Sunday, April 27, 2014

Extraordinary Claims and the Principle of Indifference

You've probably heard the saying "Extraordinary claims require extraordinary evidence." It's the starting point for perhaps the most common argument by atheists: "The existence of God is an extraordinary claim that lacks extraordinary evidence." Seems logical, right? The only problem is, how can we determine whether the evidence (or the claim) really is extraordinary?

One common definition of extraordinary is "very unusual". But the claim that God exists isn't unusual. By that definition, "There is no God" would be a more extraordinary claim. But that standard doesn't always make sense. For example, someone could make the very unusual claim that I'm currently wearing three socks, but most people wouldn't require extraordinary evidence to be convinced. Another common meaning of "extraordinary" is "very remarkable or amazing". That one brings us right back to the original problem: How can we determine how remarkable or amazing a claim is? There are other definitions of "extraordinary" but all are similarly problematic.

A much more scientific way to formulate "Extraordinary claims require extraordinary evidence" is via Bayes' theorem. In Bayesian terms, an extraordinary claim is a hypothesis with a very low prior probability (e.g., “a coin flipped 5 times will land on tails every time”, which has a prior probability of around 3%). It follows that very strong evidence is required to move the probability high enough to believe the claim. Thus, it can be shown mathematically that extraordinary claims (defined this way) do in fact require extraordinary evidence. In the above example, that evidence could be a measurement that the coin's weight is very unbalanced or an observation that it has tails on both sides.

Applying that framework to the God claim, the strength of evidence required depends on a priori assumptions about the prior probability that God exists. Theists who start with a relatively high prior probability require less evidence. Atheists who start with a low prior require more evidence. Arguments about the sufficiency of the evidence for God become circular on both sides. Thus, it's imperative that we have a good, objective way to determine the prior probability.

Because we don't have specific, definite probabilistic information about the God question, we must use an uninformative prior. The simplest and probably most common of these is the principle of indifference, which says the prior probabilities of all hypotheses are equal. In the binary case of “Does God exist?”, the prior is 50%. Starting with a 50% probability may seem crazy if the claim seems ridiculous, but it makes good sense mathematically. The evidence (or lack thereof) is probably what makes such claims seem ridiculous in the first place, and the other terms in Bayes' rule account for that. Also, if the claim seems ridiculous to most people, that fact alone is evidence that would reduce the probability.

Using the principle of indifference, presuppositions about the probability of God's existence are eliminated as determining factors. The estimate of the probability that God exists now depends entirely on the evidence. In this case, “Extraordinary claims require extraordinary evidence” is a meaningless argument. It doesn't matter how extraordinary the claim is because the evidence will tell us whether to believe it. We'll still argue about the evidence and how to assign probabilities to it, but that's a lot more useful than debating a theist's circular argument vs. an atheist's circular argument.

There are other ways to determine uninformative priors, including some that let us use the “extraordinary claims” standard. But when applied to the God claim, they generally require arbitrary assumptions that lead to self-fulfilling conclusions. That might be good enough for testing the claim that I'm wearing three socks right now, but whether or not to believe in God is a much more important question – one that I don't think should be decided (either way) by arbitrary assumptions made before examining the evidence.


  1. In science, "extraordinary" evidence is evidence that is unmistakeably in favor of (or against) the hypothesis. It is associated with a rigorous test that has been designed to exclude ambiguous results to the maximum extent, and with the extent that multiple rigorous tests fail to produce a different outcome.

    Your notion that a 50% prior probability is mathematically sensible is difficult to grasp. What does it mean to be mathematically "sensible"?

    Settling the issue of the existence of a deity certainly is a vastly more important question than your choice to wear 3 socks. If it were claimed that someone had evidence to settle a question that important, it doesn't seem at all problematic to ask that the evidence be extraordinarily convincing - at least not to me.

  2. Sorry ... change "with the extent that" to "is reinforced in situations where"

  3. Chuck, I agree with you if absolute certainty is the requirement. But faith is not a matter of absolute certainty, it's a matter of probability. Do you require "extraordinarily convincing", "unmistakable" evidence to believe that something is 51% probable?

    By "makes good sense mathematically" I mean that the proposition is logically consistent with the equation (i.e., Bayes' rule). The natural objection to the proposition is resolved via the evidence terms P(B\A) and P(B).

