Bayes’ Theorem and Faith

By Brigham

I’ll do my best to minimize the nerdiness of this post, but I make no guarantees.

Something that always puzzled me was how two rational, intelligent people could take the same information and come to opposite conclusions. Two physicists observe a universe governed by an elegant set of rules, with parameters apparently minutely tuned to allow life to develop, and one sees a Creator and the other sees  a coincidence. Is one interpreting the data wrong? Is one being fooled by wishful thinking? Not necessarily. There’s no rule that says two people seeing the same data must come to the same conclusion. In fact, there is a rule–Bayes’ Rule–that says just the opposite: the conclusion you draw from observing data depends not only on the data itself but on the beliefs you had before observing the data. So two people can draw equally valid but different conclusions from the same data, depending on what prior beliefs they are bringing to the analysis.

Here is Bayes’ rule applied to the probability that God exists, given some data we observe about the universe. This data could be the fact that the physical world seems to obey an orderly set of rules, it could be a miraculous experience, or anything else that might affect our belief in God. The probability that God exists, given some data about the universe  is equal to the probability of observing that data if God did exist, multiplied by our prior probability that God exists (our belief before observing the data), divided by the overall probability of observing the data. In mathematical form, just to keep things clean:

Pr(God exists given data) = Pr(data given God exists) x Pr(God exists) / Pr(data).

So what does this mean? It means the higher somebody’s prior probability that God exists, the higher the probability they will assign to God existing even after observing the data, and vice versa, somebody with a very low prior probability of God existing will tend to give a lower probability to God existing, even after observing the same data. In fact, if you look carefully, somebody who has a zero prior probability of God existing (an Atheist) will continue to be an atheist, no matter how much evidence he or she is confronted with, since zero times anything is … zero!

So then how do dramatic conversions happen? Wouldn’t Saul from the New Testament (before he became Paul) have had a zero prior probability that Jesus is the Son of God? Or Alma the Younger from the Book of Mormon? Does this Bayes’ Rule framework not apply here? Actually, I think it applies perfectly. It’s true that if you multiply zero by anything (finite) you get zero, but when you divide zero by zero you can get anything! So if we take our formula seriously, the only way for a recalcitrant unbeliever to be converted is to somehow divide by zero–they need to have an experience that they would have considered to have zero probability of happening. In other words, an epiphany. Saul, in hearing the voice, and Alma, in seeing the Angel, observed “data” that they would have thought had zero probability of occurring, but there it was all the same, and the result of incorporating the experience into their beliefs was conversion.

One possible criticism of Bayes’ rule as a useful framework here is the critical role the prior beliefs play in the final conclusion. After all, it says nothing about where that prior probability comes from. It could just be the subjective probability that each person chooses or is just born with, or the product of upbringing, environment, etc. And worse still, for any potential piece of evidence or data you could confront me with having some bearing on whether God exists, I could generate any final (posterior) probability of God existing by just choosing a different prior probability. So if we’re free to choose our prior beliefs, then Bayesian analysis just perpetuates whatever initial prejudices we had, right? Is this just teched-up confirmation bias?

Not quite. True, we may start out with different prior probabilities that are functions of preferences, choices, upbringing, etc., but once we start observing data and updating our beliefs, as we incorporate more and more evidence into our beliefs, the importance of the initial prior belief fades away. Indeed, you can start from any initial beliefs about God (as long as the probability you assign initially is less than one and greater than zero–unless you’re counting on an epiphany!) and eventually your beliefs will converge on the truth with more and more certainty–no  matter what your starting point was, as long as the initial probability was greater than zero, even if ever so slightly.

So the only requirement to develop faith is to start out allowing at least a shred of possibility that God exists (or whatever the spiritual truth is we are seeking faith about), and then to actively and honestly seek out and incorporate evidence into our beliefs. Easier said than done, but if Bayes was right, it’s a mathematical fact you will get to the truth!

