Bayesian Probability Theorem

Often heard in response to the arguments of historical apologetics, such as the claim that God raised Jesus of Nazareth from the dead, is the axiom ‘extraordinary claims require extraordinary evidence.’ People who use this line must be unaware this has been soundly refuted in current philosophical thought, or else persuaded to use irrational principles to satisfy the requirements of their ideological allegiances. After all, there are hundreds of extraordinary claims you come across each day, and yet have no trouble believing.

Consider the lottery reported last night on television as one such event. The chances of winning, or indeed any random sequence of numbers, is extraordinarily improbable, yet if it is true that extraordinary claims require extraordinary evidence, you should never believe it happened. Weighing the probability of the extraordinary event will swamp the reliability of the witnesses every time so that you should never believe it. Even if the programs reporting is 99.9% accurate.

This kind of thinking is really a popular hang-over of Hume’s problem with miracles, which has been thoroughly refuted. John Earman, the agnostic philosopher wrote the book called Hume’s Abject Failure, in which he argues as commonsensical as this principle sounds, it is demonstrably false. The problem that probability theorists have worked on is how one can establish highly improbable events. They realised that you also have to consider the probability that if the reported event would not have occurred that the event would have been reported as it is. 

For instance, what is the probability that the sequence of numbers reported as the lottery result, would have been reported had those numbers not been the correct result. In the case of the resurrection, what is the probability that if Jesus of Nazareth did not rise from the dead we would have the evidences of the empty tomb, the post-mortem appearances and the origin of the disciples belief, et cetera?

Thus an elegant way to assess highly improbable events was developed. The probability for hypothesis (H) on the given evidence (E) with respect to the general background knowledge (B), called Bayes’s theorem. 

bayes

How this works is you plug in values of >.5 for positive probability or <.5 for some negative probability. As the result moves towards 1 it is more likely and towards 0 it is less. In the numerator we have the intrinsic probability of H multiplied by H’s explanatory power, Pr(E/H). The intrinsic probability of H is the conditional probability of H relative to the background knowledge (B). The Pr(E/H) is the rational expectation of E given H is the case, again relative to the background knowledge (B). The background knowledge in both cases is tactically assumed. In the denominator the above product is added to the product of the intrinsic probability and explanatory power of the denial of H. If this latter product is 0 then the numerator and denominator are the same and yield a ratio equal to 1, meaning 100% probability.

Hume failed to appreciate the probability calculus which entails not only the general background knowledge of the way the world is, but also the probability that we should expect the given evidence had the proposed event not occurred. So it turns out that it could very well be the case that an extraordinary event would not require extraordinary evidence, if the evidence is highly unlikely to occur had the event not taken place. He confuses miraculousness with probability and infrequency with implausibility. That’s one reason why Hume’s argument against miracles is entirely fallacious. 

Richard Swinburne, the philosopher of science from Oxford University, after plugging in the values, gives the probability of Jesus’ resurrection from the dead as 97%. Now I’m not sure how he arrived at the values he plugged in, so I wouldn’t necessarily use Bayes’s Theorem for an analysis of philosophical hypotheses such as God raised Jesus from the dead. But I think enough has been said to show that extraordinary claims do not require extraordinary evidence.

50 replies
  1. Damian
    Damian says:

    I got as far as this and felt the need to interject:

    Consider the lottery reported last night on television as one such event. The chances of winning, or indeed any random sequence of numbers, is extraordinarily improbable, yet if it is true that extraordinary claims require extraordinary evidence, you should never believe it happened.

    If only one person purchased a ticket and happened to get the winning combination then, sure, that would be an extraordinary claim and would require extraordinary evidence.

    But because millions of tickets were purchased then the probability of someone winning rapidly approaches 1.

    Back to high school for you!

  2. Stuart
    Stuart says:

    Unfortunately you have failed to comprehend that any sequence of numbers arrived at by the lottery is extraordinarily improbable. That is why I wrote “or indeed any random sequence of numbers”.

  3. Damian
    Damian says:

    Unfortunately the failure of comprehension is all yours.

    With enough guesses the improbable becomes inevitable. We require no extraordinary evidence for the fact that someone wins the lottery.

    Nice one Stu-y but back to high school you go!

  4. Stuart
    Stuart says:

    The failure to understand is egregious. In the lottery analogy the number of contending tickets is irrelevant, as any given number in the lottery is as equally improbable as an other. And it is precisely the point of Bayes’ Theorem to show that, contrary to Hume, we do not require extraordinary evidence when someone wins the lottery.

  5. Damian
    Damian says:

    I’m not defending any ideas of Hume (I know very little about them) or arguing against Bayesian Probability.

    You seem to be setting up a strawman argument (or is it a non-sequitur, or both?) that goes along the lines of:
    – If you believe that extraordinary claims require extraordinary evidence then you should see the fact that someone (among millions) wins the lottery as an extraordinary event.
    – This type of thinking doesn’t account for Bayesian Probability.
    – Therefore: using Bayesian Probability removes the need for extraordinary evidence.

    But the reality is that people don’t find it impossible that someone wins the lottery because they naturally use an informal type of Bayesian Probability that tells them that the more people play, the more likely it is that someone will win.

    Bayes’ Theorem shows us that when one predefined person chooses the winning sequence it is an extraordinary event where as when one person among a million does it is no big deal. It helps to identify what an extraordinary event is but the demand for extraordinary evidence doesn’t go away once we’ve found that event.

    Also, I’d be interested to see the workings for that 97% probability that Jesus rose from the dead. Sounds like a work of creative genius to me. (I realise you see it as idiotic too).

  6. Stuart
    Stuart says:

    Also, I’d be interested to see the workings for that 97% probability that Jesus rose from the dead. Sounds like a work of creative genius to me. (I realise you see it as idiotic too).

    I’m also interested but I’ve yet to track down the source. Until then I’ll refrain from calling it idiotic as Richard Swineburne is an eminent scholar and philosopher.