  4. You agree with me about what? I said nothing about absolute certainty. For many believers, their faith IS absolute certainty, as they've said they wouldn't alter their faith in the face of any evidence whatsoever.

    I would require "extraordinary" evidence (as described in my comment) to accept anything as important as the existence of a deity for which there exists little or no existing evidence. What you've offered me so far is pretty weak in terms of rigor and unambiguous outcomes. If you had such evidence, I would think you'd have brought it up.

    Sorry, but I don't see how an assumption about prior probability can be logically consistent (or inconsistent) with the equation. What is it about the equation that requires the prior probability be 50%? I see how the conditionals alter the prior, of course, but I see nothing in Bayes' Theorem that requires the prior be any specific value. You're suggesting that an objective observer (if one truly exists) would say that existence and non-existence are equally probable, because it's a binary question, right?

  5. I agree with your standard of evidence (and believe it's rational) only when applied to having absolute certainty. Faith is not absolute certainty, regardless of what "many believers" might say.

    You seem to imply that the standard of evidence required for belief depends on how important the claim is to you -- which implies that your uninformative prior is based on your opinion about the claim's potential impact on your life. I don't believe that is a rational way to determine a prior.

    My point was that an indifferent prior is consistent with the equation because the equation allows for near-zero probabilities for ridiculous claims, so it's not unreasonable to start at 50%. Of course nothing about Bayes' theorem requires that the prior be indifferent. My only point was that 50% was reasonable. If you insist on a much lower prior, you'd need to provide justification for it, which you have not. I don't think "because I think it's important or extraordinary" is sufficient justification. My justification for an indifferent prior was that it's the simplest, most objective, and there's no compelling reason to use any other prior.

    I agree that the evidence I've provided so far doesn't meet your requirements. Hence, I'm not saying you should have absolute certainty that God exists. But I'm making a probabilistic claim where I only need to establish the the positive evidence outweighs the negative evidence. To demonstrate that I'm wrong, you can A) provide a better means of determining the prior, or B) provide sufficient negative evidence to negate the positive evidence (which, by the way, I've only shared a small amount of to date; there is much more).

    Before you say I'm shifting the burden of proof or asking you to disprove the existence of God, understand what my assertion is and what it isn't. It is not that hypothesis A (God exists) is true. It is that hypothesis A is *more probable* than hypothesis B (God doesn't exist).

    1. Your belief that my standard for evidence only applies in the case of having absolute certainty may be another of your beliefs, but it's also another of those beliefs I don't share. You have a habit of stating your hyperbolic interpretation of my comments in contemptuous terms and if I indeed meant them the way you interpret them, perhaps they ARE worthy of contempt.

      You bring up the topic of the additional "positive evidence" you have, but haven't mentioned what that putative evidence is ... I would have guessed if you had some "big guns", you'd already have fired them. Why hold back up to this point?

      Let me repeat my question: You're suggesting that an objective observer (if one truly exists) would say that existence and non-existence are equally probable, because it's a binary question, right? Your logical reason for a 50% prior is that it's a binary postulate? I will agree that the starting point is not very critical, however, so if you think it's reasonable to assume we know nothing then it might be reasonable to assume each hypothesis is equally likely. However, there's a lot of evidence to suggest they aren't equally likely - in the form of the ABSENCE of compelling evidence for such a deity.

      Obviously, we also disagree strongly that the evidence favors hypothesis A over hypothesis B. And we also disagree over whether or not it's rational for extraordinary claims to require extraordinary evidence. I find it somewhat difficult for me to grasp how you can deny this, but I must accept the empirical evidence. The fact that I haven't come up with an objective metric for what constitutes "extraordinary" is irrelevant. I've already discussed what sort of evidence I'm talking about, in science, and with regard to what I'd expect to be compelling evidence in favor of a deity. The fact that you brush those aside as being irrationally tough standards that it exceeds the bounds of science to require them is also irrelevant.

      I do agree that whether a claim is extraordinary or not, the evidence is what we use to judge its validity. But when a claim represents a major paradigm change, it's rational to demand more compelling evidence than when the claim is a minor adjustment to an existing paradigm. Every research scientist participates in the process of changing paradigms to a greater or lesser degree, and it's obvious (at least to me) that asking people to change the way they think is going to require you to convince them in a compelling way. We don't cast aside the consensus just because someone is skeptical - we require the skeptic to provide us with evidence that we do so.

