your page/articles – which is exactly what one would expect of someone about whom the above agenda applies. 

Have you not noticed that you always write off people as cranks, when you're the crank? Have you also noticed that cranks, like YECs, almost never do live debates either, because they don't want to be exposed? 

Even though I was being friendly, the reason you reacted unpleasantly is because all narcissists do this when they are deeply insecure and have been found out by someone who has the measure of them. Biting back aggressively is a classic sign of being found out - it’s what Waugh referred to in one of his letters as “the concealed malice of the underdog”. All that you said in your hostile response is undermined by the fact that you don’t have the courage or intellect to debate me live, where you would have to display your mind, think quickly on your feet, and not hide behind walls of manipulative text within the cosy self-incarceration of your echo chamber.  What you do, sir, is not truthseeking, and nor does it show good character – it’s a cliched Promethean cheat that will only bring you shallow and transitory pleasures – not ultimate joy, fulfilment, contentment or wisdom. When you reach the age of realisation of this, I’ll be happy to have a civilised conversation with you.

The fact that you dismissed all that in a few minutes shows me you can't be taken seriously, and just want to believe what's convenient for you.

And the same with all the attention-seeking about being godless, polyamorous, etc. The big contradiction at the heart of polyamory, I think, is that a person who is not mature and cultivated enough to comprehend how their needs can be fulfilled in the right monogamous relationship with one beloved, will also not be mature and cultivated enough to realise that their needs cannot be fulfilled through multiple relationships or sexual unions, however plentiful the number and varied the experiences. Things can still be much better for you, but not until you change and become wiser. And if those comments induce rage in you, that's because you know how much force it takes to defend a fragile house of cards.

See what I mean? Not childish at all, is it? It’s hard-hitting, sure, but it’s hard-hitting because people like Carrier need to be told that they won’t always get away with things. And there’s nothing in any of my public works that is childish or immature. It’s only what Carrier needs you to think to put the guard up against his being exposed. Carrier, on the other hand, does appear childish in virtually all his conduct – private or public. He is the one who writes off people with crass dismissiveness; peppers his blog posts with puerile Americanisms like ‘doofus’ and bozo’; isn’t even mature enough to sustain a faithful, monogamous relationship, and insults anyone who doesn’t let him get his own way. That is the epitome of childishness. 

The arguments
Let’s look further now at his attempted rebuttals, because they serve as a good lesson in how to ‘sound’ clever and competent when really it’s a litany of further errors of reasoning, bad arguments, and YEC-like distortions of data and fact. His attempted rebuttal fails at every critical level the first article does, but to an even greater extent, because this time he is trying harder to dig himself out of a bigger hole. 

He still rigs his Bayesian model from the outset by inventing reference classes that already assume Jesus is fictional, still assigns precise-looking likelihood ratios that are wholly uncalibrated, empirically ungrounded, and indistinguishable from personal intuition dressed up as mathematics; he still compounds this by multiplying dependent, culturally and textually interconnected evidence streams as if they were probabilistically independent, grotesquely inflating his results in violation of basic Bayesian requirements; he still, in classic Carrier style, responds to criticism by redefining dissent as ignorance of Bayes; and yet again, his ultimate proof of failure is that his method reliably produces fringe conclusions rejected by the historical discipline at large, not because historians fear probability theory, but because his model fundamentally misunderstands how historical evidence, sources, and uncertainty actually work.

To save wasted time going through dozens of his errors, I will just pick the ones I think come across as his best attempts at rebuttals – which are still sub-standard – but they are the ones that I think will most fool his readers, because on the surface they look like the most sophisticated. Again, I’m doing this to expose the subtle art of squalid manipulation, and show how it can easily come across as highly sophisticated, intellectual criticism. We’ll start with his comments first, then my comments underneath.

Richard Carrier 1#: Muser confuses the fact that we have evidence specific for Alexander the Great that keeps his posterior probability high with meaning his prior probability should also be high. It’s the other way around: if all we had were the wild myths of Alexander, his prior and posterior probabilities would be low, and all historians would agree—as they do for Hercules and Osiris, for example. The only reason we conclude Alexander existed is because of the evidence specific to him. Therefore, mathematically, his prior probability, which is the probability prior to considering all that evidence, cannot be affected by that fact. It therefore would not go up. It would stay low. It would remain in fact exactly what it would be if that evidence didn’t exist. That’s literally the point of the prior probability.

My comment: Oh wow, boy is this man slippery in his confusion. Most extreme people online either speak cleverly and competently or they speak foolishly and incompetently – whereas Carrier is one of those rare beasts who tries to perfect the art of sounding clever and competent while misleading you with hollow claims. I believe that’s why he draws in a certain unsuspecting following. Carrier’s claim sounds confident, but it is conceptually mistaken, and the mistake reveals exactly where his Bayesian reasoning departs from how priors actually function in historical inference. It’s basic stuff; a prior is not “what you would believe if you erased all relevant background knowledge and pretended only myths existed.” That is not what a prior is in Bayesian reasoning, and it is certainly not how priors are set in any applied field, including history. A prior is the probability of a hypothesis before considering the specific evidence under adjudication, given all relevant background knowledge. Background knowledge is not “the evidence being tested,” but the wider contextual facts that frame what kinds of hypotheses are plausible in the first place.

Carrier’s hypothetical - “if all we had were the wild myths of Alexander” – is a high school category error. If all we had were myths, then those myths would themselves become part of the evidence under evaluation, and the background knowledge would change radically. In that imaginary world, historians would also lack knowledge of Macedonian kingship, successor states, administrative continuity, geopolitical causation, and the normal sociological patterns of royal propaganda. That is not a legitimate way to define a prior; it is a counterfactual that strips away the very background conditions that inform rational priors.

For Alexander, the prior is higher than for Jesus not because of later literary attestation, but because the background facts already make his non-existence implausible (that is, empires do not appear without founders, successor states do not arise from fictional conquerors, and administrative and political continuity does not emerge from pure myth, that sort of thing). Those facts belong in the prior because they are not the disputed evidence; they are the historical context that constrains what explanations are reasonable. Ignoring them is methodological carelessness.