    You continue to be miss the point. Any possible result of the lottery is just as unlikely as the actual winning result. Thus the amount of players and the eventual winner if there is one is totally irrelevant. You seem to be convinced that an event needs specificity before it is extraordinary, but this is clearly not the case. Since you think I need to go back to high school lets do the math.

    Take the lottery as a sequence of seven from a selection of 40 balls. Thus a represents the fist in the sequence, b the second and so on for c, d, e, f and g. Any number of the 40 can be a, and any number of the remaining balls can be b through g. The probability of a = 1/40, the probability of b = 1/39, the probability of c = 1/38 and so on. To calculate the probability of the sequence on needs to multiply the fractions together. The result will be P, or the probability of every sequence of seven numbers, no matter what that sequence is, no matter matter if it is the winning number, no matter how many people bought a ticket.

    a x b x c x d x e x f x g = P

    Let us then substitute the algebra for the actual values, remembering that each value could be any random ball.

    1/40 x 1/39 x 1/38 x 1/37 x 1/36 x 1/35 x 1/34 = 1/93,963,542,400

    So P equals one chance in 93,963,542,400. Yet when the lottery result is reported on the news we have no trouble believing it, even though it is incredibly extraordinary in the sense that it is highly improbable.

    The argument in the article, contrary to your rendering, is really that failing to appreciate the probability calculus leaves you with extraordinary skepticism. Experience tells us extraordinary events do not require extraordinary evidence, and Bayes’ Theorum supports that.

  7. Damian
    Damian says:

    (A) If I throw a single die, what are the odds that I’ll throw a 6?

    (B) If six people throw a die each, what are the odds that someone amongst them will throw a 6?

    The answers to the above are 1/6 and 6/6 respectively. The reason for this is that for (B) we’ve had six chances at it and our criteria for a win is just that someone should win, not a particular pre-specified person.

    By the way, the $1M lottery that was won that you refer to is only six among 40 balls which has odds of 1/3,838,380 and the PowerBall (which wasn’t won) is 1/38,383,800. Last year Lotto along brought in $523.9M which means that, at between 60c and $1.10 per line, an average of between 9 and 17 million chances (if my sums are correct) are had each weekly draw which explains why a couple of weeks ago five people shared the $1M first division prize and why the PowerBall has jackpotted for the last few weeks.

    The fact that someone wins the Lotto when so many people have a shot at it makes it a decidedly ordinary event and therefore requires no extraordinary evidence.

    Now, what are the odds that you will see the disingenuity of your argument and admit that the rule of thumb extraordinary claims requires extraordinary evidence is a pretty good rule to live by if we are to avoid being sucked in by every passing whacky claim? 1 in 93,963,542,400?

  8. Stuart
    Stuart says:

    Damian,

    Your knowledge of Lotto exceeds mine. But its apparent your comprehension of what I have been trying to communicate is still nil. And your math is appalling.

    The chances of any random line of numbers is drawn is extraodinary, or highly improbable even if there is no winner, even if no one buys a ticket, even if no one sees the actual draw. Adding a winner to the illustration is giving specificity, but that doesn’t change the odds at all of the actual draw. Think about it – the actual draw and the test run half an hour before-hand have exactly the same odds. Thus I stated or indeed any random sequence of numbers

    Concerning the lottery I wasn’t thinking of any draw or lotto game in particular. It was just an example. As you pointed out in NZ there is no seventh ball. The odds of six balls is 1/2,763,633,600 and the numbers you give are I think because the order of the balls do not matter. However the probability there is calculated, the illustration still stands at the odds you give.

    The answer to (B) is 6/36 = 1/6
    not 6/6 = 1
    The correct parallel analogy would be what are the odds of me throwing a 6 every time in six throws. That would be 1/46,656

    As I’ve shown in the article above, the claim extraordinary claims requires extraordinary evidence is demonstrably false.

  9. Bnonn
    Bnonn says:

    I don’t think you’re listening to what Stuart is saying, Damian. Step back for a moment and think about this.

    He is not claiming that the probability of someone winning is remote. Obviously that probability is relatively high. He is saying that the probability of any given sequence of numbers obtaining is remote. Extraordinarily remote. Therefore, when a given sequence of numbers is claimed to have obtained, prima facie we should require extraordinary evidence to validate this report—if indeed it is true that extraordinary claims require extraordinary evidence.

    Commensurately, while the probability of someone winning is high, the probability that it will be any given person is extraordinarily low. Thus, prima facie, extraordinary evidence should be presented to prove that Mary Jane Smith from Pukekohe won the lottery. Obviously this is not the case, which is exactly Stuart’s point: the prima facie claim is false because it fails to account for other factors.

    Furthermore, as Steve Hays recently pointed out in ‘The onus of miracles’, the claim itself is highly questionable. For one thing, it’s excessively vague—what quality and quantity of evidence is assumed to be “extraordinary”? Furthermore, what kind of attestation does the skeptic have in mind here? Is he saying that it takes extraordinary evidence to show that an extraordinary event merely occurred; or is he saying that it takes extraordinary evidence to attest to the extraordinary nature of that event? It doesn’t seem that the latter is in view, since skeptics don’t typically argue about the nature of Christ’s resurrection (for example)—they argue about the fact of it. But the former is circular, and merely presupposes a non-Christian theory of reality. Then to say that extraordinary claims require extraordinary evidence is not really to appeal to some objective standard or some universally recognized fact, but really just to appeal to the skeptic’s own prejudices or beliefs about what is and is not implausible. So it’s just question-begging.

  10. Ken
    Ken says:

    Stuart, Damian has caught you out trying on a well known fallacy. You should just admit your mistake and face up to it rather than try to argue your way out of it by supporting your original mistake with Bafflegab – irrelevant numbers and equations.

    Its, unfortunately, a common tactic for you, Rob and Bnonn.

    No one is surprised that someone will win Lotto – its obvious that the probability of that happening is close to 1. That’s why people keep on buying the tickets (avoiding the other obvious fact that the probability that any pre-nominated individual will win a specific draw is very small).

    However, the real mistake here is more basic. Stuart is wishing to support a preconceived belief he has with what looks like (is presented as) a “scientific” argument – but is actually pseudo-scientific one. It’s a form of scientism – using science inappropriately.