      If I were to claim I had a solution to the turbulence problem in closed mathematical form, that would likely be dismissed before it was even submitted, because such a solution seems impossible. If accepted for review, every word, every equation, every bit of evidence would be scrutinized in detail, seeking to invalidate my claim. One experimental outcome would NOT be sufficient, although it likely could be enough to stimulate a search for more evidence. It would have to be confirmed by independent researchers. More rigorous experiments would be sought and conducted. Skeptics would disbelieve for decades despite mounting evidence, always nipping at the claim wherever possible.

      Is not the existence of the abrahamic deity even more extraordinarily important than the turbulence problem?

    2. Thanks for the long reply. Though I discuss evidence occasionally, my main purpose here is not to convince atheists that there is a God. My main goal is to provide tools to help people approach theological questions more rationally. I'll get into more evidence eventually, but it's pointless without first establishing a scientific framework by which to weigh the evidence.

      When you say “there's a lot of evidence to suggest they aren't equally likely”, you're talking about evidence, not priors. That belongs in the other terms of the equation. I still haven't heard a good reason why “each hypothesis is equally likely” isn't the best place to start. Again, you say “absence of compelling evidence”, but you've provided no non-arbitrary way to determine whether evidence is sufficiently compelling.

      I agree that claims that represent a major paradigm change demand more compelling evidence, but only because they demand a higher level of certainty, which requires stronger evidence. To determine the strength of evidence required for a fixed (say, 51%) probability, the potential paradigm change and practical implications of the claim are irrelevant. The required level of certainty is what determines that. Hence, I think your very high evidence requirements make sense where an extremely high level of certainty is necessary. Such certainty is not necessary here. 51% is high enough. By the way, how can something (i.e., “God exists”) be a major paradigm change if it's what most people already believe?

      You said: “We don't cast aside the consensus just because someone is skeptical - we require the skeptic to provide us with evidence that we do so.” I agree! So why can't you apply that same standard to the worldwide (current and historical) consensus regarding the existence of a God? In your turbulence example, the consensus against your claim would be evidence against it, which requires you to provide more positive evidence. But if your claim is what 84% of the world's population already believe, your evidence for it wouldn't have to be nearly as strong.

    3. OK - so the starting point for a prior is to assume nothing and make no claims regarding evidence. I believe that assumption is unjustifiable and so represents something arbitrary, as well. You're asking me to provide proof that any other starting point is arbitrary. I assert that your 50% rule is also arbitrary. Show me that it's not.

      Why is not certainty appropriate in deciding whether or not a particular deity exists? It seems to me a rather high fraction of people have a LOT of certainty regarding that issue, one way or the other.

      I already answered this:

      By the way, how can something (i.e., “God exists”) be a major paradigm change if it's what most people already believe?

      in previous comments. I think even believers would find it a sea change in their lives for compelling evidence for the existence of a deity to become available. An interesting side issue is how they might react if the evidence favors a deity different from the one they have chosen to accept on faith.

      The "consensus" regarding God is not a particularly overwhelming majority, and there's considerable differences among them regarding just what sort of deity it is that they worship. Some fraction of believers harbor doubts but they fear retribution for voicing them. Some who profess to be believers are actually atheists.

      The evidence for proving something that 84% already believe in certainly would be seen by those believers as simply reinforcing their confirmation bias. What about the other 16% (assuming your figures are correct)? Would that evidence convince them? Actually, according to this site:

      Christianity and Islam (+ a negligible number of Jews) constitute 55% of the world's population, not 84%.

      You've been going on and on about objectivity, metrics, etc. - that's all great stuff but then you have this glaringly subjective opinion that the number of people believing in something has any particular relationship to the validity of that belief. I see that as a howling inconsistency in your arguments.

      Your arguments re this consensus also underplay dramatically the effect that the religious meme has on people: many were brainwashed as children, they fear the anger of their deity, they long for meaning and purpose in their lives, they fear death, and so on. Religion is about control of people and many techniques have been learned in the last 2000 years for perpetuating the religious meme. It's obviously been a success, but it's not necessarily true as a result of that success - truth doesn't necessarily cling to the majority.