Moreover, Carrier’s statement that “that’s literally the point of the prior” reverses the concept. The point of a prior is not to simulate epistemic ignorance; it is to encode justified background expectations before updating on contested evidence. The problem is not that Carrier uses priors - obviously; it is that he defines them in a way that excludes precisely the kind of background knowledge historians are obligated to use. If you Google research Carrier’s historical disputes with others, you’ll see this plays out repeatedly in his tactics. And please do the research and see for yourself.

Carrier’s other repeated mistake here is the category error of treating priors as if they must be set in a historical vacuum stripped of all contextual background knowledge except “myth vs non-myth.” That is not how priors work in applied Bayesian reasoning, and it is certainly not how historians reason. A prior is not “what you would think if you pretended no contextual evidence existed at all”; it is what you assign before evaluating the specific evidence under dispute, given all relevant background knowledge. Carrier’s attempt to freeze the prior at a myth-only baseline misunderstands that priors legitimately incorporate historically grounded context, not just genre labels. In short, Alexander’s prior is higher than Jesus’s not because of the later literary evidence, but because the background conditions surrounding Macedonian kingship already make non-existence unlikely.

Richard Carrier 2#: Only when we add that evidence does it go up, but that’s then the posterior (not the prior) probability; and all of the change from the one to the other is in the likelihood ratio, not the prior. Alexander’s prior stays the same as for Jesus, which I set at 1 in 3 in my article on Aristotle, same as my original study (it has since been reduced to 1 in 4, as I there explain, but I set that aside to argue a fortiori). That Muser doesn’t know this shows he has no actual understanding of Bayesian reasoning or mathematics, and thus is not at all qualified to critique any application of it.

My comment: Look how he uses forceful rhetoric to make a statement that laypeople probably wouldn’t spot is just flat out conceptually wrong. Carrier is correct about one narrow point of formalism – that likelihood ratios are what update priors. But he is flatly wrong to insist that Alexander’s prior “stays the same as for Jesus.” That claim only follows if one defines priors in an artificially stripped-down way that excludes historically relevant background knowledge. Bayesian reasoning does not require identical priors across hypotheses; it requires justified priors given what is already known before examining the specific evidence at issue.

The crucial mistake is that Carrier treats the prior as though it must be set before any historically meaningful differentiation is allowed. But priors are not epistemic blank slates. In historical inference, they legitimately encode background asymmetries such as political context, sociological plausibility, institutional continuity, causal footprint, and what have you. Alexander is embedded in a well-understood class of phenomena - Hellenistic kings, imperial conquest, succession crises, and administrative aftermath - that rarely, if ever, arise from fictional individuals. Jesus, by contrast, belongs to a class of itinerant religious preachers in a highly myth-making environment where symbolic invention and theological personification are known mechanisms. These asymmetries belong in the prior, not forcibly suppressed until the likelihood stage. The fact that Jesus is so well attested is what shifts the posterior, but it does not retroactively inflate his prior; the background context keeps his prior lower than Alexander’s, even before examining the evidence.

By insisting that all differentiation must occur through likelihood ratios, Carrier is not following Bayes neutrally; he is imposing a modelling constraint that artificially equalises starting positions in order to push all discrimination into later stages where he can manipulate likelihood assignments. That is not honest work.

Finally, Carrier’s accusation that disagreement here shows “no understanding of mathematics” is equally dishonest. The objection is how he creates the parameter. One can fully understand Bayesian updating and still reject a model that defines priors in a way that deliberately excludes relevant background knowledge. That is methodological disagreement – and the fact that he thinks it’s a mathematical disagreement shows he doesn’t really understand what is being debated here, never mind where his errors lie.

In fact, I’m probably being too charitable still – because this is where Carrier’s Bayesian formalism detaches completely from historical reality. He is correct mathematically that updating happens via likelihood ratios - but he is wrong to insist that priors must therefore be equalised across radically different historical contexts. Bayesian reasoning requires justified priors, not identical priors. Treating Jesus and Alexander as starting with the same 1-in-3 prior simply because both later acquire mythic narratives ignores massive asymmetries in background knowledge: political structures, documentary cultures, social incentives for invention, administrative continuity, and the presence or absence of motive and mechanism for fabrication. Bayesian models that flatten these distinctions are not neutral; they are distorted. Carrier’s accusation that disagreement here shows “no understanding of Bayes” confuses what I would call mastery of a formula with competence in model construction. He can’t see this because he’s not on top of this subject. The sleight of hand trick is in deliberately stripping priors of historically relevant information in order to force all differentiation into the likelihood stage – well, I saysleight of hand trick’, but with Carrier it’s not always evident whether he’s being deceptive or just confused, because he is both so often.

That is not principled Bayesianism; it is model engineering - and I see the same thing with so many other guru figures that the question of honest ignorance or deliberate deception continually looms large. Carrier is more like the equivalent of YEC’s Grady McMurtry than Ken Ham. I’m convinced that Ken Ham is a simpleton as well as dishonest. I’m not sure Grady McMurtry is that simple – but he’s cunningly deceptive, and Carrier strikes me as more like the latter.

Richard Carrier 3#: This is also demonstrated by the fact that he didn’t know my likelihoods for Aristotle not existing already take into account that evidence’s dependency on “a Peripatetic tradition.” He didn’t even know that this is always the case, that it is literally what you are doing when you estimate likelihoods in Bayesian reasoning: you assume the one condition is true (“Aristotle existed”) and then assess how likely the evidence would be dependent on that being the case; then you assume the other condition is true (“it was just a Peripatetic tradition”) and assess how likely the evidence would be dependent on that being the case; and it is the ratio between those two likelihoods that tells you whether that evidence argues for or against either condition (or neither), and the size of that ratio (the difference between those two likelihoods) tells you how weak or strong that evidence is for that conclusion.