    So we get the apparent scientific formalism of the equation presented and that is the real way the argument is being justified (form not content). Ignored is the fact that the inputs to that equation are crucial. You put rubbish in – you get rubbish out.

    So, the preconceived beliefs, prejudices, influence the choice of inputs (what the hell is a “negative probability” anyway!). One will obviously choose the appropriate input values to conform to your own prejudices.

    Now in the normal scientific use of such formulae the social activity of science (peer review and critique by colleagues) will work to keep one honest. One also has the obligation to map one’s conclusions against reality – the ultimate pressure to be honest.

    But in the apologetics uses of “science” (scientism) to support the assumed and preconceived beliefs and dogma, data is selected, misrepresented and distorted. In other words- they use “science’ to argue for a preconceived conclusion (scientism) rather than using the evidence, logic and science to derive an objective conclusion – and then checking that against reality. It’s called following the evidence – not selecting it.

    Like the drunk who uses the lamppost more for support than illumination.

  11. Ken
    Ken says:

    Bnonn – your claim:

    extraordinary evidence should be presented to prove that Mary Jane Smith from Pukekohe won the lottery. Obviously this is not the case

    is clearly wrong.

    We would not go on buying Lotto tickets if there weren’t strong procedures to check both the drawn numbers and the actual ticket presented by Jane. The whole procedure is submitted to quite a rigid collection of evidence.

    Now, when anyone claiming a “miracle” can produce a similar degree of evidence we should surely see that as legitimising subsequent investigation of the alleged phenomena.

    I can produce a “miracle” from my own memory – one I am convinced I saw as a child – but in the absence of more evidence I am sure you (and most other people) would “explain away” my experience in other terms – and you would be justified in doing so.

  12. Damian
    Damian says:

    The answer to (B) is 6/36 = 1/6

    Ha! As it turns out we are both wrong: it’s 31,031/46,656 = 66.51%.

    If I tell you that each week around 10-20 million guesses are made on 6 randomly selected of 40 numbers in the NZ Lotto, what do you calculate the chances are that at least one person will win the first division prize? Do you see it as likely or unlikely that the winning combination will be chosen by someone?

  13. Stuart
    Stuart says:

    Ken,

    I don’t know why I expected better from you. Your imposing science in a field where it doesn’t belong. Assessing probability of events such as miracles is a task that belongs to the historian, the discipline of historiography and the philosopher.

    Assessing the validity of a winning lottery ticket is not the analogy advanced.

    I deny Damian’s rendering of my argument and any fallaciousness that entails, and have repeatedly given reasons why he is wrong. You need to give reasons instead of just repeating his mistakes and indicting the argument as bafflegab – which it is plainly not. It requires a small amount of understanding of simple math, and an openness to clearly consider this well-established logical formula.

  14. Ken
    Ken says:

    Stuart:

    Assessing probability of events such as miracles is a task that belongs to the historian, the discipline of historiography and the philosopher.

    So that is why you attempt to use a “scientific” argument – proposing (and supporting) the use of this formula for assessing the probability of a “miraculous” event?

    In other words, and by you own definition, you are using science inappropriately – scientism.

  15. Stuart
    Stuart says:

    Damian,

    You’re still not listening. Re-read Bnonn here,

    He is not claiming that the probability of someone winning is remote. Obviously that probability is relatively high. He is saying that the probability of any given sequence of numbers obtaining is remote. Extraordinarily remote.

    I can’t access that link you provided – my internet provider blocks it. If you could provide the reasoning here, or something similar elsewhere much appreciated. Until then your track record so far has not been exemplary so I’ll stick by (B) = 1/6

  16. Damian
    Damian says:

    Consider the lottery reported last night on television as one such event. The chances of winning, or indeed any random sequence of numbers, is extraordinarily improbable, yet if it is true that extraordinary claims require extraordinary evidence, you should never believe it happened.

    Is it an extraordinary claim that someone has won the lottery? No.
    Is it extraordinary that one particular person won it. Yes, the odds were against it.
    What extraordinary evidence should we demand? A secure ticket-dispensing system along with a proof of purchase that matches the location from which the ticket was bought along with regular auditing of the entire process should do for starters.

    If people were being asked to self-report what numbers they’d chosen without these demands for evidence and we all thought this was reasonable then you might have a point. But the fact is that we do demand extraordinary evidence for improbable events.

    For a start, we are justified in the belief that it is likely that someone should have won given the number of tickets sold. And the auditing of the ticket purchase process takes care of the remaining uncertainty. Extraordinary claims still require extraordinary evidence.

    (The formula for calculating the odds of rolling at least one 6 in n rolls is 1 – (5/6)^n)

  17. Stuart
    Stuart says:

    Damian,

    You have once again failed to recognise what is being said in the quote of mine your quoted. I ask you to not continue to post here if you insist on repeating this annoyingly repetitive misconstruing of the analogy.

    As it turns out we are both wrong: it’s 31,031/46,656 = 66.51% . . . (The formula for calculating the odds of rolling at least one 6 in n rolls is 1 – (5/6)^n

    You must have changed (B) from what it originally was.

  18. Stuart
    Stuart says:

    Ken,

    Re14:

    I have not claimed Bayesian Probability Theorem is a scientific theorem.
    By the way, ‘Scientism is more than using science inappropriately. On careful reflection I think you’ll find you read science into this article where science was not – Unless you are defining science as to include logic and math.

  19. Damian
    Damian says:

    You have once again failed to recognise what is being said in the quote of mine your quoted.

    Consider the lottery reported last night on television as one such event. The chances of winning, or indeed any random sequence of numbers, is extraordinarily improbable, yet if it is true that extraordinary claims require extraordinary evidence, you should never believe it happened. Weighing the probability of the extraordinary event will swamp the reliability of the witnesses every time so that you should never believe it. Even if the programs reporting is 99.9% accurate.

    Nope, seems pretty clear to me. You’re saying that we obviously accept extraordinary claims without extraordinary evidence when it comes to claims of people winning the lottery. If this is not what you are saying then you might want to reword it.