    4. Do you really believe it's unjustifiable and arbitrary to start with no assumptions about the relative likelihood and let the evidence decide? Really? Yet it IS justifiable to start with assumptions about the prior, based on Chuck's opinions about what is "extraordinary", and let those determine how much evidence is required? Here's justification for my method: it's the simplest method, standard practice for many years, most objective, and is repeatable. The "Chuck's opinion" method is much more complex, purely subjective, and is not repeatable by anyone other than Chuck.

      Re: Certainty -- Certainty isn't appropriate because there isn't sufficient evidence to know with 100% certainty that a particular deity exists. But there is enough to believe that one probably exists. That's why we use words like "belief" and "faith". That some people act like they're certain is irrelevant.

      Re: Paradigm -- The question is "Does God exist?". The answer of "yes" would NOT be a major paradigm change. If you change the question to "Is there undeniable evidence that proves beyond any doubt that God exists, such that belief and faith are not necessary?", perhaps "yes" would be a paradigm change, but that's not what we're talking about here.

      Re: Consensus -- Yes, Christianity, Judaism, and Islam are ~55%. My 84% figure is belief in any God.

      You brought up consensus by saying "We don't cast aside the consensus just because someone is skeptical." I'm simply asking you to apply the same standard to religion as you do to everything else. Either consensus moves the prior in the positive direction or it does not. Which is it? My answer is that it does. Your answer seems to be "It does, except when it's about God."

      I agree that consensus doesn't necessarily mean something is true. That's exactly why we use Bayes' theorem -- because it accounts for uncertainty in the interpretation of evidence. There always are other possible explanations for evidence, which is why it's imperative that we apply scientific methods like Bayes' theorem when discussing claims and evidence regarding God or lack thereof. Otherwise, it's just my confirmation bias vs. your confirmation bias, which is fruitless.

    5. More mischaracterization of my position. I'm not going to bother responding to that.

      When it comes to deciding whether or not to believe in a deity, the more certain we can be, the better ... right? If we can reach a level where the probability is 99.99+% in favor of your hypothetical deity existing (or not), then we can justify behaving with near-absolute certainty - as we do with, for example, Newton's Laws.

      Most believers in your consensus have not done anything like consider the evidence in terms of Bayes' Theorem or the scientific method. They aren't biblical scholars and many haven't even read the bible! Their confidence in the existence of their deity is not derived from evidence at all - just faith - in fact many take pride in their faith, despite the absence of convincing evidence. These are a non-trivial fraction of your (scientifically-derived?) "consensus". It might be a majority, but I see little reason to place much confidence in majority opinions when most of the sources of such opinions have so little expertise to apply to the question.

      You still have yet to show the evidence that permits you to put your conditional probability at least at 51% in favor of the existence of your god.

      Since the existing christians have no such compelling evidence that would be convincing to most non-believers, I maintain that it would indeed be a major paradigm shift for them as well as for many nonbelievers. Obviously, if compelling evidence were found for the non-existence of your deity, that would be a huge paradigm shift.

      Our discussions are going to always go down the lines of competing confirmation bias, methinks.

      I think the "consensus" of which you speak is not a nearly unanimous one, rather unlike that of, say, the consensus of global climate scientists re AGW. For reasons I've mentioned, it's a pretty weak consensus and covers a rather large spectrum of believers, not just subject matter experts. You seem to suggest that any old sort of evidence is sufficient to tilt the table in favor of existence. Suppose we say that whatever evidence you have is enough to convince *you* (as it apparently has) - does that not automatically put you on the defensive when it comes to someone who won't accept your evidence as very convincing? That's what this discourse reveals, it seems to me. You're doing your best to wrap yourself in the flag of scientific objectivity and, simultaneously, seeking to reject anything I propose as unscientific if it doesn't follow your standards religiously (pun intended).

      Your version of the "scientific methods" is not the same as mine, evidently.

    6. Chuck, 99.99+% certainty would be great, but it's not realistic with available evidence. Surely you don't require 99.99+% certainty in order to believe in something, do you? If A is 51% likely and not-A is 49% likely, I would believe A instead of not-A. Which one would you believe?

      No it's not a unanimous consensus, and they're not all experts, but 85% is substantial. Are you saying an 85% consensus shouldn't move the prior at all?

      I appreciate your other commentary & opinions and can respond to them if you'd like, but I'd prefer to stop dancing around the issue. My framework is Bayes' theorem and I explained why. If you have a better alternative, what is it and why? My prior is 50% and I explained why. You haven't yet suggested an alternative. So please tell me... What do you think the prior should be, and what is your justification for it?