My comment: Here Carrier confuses naming dependence with modelling it. While he correctly describes the formal definition of a likelihood ratio, merely imagining an alternative hypothesis (“it was just a Peripatetic tradition”) does not automatically control for evidential dependence. When multiple pieces of evidence arise from the same causal pipeline, they are correlated, and multiplying likelihood ratios still double-counts information unless that correlation is explicitly modelled. Carrier does not do this. He assigns separate likelihood ratios to Aristotle’s writings, testimonies, inscriptions, and later historians, and multiplies them anyway; reducing the numbers does not fix the inflation. Worse, he applies this leniency asymmetrically: dependence is “handled” for Aristotle but used to collapse multiple Jesus traditions into near-total dependence. Again, is it confusion or dishonesty? Either way, it is not principled Bayesian reasoning; it is informal judgment expressed with the appearance of rigour but none of the control. The issue is not whether he conceptually acknowledges dependency, but whether his numerical assignments properly discount correlated evidence streams. In the Aristotle case, nearly all the evidence flows through the same Peripatetic–Hellenistic transmission networks. Treating each as yielding an independent likelihood ratio (even a reduced one) still double-counts shared provenance unless explicitly modelled as conditional on the same causal chain. Carrier never provides such a model; he simply assigns smaller numbers and multiplies anyway. That is not Bayesian rigour, and his readers don’t spot this. Worse, he applies the exact opposite standard to Jesus, collapsing multiple traditions into near-total dependence even when they demonstrably diverge in theology, chronology, and Biblical setting. This asymmetry shows him up for what he is - that he has an agenda, and that his “dependency adjustment” is not principled, it is outcome-driven (which is exactly what my first article expressed too).

Richard Carrier 4#: Every probability in every application of Bayes’ Theorem is a dependent probability. And item by item you can run strict dependency in likelihoods by iterating for each item of evidence in chronological order. But when this has no effect on the outcome, there is no reason to do it, because it requires needlessly complex arithmetic, to no improved result. There is only a dependency on each hypothesis, which is already baked into Bayes’ Theorem. That is precisely the dependency relation we are calculating when we derive the likelihood for each hypothesis: the probability of that evidence dependent on each hypothesis being true—and dependent on all human background knowledge, because each likelihood is P(e|h.b), where the probability of e is dependent on the truth of h and the entire contents of b (such as all inscriptions from Greco-Roman antiquity and their empirically observed frequency of forgery). This is Bayes’ Theorem.

This no dependency on “a Peripatetic tradition” has any effect on this estimate, because the question is whether Aristoxenus is fake or a liar, not where he got his notions from. On Bayesian reasoning we are assuming Aristoxenus got it from just ‘a Paripatetic tradition’ on the contrary hypothesis, and so we are already accounting for its dependency on that tradition. Our likelihood is a dependent probability in that sense, because it has to be: that’s what Bayes’ Theorem is doing. We then put that in ratio to the other hypothesis, where we assume Aristoxenus got it from being an eyewitness partner of Aristotle, and asking how likely the text we have is then. Those two probabilities then sit in a ratio, to give us the weight this evidence has in attesting that Aristotle existed or not.

My comment: Again, there are real, substantive mistakes here, not in the algebra of Bayes’ theorem (though I don’t know if that’s a mere fluke on Carrier’s part), but in how dependence, background knowledge, and evidence aggregation are being handled. At first glance, Carrier doesn’t appear to make any mistakes about Bayes’ theorem in the abstract in this section, but he misapplies it in ways that matter.

Here’s where it goes wrong. He equivocates between “conditional on a hypothesis” and “independent pieces of evidence”. It is true that every likelihood is conditional as per the correct formula, but that does not mean that multiple pieces of evidence are automatically independent given the hypothesis. Conditional dependence between evidence items is a separate issue. Bayes does not magically bake in evidential correlations. And when Carrier says “You can ignore dependency if it doesn’t change the outcome” - that is methodologically backwards. Whether dependency changes the outcome is exactly what must be shown, not asserted. You can’t justify skipping dependency modelling by claiming in advance that it won’t matter - especially when the central question is whether the evidence represents multiple independent attestations or repeated recycling of the same source. In Bayesian terms, this is equivalent to assuming conditional independence because it is convenient, not because it has been demonstrated.

Carrier is right that likelihoods are conditioned on background knowledge. But bundling things like inscription frequency, literary survival, institutional transmission, and genre practice into a sloppy group does not solve dependence - it obscures it. Background knowledge constrains plausibility; it does not license treating correlated signals as separate evidential hits. Carrier actually grossly misstates what the dependency objection actually is. The issue is not merely where Aristoxenus got his information, but whether multiple attestations reduce to a small number of causal sources. If they do, then each new item adds far less information than Carrier’s multiplication assumes. Saying “we already assume tradition on the null hypothesis” does not address whether the same tradition is being counted multiple times as if it were independent confirmation. And as I said above, he also applies dependency asymmetrically, especially when Jesus comes into the equation.

Conditioning on a hypothesis is not the same thing as modelling correlation between evidence items, and Bayes’ theorem does not excuse you from that work. Bayes’ theorem does not license multiplying likelihood ratios just because each likelihood is conditioned on the hypothesis. If you could see this, you’d see that Carrier’s “smaller numbers” defense is mathematically irrelevant. Reducing likelihood ratios does not solve dependence. Whether the multiplier is 100:1 or 2:1, multiplying correlated evidence still inflates confidence. Or to put it another way, dependency is structural, not scalar. You fix it by modelling joint likelihoods or collapsing evidence streams - not by eyeballing smaller ratios. If you are unsure who to believe, simply show this blog post to your smartest friend, those who are most apprised of these subjects, and I’ll wager you’ll hear them agree with me. Or do the research yourself – if you do it honestly and pay close attention, you will come to the same conclusion.