    You must have changed (B) from what it originally was.

    I’ve somehow hacked this blog to retrospectively edit my entry? Nope; that’s what it said all along.

  20. Ken
    Ken says:

    Yes I do (I used logic, maths and statistical analysis all the time in my scientific work) – and face up to it Stuart – you do too. You were trying to give a “scientific” veneer to biased conclusion justifying a preconceived belief. It’s pseudo-science – and does come under the definition of scientism as using science where it shouldn’t be.

    Why else present that formula?

  21. Stuart
    Stuart says:

    Ken,

    I think you are conflating what science is with what science uses and has to assume for its success. Bayes’ Theorum is a philosophical, specifically logical proof for assessing the plausibility of highly improbable events.

    Listen, you need to give reasons why the use of the formula is invalid, not impugn my motivations for doing so. This is fallacious reasoning – the genetic fallacy – and deserves no further response until it can be redressed.

  22. Stuart
    Stuart says:

    Damian,

    Lets say we have one witness who is accurate 99.9% of the time. The probability of the actual lottery numbers (I’ll use your numbers, even though they are for the probability of someone winning and not for the actual numbers) falling the way they did is 1/3,838,380 which is 0.00000025%. So obviously the probability of the lottery numbers falling the way they did is dwarfed by the reliability of the news reporter, who is wrong only – a generous – 0.1% of the time.

    If it were true that extraordinary events require extraordinary evidence, we should never believe the report of the lottery on the news, for the evidence available to us then is far from extraordinary. Since we don’t have trouble believing the news report, the axiom must be false. Bayes’ Theorem helps to tell us why.

  23. Ken
    Ken says:

    Stuart – stop avoiding the issue. No-one sees news reporters as reliable. But the whole authentication procedures around lotteries does amount to collection of reliable evidence. Similarly, any myth like the resurrection of Christ could be based on an unreliable reporter – science would demand more evidence than that. Just as we do for Lotto.

    The formula itself is not at fault – the input is.

    As an aside – Have a look at Dawkins’ comments in The God Delusion where he describes Unwin’s use of the formula to judge the probability of the existence of a god. Unwin got 67% originally, but then went back to doctor his input to get 99% (by inserting “faith).

    The point is the outcome is purely a reflection of the users prejudices – which determine the input. It’s just dressing up one’s prejudices in “scientific” formalism – inappropriate use of science or scientism.

    The formula works, and is valuable, when you have reliable input data.You don’t in your case – only preconceived belief and faith.

    I think the problem here is not my conflating – its your confabulating.

    Rob – my backgorund. My original training is in Chemistry, Maths, Physics (we really didn’t have the other interesting choices in my day, unfortunately); M. Sc. Hons and Ph. D. degrees in surface chemistry and most of my research career in soil and agricultural chemistry. I am now retired and finding the free time a great opportunity to catch up on the advances in cosmology, biological science and recent advances in the understanding of “matter.” Its been 40 years since I could do justice to those interests. I have always had an interest in the philosophy of science and found research experience invaluable in achieving a sophisticated understanding of that.

    So Rob – what branch of science are you interested/working in?

  24. Stuart
    Stuart says:

    Ken,

    But the whole authentication procedures around lotteries does amount to collection of reliable evidence.

    You are missing the analogy. See comment 22 for clarification. As I’ve repeated myself all too often I’ve come to the conclusion that further discussion with you is fruitless.

    The formula itself is not at fault – the input is.

    But Ken, I haven’t put anything into the formula! I haven’t even given any evidence for a particular miracle claim. My argument has been focusing on the the axiom ‘extraordinary claims require extraordinary evidence’ to which I conclude is demonstrable false. The lottery illustration rendered correctly is a demonstration of that conclusion. If as you say the formula works, then my conclusion should be your conclusion as well.

    It’s just dressing up one’s prejudices in “scientific” formalism – inappropriate use of science or scientism.

    Ken, I’m the one arguing that Bases’ Theorem isn’t science. You’re the one claiming it is, and that is using science where it is inappropriate, hence the scientism is yours.

  25. Ken
    Ken says:

    [Go back and read the actual discussion and make sure you understand it before you accuse anyone here of dishonesty again. —Bnonn]

  26. Bnonn
    Bnonn says:

    Damian said,

    Is it extraordinary that one particular person won it. Yes, the odds were against it.
    What extraordinary evidence should we demand? A secure ticket-dispensing system along with a proof of purchase that matches the location from which the ticket was bought along with regular auditing of the entire process should do for starters.

    If that is what you consider extraordinary evidence, I am left wondering what ordinary evidence is. What’s the difference? There’s nothing particularly extraordinary about the evidence you cite. What matters is that the person has an authentic proof of purchase with the same numbers on it as the numbers which came up in the draw. That’s really all any reasonable person actually requires as evidence. A piece of paper. What’s the Bible made of?

    Extraordinary claims still require extraordinary evidence.

    According to a highly arbitrary, ad hoc definition of “extraordinary evidence” perhaps. I don’t see anything extraordinary about the evidence you deem necessary. Which merely confirms the objections I raised to this insipid “rule” in my previous post.

  27. Damian
    Damian says:

    If that is what you consider extraordinary evidence, I am left wondering what ordinary evidence is. What’s the difference?

    The fact that someone wins is expected and so the procedures in place to check that the correct person is making the claim is about perfectly matched in extraordinariness to the claim.

    That someone overcame incredible odds to pick the correct combination is explained (i.e. evidenced) by the fact that so many people played. If only three people played each week and someone successfully picked the winning combination each week then our demand for evidence would be extraordinary indeed.

    (BTW, I really don’t like your tendency to censor people like Ken’s posts – I find it rude. It’s your blog so you can do what you want but thought you should at least know how some of us feel about it.)

  28. Stuart
    Stuart says:

    Damian,

    I can imagine how it feels to be censored. But I did receive what Ken said in the email and I do think Bnonn was correct to do so on this occasion. I think it more a mercy for Ken that people not see what he wrote. To sum it up I was accused of dishonesty, clinging to mistakes, and using a formulae as decoration to protect preconceived beliefs. Nothing was there he was not regurgitating, and after repeated warnings there was only accusation – no actual refutation. Contributions to the discussion (positive and negative), questions, and opinions supported with reasons are welcome.