  6. Matt--I completely agree that the evidence for God required by followers of any theistic faith is considerably less than is demanded by atheists. Empirical observation has made this abundantly clear, and the pattern is very consistent. Agnostics seem to sit astride the fence, on a broad spectrum between intense curiosity and pure apathy. Those agnostics not too far on the apathetic end of the spectrum are 1) open to evidence for, or against, that they haven't seen yet, or 2) evidence that they have seen but not comprehended sufficiently, all the while waiting for a tipping point of some sort.

    Another aspect is that for many believers, the evidence for God lies largely in intangibles such as profound spiritual experiences, answered prayers or personal miracles. These lie outside the realm of pure (or to some, "mere") science; they cannot spin an anemometer, be poured into a beaker or get blasted with protons in a particle accelerator. Yet for those who have such experiences, there's no turning back; the evidence to them is overwhelming, even if it isn't mathematical. Legitimate realms of thought do exist outside the tangible and measurable.

    I want to say that this is a very, very fascinating and stimulating BLOG, of which I am a regular reader. I haven't chimed in much yet, or had a lot to say, simply because I'm reading, learning, sponging this stuff up, absorbing your analyses and discussions as they come. Keep the insightful discussions coming.

    1. Thanks very much, Roger! I really appreciate the encouraging words.

      Interesting point about intangible evidence. I generally focus (maybe too much) on the tangible and scientific stuff for this blog, but I am intrigued by the intangibles. I agree that it doesn't have to be quantitative or tangible to be evidence. There also are philosophical arguments and 'evidence' for God (some I agree with, some I don't), which I haven't touched yet but might. If I do, it'll probably be from a scientific perspective as I'm not qualified to say much about philosophy.

    2. There's no way that intangible evidence can ever be considered evidence by any reasonable scientific standard. It may be compelling to those who experienced these things, but it's quite comparable to alien abductions or sasquatch encounters ...

    3. The scientific standard of evidence is whether it is more likely to be observed if the hypothesis is true than if it is not true. Granted that's difficult to estimate and uncertainty is relatively high for intangible evidence, but if it meets that standard, how can you possibly justify rejecting it as evidence?

    4. Wow! You have an interesting view on what is the "scientific" standard for evidence, since it seems directly linked to the validity of the claim. I've already discussed this in earlier comments elsewhere: the evidence must be reproducible, offer an outcome as unambiguous as possible in favor of or against the hypothesis, must exclude the possibility of alternative hypotheses to the maximum extent possible, must account for as many relevant variables as possible, etc. Intangible evidence simply doesn't permit any of that.

    5. That's not my view, it's what directly follows from Bayes' theorem. If you think you can disprove it or can find a loophole, you should write a paper about it. That would be a major paradigm change indeed.

      I don't disagree with "as unambiguous as possible", "to the maximum extent possible", and "as many relevant variables as possible", but they don't mean much in this context, because "as ____ as possible" is a very small amount (whether for or against). Situations like this are why we have probability theory. It makes accurate predictions in the real world, even when the available evidence is relatively weak, ambiguous, non-reproducible, etc.

    6. It's not YOUR view? Then why has that never been mentioned in any of my education and 40 years of research experience ... until now? I see "intangible evidence" as containing a near-100% certainty of not being useful in any sort of rational search for evidence.

    7. I never heard about it in school either, but I think meteorology schools should teach it. It's very useful when building probabilistic models, especially when there's a lot of uncertainty in the input data.

      There's no "tangibility" term in Bayes' rule. The only requirement for evidence is that it not be equally probable for every hypothesis. Do you believe intangible evidence is always equally probable for every hypothesis? I think that'd be easy to disprove.

  7. I've never seen a scientist use intangible evidence in a peer-reviewed paper, ever. Do you have any examples?

    "The only requirement for evidence is that it not be equally probable for every hypothesis." Where do you come up with these rules I've never heard any scientist use, ever? Are they part of some personal vision of how science works?

    1. Here's a good paper about it from Annals of Internal Medicine. It also mentions indifferent priors.

      Here's a good quote from it: "Every hypothesis under which the observed data are not impossible can be said to have some evidence for it. The strength of this evidence is proportional to the probability of the data under that hypothesis"