You’ll also see that Carrier conflates “acknowledging dependence” with “accounting for it.” Carrier repeatedly says he “assumes tradition” on the null hypothesis and therefore has already handled dependence. But that only specifies a story about causation; it does not mathematically prevent multiple evidence items generated by that same story from being counted repeatedly. Bayesian rigour requires showing how much new information each item contributes given the others. Carrier does not do that. Again, I don’t know if it’s dishonesty or deception, but Carrier aggressively collapses Jesus traditions into near-total dependence while allowing Aristotle’s evidence - much of it flowing through the same educational, textual, and institutional channels I mentioned in my first Blog post - to count as multiple semi-independent hits. That is a discretionary modelling choice that should fool no one.

Carrier seems to understand that all probabilities are conditional - but he is damn wrong to imply that this dissolves evidential correlation. Bayes’ theorem requires you to deal with dependence; it does not excuse you from it. Treating “background knowledge” as a black box does not magically absorb correlated causal structure. Carrier seems to comprehend a bit of Bayes’ formula, but he misuses it by treating correlated historical evidence as quasi-independent without a formal dependency model, then defending the move rhetorically rather than mathematically.

Richard Carrier 5#: This is an equivocation fallacy, where Muser has switched illicitly between two different meanings of “mythologized,” and ignored the actual data-driven process he is incorrectly describing. First “therefore Jesus is mythologized” isn’t a statement that he didn’t exist (because historical people are also mythologized). So “Jesus is mythologised therefore Jesus is mythologized” is not a circular argument but a tautology. It’s as true as “Jesus is a man, therefore Jesus is a man.” That Jesus is heavily (not merely) mythologized (unlike Aristotle) is a fact (not an argument capable of being circular). It is a thoroughly, professionally documented fact. The consequences of that fact are that Jesus starts out as likely to exist as anyone else heavily (not merely) mythologized (unlike Aristotle), which is empirically observed to be no more than 1 in 3 times. Muser is thus just not paying attention. He doesn’t care about what I am actually saying or referencing in my article. He wants to emotionally rewrite reality so that 1 in 3 is “very low” when in fact it’s generously high, that this predetermines the outcome when it routinely doesn’t (and indeed couldn’t), that heavily mythologized heroes exist just as often as mundane people when obviously they don’t, and that an empirically documented frequency is a “circular argument” and “not grounded in history,” all while completely ignoring (and dishonestly failing to inform his readers of) what I actually said, argued, and demonstrated professionally with evidence.

The likelihood of a certain cause (e.g. a mere legendary person, or a historical person, the two hypotheses being predictively compared against the evidence here) derives from the background facts of the frequency of such causes producing such effects. Real revered scientists always produce more or less this kind of evidence (even if medieval selection destroyed most of it). Merely legendary scientists almost never do (the frequency of that in the available massive datasets is not always zero, but is definitely low). Muser does not appear to understand what likelihoods measure, or how they derive from extensive field-generated background evidence of what has typically happened in the relevant period.

My comment: No, no, no. Carrier’s reply dodges the real objection by sloppily redefining it. The charge is not that “mythologised” means non-existent, but that degree of mythologisation is being used as a proxy for existence in a way that quietly bakes the conclusion into the setup. Calling Jesus “heavily mythologised” is not a neutral fact like “Jesus is a man”; it is a contested classification whose boundaries depend on the very historical judgments at issue. Declaring that class to have a fixed empirical base rate (1 in 3) simply shifts the argument along rather than dissolving the circularity. Moreover, Carrier conflates frequency with explanation again: observing that legendary figures “often don’t exist” does not license treating mythic density as an independent causal variable, especially when mythologisation itself can be an effect of later reception rather than evidence about origins. His appeal to “massive datasets” glosses over genre, transmission, selection effects, survivorship bias, and what have you - all of which historians normally weigh qualitatively rather than collapse into a single scalar. Clearly, Carrier has some experience with historical analyses, so quite why he treats a debatable typology as settled data, then treats disagreement with that move as confusion about Bayes rather than a substantive historical critique, is beyond me. Once again, is it confusion or deception? I suppose even being a seasoned studier of history doesn’t automatically mean competence in Bayesian analysis.

Richard Carrier 6#: “My reference classes don’t predetermine the outcome. In that article I am explicit that I use a fortiori priors that are actually very generous to the alternative, estimating the prior odds on Aristotle of being historical way lower than is realistic and for Jesus being way higher than is realistic, to prevent predetermining the outcome. My exact words (which Muser evidently ignored): “In reality the prior for Aristotle would be higher (here I am using the most unfavorable lower bound of the error margin) while for Jesus it would be lower (I am using the most favorable upper bound … ).” Muser never defends any alternative priors nor explains how my selection of priors “predetermines” the outcome either way. I get the impression he doesn’t care if that is even true. He just wants to say this false thing because he is all emotional about it.”

My comment: No, sorry, not good enough again. Carrier’s reply completely misses the point. The problem is not how generous he claims his numerical priors are, but how he defines the reference classes that generate those priors in the first place. Even a “generous” number is still methodologically corrupt if it is derived from a category that already assumes the conclusion. And lowering or raising the numerical values afterward does not fix this circularity; it merely softens a rigged scale. A prior derived from a theory-laden classification is still theory-laden, even if its numerical value is deliberately moderated. Moreover, demanding that critics supply alternative priors is a red herring; if you really want to postulate such an outlandish theory that Jesus didn’t exist, the burden is on you, Carrier, to justify why his reference classes are historically legitimate rather than ad hoc constructs invented for Bayesian theatre. Historians do not infer non-existence from later mythologisation, nor do they treat “mythic” literary development as probabilistic evidence against an underlying person. Carrier’s priors therefore predetermine the outcome, not because the numbers are extreme, but because the conceptual machinery generating them already presupposes Jesus’ improbability and Aristotle’s reality. Calling critics “emotional” is not a rebuttal – he just hates the fact that I can see through his fatal methodological flaw.

Richard Carrier 7#:  “This is embarrassingly confused. Of course my article does give “historically grounded” priors, entirely based on data that has passed peer review multiple times now. It is not circular to observe the fact that “heavily mythologized” persons usually don’t exist. We can count them and develop an actual data-driven frequency of this; and that’s exactly what I did in my original study, with a generous margin of error placing the prior odds on heroes like Jesus existing over a whole third of the entire probability space. The Gregor team’s study improved my results by narrowing that margin, yet ending up within (and thus confirming) my original study’s tolerances, by perfecting its empirical data. Their results now set the frequency of heavily mythologized persons who plausibly existed at 1 in 4, even lower than I assigned in my article on Aristotle, as I demonstrate in OPH.”