    Re the rest of your comment:
    I read a fiction book once about a man who predicted the winning lottery number, not once but twice in a row. The procedure for checking his ticket the first time was marked with casual interest. The procedure for checking his ticket the second time was the same, though marked with incredulity. The point – the checking procedure each time was sufficient to establish the ticket was genuine. His winning was not extraordinary when placed in the context that someone had to win. His winning the second time could be passed off as a fluke, a statistical anomaly.

    But lets imagine this guy, before he won anything claimed he had a system, and moreover the system worked. ‘In the next two weeks running,’ he says to the boys having drinks in the bar, ‘I’ve decided to win the lottery.’ He then goes on to explain how he can see the future. That is an extraordinary claim. Suppose further this boast was reported to you after he had won both draws consecutively. Would you believe it?

    The question is a good one. We have an extraordinary claim. We have the evidence of the winnings and press reports of him, the “double winner” to support the validity of his incredible claim. But do we have reliable evidence that he made the claim itself? There are all sorts of reasons why we should never believe he made that claim, let alone that he has developed pre-cognitive abilities; the people he told were drinking, the people he told are unreliable and predisposed to lying, the bars security video and audio could have been doctored, pre-cognition is a fanciful fiction, etc.

    In order to assess the plausibility that this extraordinary claim was made before he won the lottery twice, is it true that we require extraordinary evidence? Hume would say yes, the miraculousness entailed by such a claim should never by believed because the probability of winning twice swamps the reliability of the witnesses and all evidence to his boast. But Hume fails to appreciate the probability calculus. In all fairness to him it wasn’t yet developed. He focuses exclusively on Pr(M|B), which is the intrinsic probability of M. One needs to also to weigh the other side of the equation Pr(not-M|E&B) or the probability of him not boasting of pre-cognitive abilities with respect to the cumulative evidence and background knowledge of the world. In other words, the expectation we would have the evidence we do have had he not given the boast.

    No matter what non-zero value Pr(M|B) is assigned the probability of M can very good if Pr(not-M|E&B) is sufficiently large. So, in fact, it is not the case that extraordinary events require extraordinary evidence.

  29. Damian
    Damian says:

    That’s a good example Stewart but I would say that the scenario where a person predicts that he will do an extraordinary thing and follow through with one or two proofs of that claim then this constitutes extraordinary evidence.

    If I say I’m going to roll a 6 and subsequently do so some of your (justifiable) scepticism would be met. If I went on to repeat it time and time again you would have your demand for extraordinary met for the claim that I can throw a 6. Remember, this says nothing about any side claims (like, that I use magic) just that I can throw a six.

    But if I throw a 6 and then tell you that I intended to throw a six you’d be well within your rights to demand further proof. You can choose to believe me straight away if you like but if you have a tendency to believe this you open yourself to other potential untruths.

    I don’t know what Hume’s arguments are but in the example of me throwing a 6 it’s the repetition of the event that constitutes extraordinary evidence. Before I demonstrated my claim the demand for evidence should have been at least inversely proportional to 1/6. As I continue to beat the odds the demand for evidence is met.

    When it comes to one-off historical extraordinary claims I think the best we can do is balance them against what we know to be possible (keeping in mind that ‘possible’ isn’t necessarily restricted to what we think) and try to factor in as many pros (reliable witnesses, etc) and cons (the tendency to exaggerate, etc) as we honestly can before choosing whether to believe or not. Of course, it is totally valid to remain agnostic while waiting for further evidence.

    I think that extraordinary evidence should be a rule of thumb rather than an axiom because life’s interactions are often too complicated to be put into a formula. Agnosticism is technically the most honest approach to almost everything but we often have to make a choice one way or the other for practical purposes.

    I think that most people (including yourselves) most of the time require extraordinary evidence for extraordinary claims but that we let our guard down for issues that either don’t matter all that much or that we have already invested a certain degree of belief in.

    And, to the point, I think that this attempt to take the extraordinary evidence guide down a notch or two fits into the latter where you have beliefs that you’ve already invested much into that, in other circumstances, would require extraordinary evidence. I don’t think this is the best way of finding the truth.

  30. Stuart
    Stuart says:

    Damian,

    That’s a good example Stewart [sic] but I would say that the scenario where a person predicts that he will do an extraordinary thing and follow through with one or two proofs of that claim then this constitutes extraordinary evidence.

    Of course, that is not the illustrations claim or extraordinary evidence. The question was, how do assess the validity of his boast of pre-cognitive abilities after the lotto is won?

    As rolling a six on a six-sided die is by no means extraordinary, I fail to see the usefulness of such an illustration. If the claim was “I can throw a 6, six times consecutively,” a demonstration would satisfy me, I’d distrust the die and cry foul, but there I’d have it. The correct analogy would be “I can throw a 6, six times consecutively with a fair die. In fact, I did so last night.” What evidence would be required then?

    I think you do comprehend that I am concerned with one-off historical extraordinary claims. I don’t understand why you can’t see that all the die casting analogies and lotto ticket verification analogies you give are pointless. I agree that we should weight the evidence, but that’s exactly the hole of skepticism that Hume fell into. Remaining agnostic is, as you say, “impractical,” so why not give up the agnosticism?

    Take for instance the girl whose five tickets she was given won the daily Keno draw Monday, Tuesday, Wednesday, Thursday and Friday all in that week. She’d never played before and her first five tickets are all winners. If extraordinary claims require extraordinary evidence she should never believe that she had won. The odds are so incredibly against such an occurrence that they far outweigh the incredible odds that her senses are deceiving her and she is actually in a drug induced coma dreaming she’s won.

    But of course, remaining skeptical or agnostic is plainly ridiculous. The odds are so incredibly against such a circumstance that one immediately suspects capricious design. Exactly the sort of inference drawn when considering Pr(not-M|E&B). Failing to appreciate this half of the equation leads to either rejection of or obstinate refusal to accept things we really should, when the evidence (pro and con) is weighed and the background knowledge considered.