My comment: Carrier’s up to his old tricks again because he equivocates between counting myths and establishing historically relevant reference classes. The core error is not whether one can “count” heavily mythologised figures, but whether the set he counts is methodologically comparable to Jesus’ case in the first place. His dataset conflates fundamentally different phenomena: purely literary gods, folkloric heroes, euhemerised deities, composite figures, and later legendary accretions onto real people, then treats them as a single frequency class. That move already smuggles in the conclusion, because Jesus is classified as “heavily mythologised” on the basis of the very sources whose historicity is in question. This is textbook circularity: the evidence under evaluation is used to determine the prior probability governing its own evaluation. No amount of Carrier’s so-called “peer review” rescues that error. Moreover, Carrier repeatedly asserts that mythologised founders “usually don’t exist,” but this claim collapses once the comparison set is restricted to known historical categories: named teachers, executed sect leaders, or founders of movements in the Hellenistic–Roman world. In that properly constrained class, existence is the norm, not the exception - even when extensive myth-making follows. And, I mean, really – desperate appeals to this Gregor team does not save the argument, as refining error margins within a flawed classification scheme does not validate the scheme itself, and more than it does within the circle of young earth creationists. Narrowing the confidence interval around an invalid prior does not make it historically grounded; it merely makes the mistake more precise.

I think that’ll do for now. After you’ve read this, if you’re one of those still convinced by Richard Carrier’s tricks, I encourage you to do what he’d absolutely hate you to do; look behind the stage door; check not just his sources but the sources of his sources, and the sources of those sources in context, and so on, and see what they really amount to. See those sources in their full context, not in the selectively trimmed form he presents. Ask whether the citations actually support the claims he says they do, or whether they are being leaned on, stretched, or quietly repurposed in typical Carrier style. Look closely at what he calls “peer review,” which – like young earth creationism - frequently turns out to be review by sympathetic presses or friendly associates rather than sustained engagement by specialists in the relevant field. Notice how frequently quotation replaces argument, how dissenting scholarship is collapsed into caricature, and how methodological objections are waved away with appeals to authority, metrics, or imagined consensus.

If you don’t want to just take my word for it when it comes or his understanding of probability theory and Bayesian methods, find someone you know who is an expert in the field and ask them to evaluate both sets of arguments for you – I am immensely confident they will tell you the same things I am about Carrier’s lack of competence. The reality is, if you hold him to account rigorously and honestly, the spell breaks - because the scaffolding holding up the Carrier structure turns out to be far more fragile and undependable than he proclaims. The best advice I can give you on Richard Carrier is to give him a wide berth.

Edit to add: For readers of Richard Carrier’s blog who have got this far, I tried to reply on his article to elaborate but won’t be able to comment on his blog at all anymore on account of being blocked for apparent ‘sock-puppeting’ because three additional comments using anonymous names came from my IP address. This is another one of Carrier’s ploys to silence criticism. I was not 'sock-puppeting', I was sharing comments I received from other people but asked them if I could share anonymously on is article and changed their name. At that point, I was trying to ascertain whether Carrier was just refusing to publish my criticisms of him because it’s me, and he knows I've got the measure of him, or whether he would refuse to publish any criticisms of him at all from anyone. And I got my answer, and I’ll bet you can guess what it is

Further edit to add: In the spirit of fairness, I should add that a few days later he has now published my comment with the link to this article, but immediately used the comment to make the inaccurate allegations above - which suggests he only wanted to further alienate his readers from direct criticism of his work by encouraging them not to take the author seriously. Once again, such is his contempt for his readers that his reply had the audacity to say that this article 'ignores all his actual arguments' - where, even a cursory glance above shows that is another blatant lie. 

Perhaps the ultimate take home message at the end of this, and one with which various friends of mine who've also publicly criticised Carrier have concurred, is that, while these critiques are useful for helping others spot frauds and enhance their own critical thinking, it is usually not the best use of our time engaging in direct arguments with the offenders (Carrier being a particularly dishonest and deceptive type), because their modus operandi is usually to distort and lie, and become hostile and manipulative in the face of criticism. 


Wednesday, 11 June 2025

It Would Be Stranger If The Bible Is False

 

When you look at everything contained in the Bible that purports to show the truth of God – the prophecies, the miracles, the testimonies, the Incarnation, the resurrection, the eyewitness accounts, and so forth – you are being asked to consider the probability that it speaks true propositions about God. You have to decide which is more probable, that when interpreted correctly the Bible A) does speak the truth about God, or B) that it does not. Here is why A is much more likely than B. 

Every time someone wins the lottery, the sequence of six balls is highly improbable, yet someone wins regularly. The chance of guessing the six winning numbers is about 1 in 45 million - extremely unlikely. Still, we don’t dismiss the outcome simply because it’s improbable; we look at the context and evidence. The improbability of an event does not automatically disqualify its truth. The same principle applies when evaluating claims in the Bible: the question is not whether the events are rare, but whether the surrounding context makes them credible. Faking the winning lottery by guessing the correct numbers beforehand is just as unlikely as actually winning. Likewise, the idea that the Bible’s unified narrative, fulfilled prophecies, and theological depth were all fabricated - and yet converged so seamlessly over centuries - strikes me as at least as improbable as the proposition that it is, in fact, the divinely inspired word of God.