  31. Damian
    Damian says:

    The question was, how do assess the validity of his boast of pre-cognitive abilities after the lotto is won?

    I think that if you’ve never seen evidence of precognition and precognition is unable to be subsequently demonstrated and you can think of no explanation for how precognition might work then you should be sceptical or, if you are feeling generous, agnostic.

    If there is video tape of him holding a dated newspaper prior to the event whilst making the prediction then I think the first approach would be to thoroughly investigate the integrity of the video because we at least have experience of video forgery and special effects.

    If we are shown irrefutable evidence that the video is not a forgery then we should creatively try to exhaust as many other possible explanations for how such a claim could be made. Involve people who have experience in deception or forgery because most of the rest of us are easily deceived as is evidenced by magicians’ tricks and optical illusions.

    In the absence of a natural explanation or of further proof of precognition I would then choose agnosticism over belief or disbelief and await further evidence because my need for evidence hadn’t been met. If he is able to subsequently repeat demonstrations of precognition I would count that as strong evidence and would tend to believe but would hold a small reserve of scepticism in reserve in the absence of an explanation for the mechanisms of precognition. (Which is what I currently do with hypotheses of the origins of the universe, life and the workings of quanta and so on).

    What would your approach be in this situation? At what point would you believe in this person’s claims of precognition?

  32. Stuart
    Stuart says:

    Damian,

    What would your approach be in this situation? At what point would you believe in this person’s claims of precognition?

    I think, based on your last comment, we have very similar approaches. First, into the pool of live options go all the different hypotheses. Second step, we sift through, collect and test the available evidence. Third, hold up the different explanations in the light of that evidence.

    I agree that naturalistic explanations should be preferred. As a rule of thumb that is a good one. But here your method breaks down.

    In the absence of a natural explanation or of further proof of precognition I would then choose agnosticism over belief or disbelief and await further evidence because my need for evidence hadn’t been met.

    In the absence of a naturalistic explanation, the only explanation left is the one proffered by our lottery winner. Why is this explanation met with such a dogged refusal of acceptance? Why is the fall-back position agnosticism if there is only one explanation that remains? On the assumption that you do not hold an anti-precognition bias, Bayes’ Theorem will help bring clarity to the issue.

    It tells us we should not exclude any hypothesis on the basis of the evidence alone. We also need to (i) consider the general background knowledge of the world apart from the specific evidence, and (ii) the rational expectation that we would have the evidence we do have had he not given the boast of pre-cognitive abilities. The hypothesis advocated by our two-time consecutive lottery winner will become more plausible if our background knowledge of the world includes certain psychic phenomenon, or some view of cosmic karma for instance. Conversely it will become less plausible if our general background knowledge of the world excludes these things (of course one would have to offer arguments in favour of excluding those things if they are to be more than mere assertions.)

    Considering (ii)—the rational expectation that we would have the evidence we do have had he not given the boast of pre-cognitive abilities—it will be helpful if I extend the illustration a little. Suppose our winner—Larry Longbottom—was obstinate about his abilities and additionally claimed he had exhausted his abilities on his efforts to win the lotto—twice. Because of his claim he was first seen as eccentric, but because he stuck to his story people started viewing him as off-the-wall. His refusal to retract his claim, and the seriousness with which he continued to insist that it was true attracted to him a great deal of mockery. This escalated eventually to heavy persecution and in defending and protecting himself against this he died, old but alone and miserable.

    The probability calculus asks us to consider the rational expectation we would have the evidence of Larry’s life-long insistence if Larry did not in fact have pre-cognitive abilities. What is the probability that we would have the bars security surveillance evidence had he be telling a lie about his making the claim before he won? What is the likelihood of the witnesses who were at the bar would all stick to Longbottom’s story if he hadn’t given his predictions?

    Plug in those probabilities to the equation and it may well be the case that we are well within our rational rights to believe that Longbottom once had precognitive abilities. Now we can hold that view lightly, expecting further evidence to be revealed to tip the scales, but I don’t see how we are justified in remaining agnostic if the only hypothesis that explained all the evidence, conformed with (i) and accounted for (ii) was Larry Longbottom’s claim.

  33. Stuart
    Stuart says:

    Damian,

    If he is able to subsequently repeat demonstrations of precognition I would count that as strong evidence and would tend to believe but would hold a small reserve of scepticism in reserve…

    Agreed up to this part

    …in the absence of an explanation for the mechanisms of precognition.

    One doesn’t need to know how something works for it to be the best explanation. I may not know how light-bulb works, but can be justified in believing in electricity runs it. I may have never seen lightening before and have no idea how it works, but on the basis of a tree ripped apart and singed by fire, I can be justified in believing in lightening. If someone I knew was on the other side of the world was to suddenly appear next to me, I would be justified in thinking they were there. An explanation does not need an explanation in order for it to be the best explanation.

  34. Heather
    Heather says:

    Forgive me for interjecting, but it really sounds like everyone is getting so caught up in the numbers and statistical probabilities that the point is being missed. The claim Stuart is attempting to refute is that extraordinary claims require extraordinary evidence. The lottery example is just causing everyone to veer far away far away from the conclusion. Let’s leave the numbers and statistics out of it for a minute and look at a different example, not randomization of numbers and laws of probability:

    A commercial airplane recently crashed in the Hudson river in New York City. All passengers and crew on board survived. This is widely recognized (at least in the U.S.) as an extraordinary event.

    Different rules apply than the lottery model; whereas you could say with the lotto that with so many people buying tickets someone was bound to win, you can’t really say that with so many planes flying, one was bound to crash in the Hudson River with no fatalities.

    Having recorded accounts from eyewitnesses to the event is considered sufficient proof that the event occurred, just as that is sufficient proof for more mundane events. It doesn’t require any more evidence than watching the plane go down security camera footage and hearing reports that all survived. Even though I did not personally interview all the survivors, if enough sources report that all survived you should be able to believe it with reasonable certainty.