To be crystal clear – I’m not making the non-sequitur that says “The Bible is highly unique; Therefore, it's unlikely to be false; Therefore, it is probably from God”, and nor am I saying that someone couldn’t easily make up a story that they correctly guessed the winning lottery numbers, but didn’t really. I also acknowledge that this is a subjective assessment where the priors are not numeric like they are with the lottery win. But the point is, the Bible is such a remarkable book, with such a completely unique set of stories, prophecies, linked text, historical accounts and integrated composition – all cohesively tied up with the proposition that Jesus is God (call this proposition x) – that the theory that it’s not the word of God is much less likely than the proposition that it is. In other words, I think the Bible has a lower probability that it’s all fabricated, made up, false, inaccurate, or a mixture of the four, than the proposition that it’s a true account of Jesus being God. It would be more remarkable and improbable if it isn’t from God than if it is. It would be stranger if the Bible were false than true.

If you want to proceed in Bayesian terms (as I did here, here and here), then we begin with a prior probability that Christianity is true (call that hypothesis A), and we then ask whether the evidence in the Bible makes that hypothesis more likely than its alternatives (that it's false, inaccurate or fabricated - call that B). I’m saying proposition x so greatly increases the probability of A over B that it would far more remarkable if B is true not A. In other words, such a remarkably consistent narrative written over many centuries by dozens of authors from different backgrounds; the fulfilled prophecies; a compelling account of the resurrection, with multiple witnesses, early proclamations, and a dramatically transformed early Christian community; and a coherent theological system centred around Jesus Christ, whose life and teachings have exerted a more profound and lasting influence on the world than anyone else - together, they form such a strong cumulative case that it would be far stranger if B was true not A.

Footnote: If you disagree that Christianity is true, you may have some objections on similar grounds to what I would have if someone used Bayesian reasoning and a lottery analogy to make a case for something I thought was untrue. So, let me consider the objections I’d offer to that person, and tackle them on my own article.

Objection 1: James, you're using probability theory without any numbers, models, or defined probability space, so a Bayesian analysis is difficult for Bible analysis.

My comment: Indeed, and I too am cautious about the use of Bayes to "prove" God, but the hyperlinked blog posts above show that if we are simply looking for evidence, then it can be applied to increase the probability. Under the frequentist interpretation of probability, calculating the probability of an event requires defining the broader context or reference class within which that event occurs – and if we nest all those contexts together surrounding reasons to believe, and priors, the weight of evidence is stacked in favour of Christianity. It’s fine to use a Bayesian structure to frame reasoning even when precise values are unavailable, as long as we are clear and transparent about what we are doing.

Objection 2: James, the lottery analogy is problematic. Lottery outcomes are random and well-modelled, whereas Biblical authorship and theology are historical, intentional, and deeply contextual.

My comment: Don’t worry, it’s fine. The point of the analogy is epistemic, in that it shows that improbability alone doesn't justify disbelief, especially when there’s a known context that supports the outcome. I acknowledge that the subjectivity and imprecision in such cases are not exactly the same as the lottery, but then analogies are not exactly the same as the real thing. The lottery analogy is good for showing that the probabilistic framework still helps articulate the idea that certain types of evidence (like cumulative coherence, fulfilment of prophecy, and unique historical influence) shift the balance in favour of one hypothesis over another.

Objection 3: James, you're arguing that the evidence strongly supports Christianity, but you have neglected to mention the base rate.

My comment: Indeed, and in many blog posts you’ll find me making the same criticism of other writers who neglect the base rate in a way that undermines their argument. But in this case, it’s difficult to meaningfully assign base rates to metaphysical events like divine revelation, because such events are one-off and unique, not repeatable in a way that makes statistical base rates meaningful. In this case, priors must be informed by the weight and coherence of the available evidence. And as per comment 1, it’s absolutely fine, as long as we establish what we are doing. And what we are doing is saying, over several blog posts, that Christianity justifies significantly updating our priors in huge favour to its truth.

Objection 4: James, your category B “Not from God” is too broad, and you risk lumping all alternatives (fabricated, inaccurate, partly true, culturally evolved) into one single hypothesis B, which oversimplifies the spectrum of possible explanations.

My comment: It’s not a problem, because we are basically saying either the Bible is the word of God or it isn’t. It’s a binary proposition. This highlights that none of the alternative explanations, taken together or separately, account for the cumulative complexity, depth, and consistency of the Biblical narrative – so either Christianity is the truth (with a correct interpretation - especially of the fundamentals) or it isn’t. There can’t be a “It’s sort of true’ or “It’s partly true” because the whole creation story hangs on it being the truth or not. In fact, the multiplicity of alternative theories weakens them, because they lack the unified explanatory power that the divine authorship hypothesis has. Bayesian reasoning often involves comparing clusters of alternative hypotheses with a single well-defined theory when the alternatives individually lack explanatory strength, and that is what we are doing here.

Objection 5: James, aren’t you in danger of selectively emphasising the positive features of the Bible (coherence, influence, prophecy) and downplaying purported counterevidence (apparent contradictions, moral difficulties, historical issues).

My comment: Well, every worldview involves a degree of interpretive weighting of the evidence - but that doesn’t invalidate the process of evaluating cumulative credibility. I am familiar with all the purported counterarguments, and I’m not ignoring them; I’m saying that in totality, the affirmative evidence for Christianity not just outweighs the objections, but actually renders those kinds of counterarguments either moot or easily explainable through the lens of Biblical truth and justification – especially once we realise that if a document is 2,000+ years old, written in multiple languages across centuries, some tension points are inevitable. What’s surprising isn’t that these exist, but how consistently the central message holds together despite them – which is why the positive features are strong enough to frame the purported counterevidence in their right place, and in a way that actually reinforces the Bible’s authenticity, rather than undermines it.

Wednesday, 28 May 2025

Playing Dice With God Probabilities


I just watched this debate about God’s existence between prominent atheist Matt Dillahunty and Matthew Adelstein (Bentham’s Bulldog). They got into a fine-tuning-based dispute based on the following claim “Just because A is likelier given B than not-B, that doesn’t mean B is likelier given A than not-A”, disputing the extent to which it’s true. They argued this back and forth for ages – using this illustration “John walks up to Jack and sees two dice with two 6s on the table. Is it more likely in terms of probability that Jack placed them like that or rolled them?” – and they couldn’t resolve the dispute because both were partially wrong in their analysis.

The proposition is actually mathematically valid - just because A is more likely given B than given not-B, that doesn’t mean B is more likely given A than not-A. But in some specific scenarios, the probabilities may line up in such a way that both directions appear to hold - even though the principle still warns us not to assume that. I think that was at the heart of why Matthew and Matt talked past each other on this (see footnote*).

The footnote has a slight elaboration on why “Just because A is likelier given B than not-B, that doesn’t mean B is likelier given A than not-A” is not actually true, but it’s easy to think of very simple examples that warn against the error of hastily flipping conditional probabilities. For example, just because A (smoke) is more likely if B (a house on fire) happens, does not automatically mean that the presence of A makes B more likely - unless you consider the whole context, because house fires are quite rare and other sources of smoke are common. Or an even more obvious example, just because A (clouds) is more likely if B (it's raining) happens, does not automatically mean that the presence of A makes B more likely - unless you consider the whole context, because clouds often appear without rain. 

Let’s move on to apply this to God. Here is an easy example from the dice rolling scenario the debaters used. Imagine you're rolling two fair six-sided dice. Let’s define two events: Event A is that the total of the two dice is 7, and Event B is that the first die shows a 3. Now, if you know the first die is a 3, the only way to get a total of 7 is for the second die to be a 4. Since the second die is fair, there's a 1 in 6 chance of that happening. So the chance of getting a total of 7 given that the first die is a 3 is 1 in 6. Now suppose the first die is not a 3. That leaves 30 possible combinations (that is, suppose first die is not a 3. That means it could be 1, 2, 4, 5, or 6 - giving 5 × 6 = 30 possible outcomes). Out of those, there are 5 that total to 7 - like (1,6), (2,5), (4,3), (5,2), and (6,1). So again, the chance of getting a total of 7 given that the first die is not 3 is also 1 in 6. This means that even though the probability of A (getting a 7) is the same whether or not B (first die being 3) is true, knowing that A happened doesn't actually tell us anything about whether B is more or less likely. So, in this case, even though A is equally likely given B or not-B, that doesn’t make B more likely given A - which is exactly the kind of situation that shows the original statement is true.

Now we’ve cleared that up, let’s go back to the initial question. John walks up to Jack and sees two dice with two 6s on the table. Is it more likely in terms of probability that Jack placed them like that or rolled them? On the surface, the answer seems obvious – it’s more probable that he placed them. Here’s why. The probability of rolling two sixes in one roll is 1 in 36 (1 in 6 x 1 in 6). But if Jack intentionally placed them, the probability of seeing double sixes is 1, because he could simply set them that way. So, the evidence (double sixes) is 36 times more likely under the “placed” hypothesis than under the “rolled” hypothesis.

Doesn’t that mean we should believe Jack placed them? No, not quite. As any good Bayesian will tell you, you don’t get to a posterior belief without a prior. In this case, we must also consider; how likely is it in general that Jack would place the dice instead of rolling them? What if Jack is known to always play fair? What if he’s never been seen to manipulate dice, and we have no prior evidence suggesting deception? What if Jack is just a thoroughly honest and decent guy all round, with almost unimpeachable character and integrity? Then our prior probability that Jack placed the dice may be extremely low - say, 0.001. And despite the evidence (double sixes) being better explained by "placed," the posterior probability may still favour "rolled" unless the likelihood difference is sufficient to overcome the low prior. Here, posterior probability means your updated belief in a hypothesis after taking the evidence into account. You need likelihood x prior to determine the posterior (your actual belief after evidence).This is why Bayesian reasoning is so powerful - it respects both what you believed before and what you just learned, combining them to form a rational belief.

As I’ve written in previous posts and articles, this is the right way to consider the probability of Christianity being true. The existence of a finely tuned, life-permitting universe has often been likened to seeing dice all land on six (although in this case, across an entire world of casinos). The physical constants appear finely calibrated, as though they are set by a cosmic Mind with astronomical purpose and intelligence. But we should proceed with caution there, because even if this state of the universe is vastly more probable given the hypothesis of God than under a godless process, having sufficient prior probability to believe in God in the first place requires a little more, and a lot more if we specify the Christian God.

You may recall my probabilistic model proposed in my Mathematical Bias Theory, in which I framed the universe's apparent structure in terms of a “biased random walk.” As outlined in that blog post, imagine all possible physical configurations forming a vast probability space, with a neutral "random walk" through them generating the high order we see before us, without which such order would be astronomically unlikely. Given a mathematical bias towards law-like coherence - a structure that systematically tilts the cosmic narrative toward order - then such a bias suggests that our universe is the product of an ingenious Cosmic Mathematician (God).

Apply that to the dice model above. Even if your prior for theism is low, then according to the exquisite degree of structure, order, and life-permitting constants we see in the biased random walk - all of which are vastly more likely under theism than atheism - then just as with the dice, our posterior belief in God should increase substantially. That would get us to deism or even theism, but we must keep going to get to Christianity. To keep moving from theism to Christianity we must go further along a Bayesian staircase - a sequence of probability updates, each supported by its own priors and evidential strength. Once you factor in the cumulative evidence for Christianity’s truth – Biblical evidence, Christ’s teachings, credibility of the Resurrection, Biblical prophecies, miracles, healings, personal testimonies, to name just a few - each of which incrementally increases the plausibility of Christianity to such an extent that we have every reason to believe it is true.

For some people, Christianity, like the hypothesis that Jack rolled the dice, may begin with a low prior. But through the accumulation of evidences, each step increasing the probability that the hypothesis is true, it becomes more and more rational to believe Christianity is the truth, as Bayesian reasoning provides a clear path for belief that is proportional to evidence - a climb up a probabilistic staircase, if you like, where every step is supported by reason, evidence, and explanatory power. 

*Footnote: In case it’s still not clearer, let’s break it down further Remember, it went like this:

Matt: “Just because A is likelier given B than not B, that doesn’t mean B is likelier given A than not A.”

Matthew: “Problem, it does.”