    Similarly there were several eye witnesses to Jesus’ death and resurrection that made a record of the event. An extraordinary event, yes, but it should not require a significantly greater amount of evidence than other events in order to believe it occurred with reasonable certainty.

  35. Damian
    Damian says:

    Stuart, I was choosing agnosticism in the absence of an explanation for the mechanism for precognition and while precognition had still not been repeatedly demonstrated.

    Are you saying that if everyone playing this week’s lottery were secretly record themselves with a current newspaper and claiming precognition, you’d believe the claim of the winner when his video evidence is presented? I think scepticism is still the most reasonable option in this scenario given the lack of evidence and proof by repetition or mechanism. (Remember, in my example the guy only wins Lotto once).

    Heather, there is a significant difference between the belief in the safe landing of an airplane and belief in the resurrection of a person 2000 years ago.

  36. Stuart
    Stuart says:

    Damian,

    It’s clear to me at least from your comment 32 you were advocating agnosticism to avoid accepting the hypothesis of pre-cognition. It looked as if you’d ruled out that hypothesis a priori. You say the evidence is not sufficient to persuade you of Larry’s former pre-cognitive abilities, but what sort of evidence would be required in order for you to abandon your bias? Extraordinary evidence?

    There are two problems with that. (1) You’d need extraordinary evidence to establish your extraordinary evidence as well, and extraordinary evidence to back up that extraordinary evidence, and so on ad infinitum. (2) This is the point I’ve been hammering – if extraordinary evidence was indeed required then we should believe very little day to day. But we do believe things like those were the numbers that rolled out in the draw, and that plane did land safely on the Hudson river.

    I’m deliberately using an illustration where repeatable demonstration is not an option because it is past non-repeatable phenomenon like miraculous events (including the resurrection of Jesus) that Bayes’ theorem can help to bring clarity to when trying to establish its historicity. The point in the article is that cavalierly dismissing the evidence if it fails to convince you of a certain hypothesis by using the phrase “extraordinary events require extraordinary evidence” is poor reasoning and should be avoided. As a axiom or principle – even as a rule of thumb – it has been shown to be demonstrably false. On the basis of the Bayesian probability calculus what is characterised as highly improbable, may not necessarily be implausible.

    . . . there is a significant difference between the belief in the safe landing of an airplane and belief in the resurrection of a person 2000 years ago.

    It seems to me that there are at least two differences. The plane can be put down to coincidence and the mighty good fortune of all the natural phenomena coming together – sound like providence – and the other is impossible naturally. It breaks the natural expectation that dead men stay dead and do not come back to life. The second difference is the elapsed time between the events.

    On reflection these differences don’t diminish Heather’s point, which I take to be for this extraordinary event we have good, ordinary evidence, and are justified in believing it as historical.

    The point I make is the principle “extraordinary events require extraordinary evidence” does not apply to the hypothesis “God raised Jesus from the dead”, just as it does not apply when you see the lottery numbers reported even though they are, as a sequence, highly unlikely.

  37. James
    James says:

    Given the problem of induction (that past events can not rationally predict future events, or that particular events can’t be given universal status) I don’t see how probability theories (based on induction) help or hinder the Christian claims.

    So the claim that extraordinary claims require extraordinary evidence is based on what? And extraordinary claim is extraordinary compared to what?

  38. Stuart
    Stuart says:

    Bayes’ Probability theorem will help justify truth-claims that are historical in nature, like Christ’s ministry of miracles, and particularly His resurrection.

    The claim that extraordinary claims require extraordinary evidence is based on Hume’s claim that no amount of evidence of a reported miracle is enough to overcome its intrinsic improbability.

    An extraordinary claim I take to mean something which is out of the ordinary, which loosely fits the description of a miracle defined by Hume and thought soundly refuted. But as I’ve said above, in part by failing to appreciate the probability calculus he confused miraculousness with probability and frequency with implausibility.

  39. Jim
    Jim says:

    The problem with the lottery example is that the “claim” has more meaning to the average human mind than you’re giving credit for.

    The true “claim” (in it’s entirety) is thus: (From the newspaper) Ladies and gentlemen, on Tuesday, someone had the winning ticket for the lottery. The chances for winning with any given ticket are 1 in 45 million. Over the past 3 weeks, 175 million tickets were sold.

    I think THIS is why the average human does not find it extraordinary. It’s from human EXPERIENCE that if we flip a coin, about half the time we get heads and half the time we get tails. We extrapolate from this experience that with enough trials, even hugely improbable events are basically inevitable.

    The 1 in 45 million is the claim.
    The 175 million tickets sold is the evidence. (We witness people buying gobs of lottery tickets at the store)

    Extraordinary claim met by extraordinary evidence.

    The claim that Jesus was Resurrected–Extraordinary? (except for many religions prior to Christianity)
    The evidence that Jesus was Resurrected–some ink on some paper by one or two guys that had an agenda to start a new religion (similar to Joseph Smith, L Ron Hubbard, et al)

    I don’t need Bayes Theorem to tell me that more than one of these religions is wrong. And they all have the same amount of evidence. Ink on paper.

  40. Stuart
    Stuart says:

    Jim,

    I will leave your poor analysis of the historical credibility of the resurrection and the origin of the Christian faith aside for now. (I hope to write on this in the near future.)

    I’ve continually maintained that someone winning the lottery is not the extraordinary event. Rather it is the actual numbers. Given the complete randomisation of 40 balls, falling into any sequence of six is 1:2,763,633,600. Because experience tells us we believe the report of that sequence there must be something wrong with the phrase “extraordinary events require extraordinary evidence.” Bayes’ Theorem helps tell us what exactly.

  41. Jim
    Jim says:

    Hey guys,

    From your original post:

    >>what is the probability that if Jesus of Nazareth did not rise from the dead we would have the evidences of the empty tomb, the post-mortem appearances and the origin of the disciples belief, et cetera?<<

    What’s the probability that if there really were no golden tablets and the Angel Moroni hadn’t led Joseph Smith to them that we would still have the Book of Mormon and 10 million+ mormon belivers?