Matthew was right to say more likely, but misled by a symmetrical result in a simple example if he thinks it's certain - a situation where the numbers happen to align. But this symmetry doesn’t hold every time, and as I said above, Bayes’ theorem explicitly tells us that you cannot reverse conditional probabilities without re-evaluating all terms. Mistaking symmetry for a universal rule is a common intuitive trap in certain examples (like fair dice) but it is easy to mistake a coincidence for a law. Bayes' Theorem governs how P(B given A) behaves. You cannot reverse the inequality in P(A given B) > P(A given not-B) without evaluating all the terms. So we have to guard against misapplying conditional probability and assuming that an implication works both ways - which it mathematically does not, unless under special constraints. It’s a shame because Matthew has the better arguments generally, but this error proved to be an impediment to Matt seeing the broader point – although, personally, I don’t think Matt Dillahunty is very open or particularly teachable.

 Let me try to illustrate the error in simple terms. Take a fair 6-sided die. If I tell Matthew the number is 4, he knows it’s even. If I say it’s not 4, it’s less likely to be even. Therefore, “even” is more likely when it’s 4 than when it’s not 4. And if I tell Matthew it’s even, there’s a better chance it might be 4 than if I told him it’s odd. So, from Matthew’s perspective, that seems symmetrical, and I suspect that’s what fooled him. The trouble is, that symmetry is far less typical than the mathematical rule that supports the counter.  In fact, that symmetry is just a coincidence in the simple example I gave - it doesn’t work in general. There are many situations where A is more likely when B happens, but B is less likely when A happens, and I showed this with the examples I used in the blog post above. Conditional probabilities like that are not there to be flipped at will.

Edit to add: I think the distinction we must make is between more likely and guaranteed. The statement "the fact that X is more likely on Y means that the existence of X makes Y more likely" is true under standard probability assumptions. This is because Bayes' rule directly relates the two conditional probabilities, and if observing Y increases the likelihood of X, then it also increases the likelihood of Y when X is observed - though the strength of this effect depends on the base rates of X and Y. So, while this does not imply certainty or causation, it does imply a directional probabilistic dependence, making the original statement valid in the probabilistic sense, but not logically or causally guaranteed.

Friday, 21 February 2025

The Resurrection and Bayesian Reasoning

 

A friend asked me, in terms of Bayesian probability, what are the chances that Jesus actually rose from the dead? My response:

I'll write this in the most accessible way I can. Bayesian probability helps us update our beliefs based on new evidence - so the more supporting evidence we have, the more rational it becomes to believe a claim. We have to apply some caution when using Bayesian probability in relation to God, because Bayesian probability typically requires a prior probability (P(H)) that is determined independently of the specific evidence being evaluated (this is known as a common prior) - and this is tricky without a clear statistical or empirical basis to quantify it (it's not like calculating the probability of rolling a 12 with two dice). But as long as we keep the above caveats in mind, we can have a go. So we are considering four primary things:

P(H | E) = Probability of the hypothesis (resurrection) given the evidence
P(E | H) = Probability of the evidence if the resurrection happened
P(H) = Prior probability of the resurrection (before considering the evidence)
P(E) = Probability of the evidence occurring in general

P is probability, H is hypothesis and E is evidence - and obviously from a naturalistic perspective, if there is no God, the probability that anyone would rise from the dead is near-zero. But Bayesian probability from a scientific perspective changes radically as/if we know there is a God who loves us, and was willing to suffer and die for us, and rise from the dead to give us eternal life. Therefore, this won't be easily measurable in terms of the Bayesian ratio of favourable cases to total possible cases. The P(H) (prior probability of resurrection) is extremely low in naturalistic terms, but high if we believe Jesus is who He says He is; and the P(E | H) (likelihood of the evidence if resurrection happened) is high, because if it happened it would be the most remarkable, earth-shattering, life changing event in world history (and it is).

I'd also add that, if there is a God, we’d expect Him to make Himself known – which is what Christianity claims. It’s the only religion which claims with any justification that God Himself has made Himself known in person (in Christ, and through the Holy Spirit). And we know from 1 Corinthians 15 that the resurrected Christ appeared to over 500 people. It's not easily conceivable that Christianity would have been the biggest and most important alteration of human history if Christ had not died and risen, especially under the conditions and culture of the time, where it was very much not in people's interest to promote Christianity, and in many cases, doing so presented a danger to their own life. In fact, from the impression I get from reading New Testament history over the years, I think I would be bold enough to state it even more stridently - not only did Christianity spread, but in worldly terms it did so among people who had little to gain and so much to lose - domestically, socially, politically, physically, you name it. Unlike other movements that thrived through military conquest or political weight, Christianity grew in the most remarkable way - through self-sacrifice and love, radically defying the patterns of history that went before or since.

Moreover, in terms of the credibility of what Jesus actually said and did, many leaders and cult figures have deceived many willing adherents over the years, but Jesus demonstrated all the opposite traits of typical false leaders. False leaders usually seek power, status, and control in a way that serves themselves. Jesus showed His humility and power by serving others, giving up His life for us. False leaders use lies, manipulation and falsehoods to maintain their power and influence, but Jesus spoke only the truth. False leaders operate through promoting division, separateness and fear, but Jesus preached love, inclusivity and togetherness. Jesus spoke the wisest and truest words ever spoken about God, because He is God. He is unique in history and in personally revealing God's love to humankind, and that is why Christianity changed the world in the way that it did, in a way that's unparalleled in human history. Not only is there nothing like it, there is nothing that can even be spoke of using the same type of language. It's as different from the false religions as the sun is different to a candle.

So, in summary, Christians have confidence that God exists, that He has revealed Himself in history, that He loves us, that He was willing to die for us, and that when all the evidence for His resurrection is considered in light of that (namely, the sheer weight of historical testimony, the transformation of Jesus’ followers, and the unparalleled impact of Christianity), Bayesian reasoning points to the resurrection being a historical event, not a made up one. And, of course, this logic works in reverse too - if it was merely made up, it would be far more improbable, contrived, historically unexplainable and philosophically unsatisfying than if it really happened.

/>