    What’s the probability that if there really were no DC-8 flying through space with souls on board and delivering them to earth that we would still have Scientology.

    The answer is obviously “1” since these religions really do exist–and they can’t both be right.

    So the answer to your initial question above can reasonably be “1.” No Bayes Theorem required.

    Jim

  42. Jim
    Jim says:

    O.K., I was just reading the transcript of the debate between W. Craig and Bart Ehrmann and saw what you you’re talking about–Craig uses the Bayes Theorem equation to (unsuccessfully) make his point.

    He says the first term in the equation [you use pr(H)] is “intrinsic probability of the resurrection. It tells how probable the resurrection is given our general knowledge of the world.”

    I think I would give that a “0” since what I “know about the world” says that people don’t rise from the dead?

    With a “0” in the numerator, the rest is irrelevant.

    What am I missing?

    Jim

  43. Stuart
    Stuart says:

    Jim,

    Re43:

    First off, if the probability of them all is 1 then they are all true, and as—as you say—all religions cannot be true, then something must be wrong with your analysis.

    I do not see how your two analogies correspond to the historicity for the resurrection of Christ. The empty tomb for instance is not in itself supernatural, miraculous or extraordinary. It is empirically verifiable for the contemporaneous witnesses in the strict sense of scientific investigation, and in the secondary sense of historical investigation for any later historian. As neither DC-8’s nor the golden tablets and Angel Moroni are empirically verifiable by, nor accessible to, outside observers they can hardly count as evidence towards the truth or falsification of said belief.

    Further, it is not the religion and its many adherents that is evidence for their claims. For Christianity it is the very origin of the disciples belief the God had raised Jesus from the dead, that stands as evidence. Jewish messianic expectation had no concept of a dying, much less a rising, messiah. For those disciples to subsequently be preaching boldly and proclaiming the resurrection a matter of weeks afterwards, and for the resurrection to be the centre-point of that early Christian movement, this stands as powerful evidence that something must have happened. Now what else could have happened to make them so willing to die for that belief?

  44. Stuart
    Stuart says:

    Jim,

    Re44:

    I think I would give that a “0? since what I “know about the world” says that people don’t rise from the dead?

    First, if H is the hypothesis God raised Jesus from the dead, you can’t assume then that “people don’t rise from the dead” as a brute fact before you’ve even started. That is the very thing trying to be established.

    Second, the general background knowledge of the world would be that dead men don’t rise naturally from the dead – but that’s not the hypothesis is it.

    Third, science works from observation of the specific to postulate outwards to the general, i.e., in an inductive fashion. But science and observation itself cannot rule out something in the deductive way you seem to do here. Science cannot disprove an acceptation to a pattern, it can only confirm and predict the continuance of a pattern. So knowing that no one has ever been risen from the dead can never be proven and strikes me as wildly arrogant.

    Fourth, Craig comments on that same debate in Reasonable Faith, Third Ed, p. 280;

    When I pointed out the faux pas to Erhmann in our 2006 debate on the resurrection at Holy Cross, rather than correct his mistake he pooh-pooher my explanation of the probability calculus as a “mathematical proof for the existence of God.” He did not seem to understand that I was not using Bayes’ Theorem to prove God’s existence or even Jesus’ resurrection but rather to explain to him why his own argument based on the improbability of miracles is demonstrably mistaken. It was clear that he understood neither Hume nor Bayes’ Theorem. Ironically, Ehrmann sought to defend his position by claiming that because the hypothesis “God raised Jesus from the dead” is a statement about God, it “is a theological conclusion . . . not a historical one.” Since “historians have no access to God,” they “are unable to establish what God does.” This claim, whatever it is worth, is logically contradictory with his claim that the resurrection is intrinsically improbable. For if the historian cannot say anything about God, neither can he say that it is improbable that God raised Jesus. The historian would have to say that the probability of Jesus’ resurrection is simply inscrutable. Thus, Ehrmann’s position is literally self defeating.

  45. Jim
    Jim says:

    In the transcript for the Ehrmann-Craig debate, Dr. Craig states:

    >>Specifically, Dr. Ehrman just ignores the crucial factors of the probability of the naturalistic alternatives to the resurrection [Pr(not-R/B) × Pr(E/B& not-R)]<>“Ananus thought he had a favorable opportunity…And so he convened the judges of the Sanhedrin and brought before them a man named James, the brother of Jesus who was called the Christ, and certain others. He accused them of having transgressed the law and delivered them up to be stoned.”<<

    The text just labeled him as the “brother of Jesus who was called the Christ.” It doesn’t say he was stoned BECAUSE of his Christianity. Hector Avalos chastised Dr. Craig’s “license” to imply some meaning above and beyond the text. The rest of the martyrdoms (again, as I understand it) are all self-contained in the Bible, which is using the Bible to prove itself.

    Cheers,

    Jim

  46. Stuart
    Stuart says:

    Jim,

    When historians assess the texts now contained in the Bible, they do not assume any sort of inspiration, inerrancy or reliability. Nevertheless, when assessed for historical value these books are given very favourable marks. So discounting the books of the Bible because they are contained within the Bible is very poor historical practice.

    We’re getting a little far from the field. I don’t really want to address the historical credentials of the content of the NT and the resurrection of Christ here. I’ll just quickly mention…

    James’ martyrdom does not rely on the quoted passage from Josephus alone. But on that passage alone it is reasonable to think that James was killed because he was a Christian. The historicity for Peter and Paul’s martyrdom is also very well established. And there are plenty of extra-biblical sources that speak of other martyrdoms, including at least three early secular historians. And virtually no scholar would deny that to preach Christ in the early church context was to expose yourself to the possibility of death.

  47. Matthew Flannagan
    Matthew Flannagan says:

    “The answer is obviously “1? since these religions really do exist–and they can’t both be right.

    So the answer to your initial question above can reasonably be “1.” No Bayes Theorem required.”

    Yes but the same line of reasoing shows that atheism is improbable. Athiesm, is incompatibel with Islam, Christainity, Hindusim, Judaism, Deism, they can’t all be correct so the probability that atheism is correct is low.

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