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33 thoughts on “Bayes’ Theorem and Faith

  1. Jason says:

    Two (multi-part) follow-up questions. Why believe anyone could replicate the process for themselves and have similar experiences? Let me distinguish between the physiology of the experience (e.g. we may have drugs which activate a similar region of the brain as when people pray) and the religious interpretation of those experiences. I agree it is quite plausible that all or most humans can experience something which is physiologically akin to spiritual ecstasy (this could be drug induced), but I think you are saying something which is more closely bundled with an interpretation – not merely that people have a particular emotion, but that they reorient their perspective towards the events in their life in a way that would convince them of the reality of a God external to them. But in what sense is it meaningful to say that anyone could do that? There have been many people who have wanted dearly to have a relationship with God and then found themselves compelled by personal experience – and by the apparent absence of God in their lives – to abandon their faith (You could say, “Like Job, they could have chosen not to” but you can’t really choose what beliefs you think are warranted). Not to mention many others who have grown up in secular societies and for whom the idea of a loving God with whom they could have a personal relationship is no more a live hypothesis than the idea that mummification is required to gain passage into the afterlife. In what sense is it meaningful to say that such people “could have similar experiences” other than the trivial claim that if their lives had gone differently they might believe different things?

    Second, my account of the alien abduction stories would be that no matter how compelling their subjective experiences, people who believe they have been abducted by aliens are making an error because they overweight their (erroneous) interpretation of these experiences relative to the overwhelming objective evidence which suggests that this interpretation is almost certainly wrong. Do you agree with this account of the alien abduction stories? If so, what do you see as the disanalogy with your religious beliefs? Is the issue that the objective evidence against the alien abduction hypothesis is stronger than the objective evidence against God, or would you say that there is no disanalogy and I just give too little credence to the personal experience of those who claim to have been abducted?

  2. Jason says:

    That is a helpful clarification, and I don’t disagree with your last point. Let me make the following distinction: some believers put forward objective arguments for God’s existence – e.g. arguments based on the origin of the universe, design in the physical or biological world, historical evidence for the truth of biblical claims, etc… One can contrast these with subjective arguments based on one’s experiences (construed holistically, to include things like how one sees God’s plan in the events of one’s life).

    Are you saying that your believe in God largely on the basis of what I am calling subjective arguments? In particular, suppose your personal experiences were different than they are – you didn’t feel that God answered your prayers, you didn’t see God’s plan at work in your life, etc… would you then view God’s existence as unlikely given everything you know about the world? (which as you say, is not the same as saying that God does not exist).

    1. Brigham says:

      Excellent question, and one I think all believers and nonbelievers should put to themselves. My answer is that my belief in God originates from what you’re calling subjective arguments. I should add, though, that despite being subjective I believe anybody could replicate the process for themselves and have similar experiences. If you like, these subjective experiences formed my prior about God which affects how I interpret objective data on whether or not God exists.

  3. Jason says:

    Brigham,

    I actually didn’t know that is what Mormons believe. Is this an accurate summary of your position: “From an objective standpoint, my faith in God is no better substantiated than the beliefs of people who insist that they have been abducted by aliens. From my point of view, it seems unlikely that they were abducted by aliens, although perhaps from their point of view God’s existence is equally implausible. We are both justified in believing what we believe, and we both can do no more than to trust our experiences since if we don’t trust those, why should we believe anything?”

    Here is an opposing view (closer to my own). There is a truth of the matter – either God exists or he doesn’t. Either aliens really have come to Earth and abducted people or they have not. If someone claims they have been abducted by aliens, they might be right (it’s not impossible that aliens have come to Earth), but they are almost certainly objectively wrong. Prior to 1950, people reported experiences just like alien abduction, but instead of aliens, they vividly describe being visited by witches or ghosts or whatever idea of this type was prevalent in their culture at the time. We know people hallucinate and rationalize to trick themselves all the time, so this is a much more parsimonious explanation than the claim that aliens for which we have no independent evidence have flown thousands of light-years across the galaxy to probe some rural Americans, concealed themselves from human civilization almost perfectly but occasionally slip up because their anesthesiologists aren’t up to snuff, and by the way, they happen to look uncannily like the illustrations conjured up by science fiction writers of the era. This position could be fully confirmed if we could understand the conditions which lead to abduction beliefs well enough to recreate them in a controlled setting where we could document what was occurring. This would establish objectively to near certainty that the experiences are not evidence of actual aliens.

    So maybe I’m still not understanding your position. I don’t think you are denying the possibility that our sensory experiences can just be mistaken. Any optical illusion is proof of this (you might think the blue and green here are different colors, but they are objectively the same: http://bit.ly/JICuB).

    I am NOT arguing that in order to believe anything you must deductively derive your beliefs from first principles a la Descartes without any reliance on personal experience. I’m just saying that the inferences you make from any given personal experience can be called questioned and must be judged in light of everything else you know about the world, so it really does matter whether your beliefs about God are well-substantiated by other lines of thought or undermined like those of the alien abductees.

    1. Brigham says:

      That’s an awesome optical illusion. I’d never seen that one.

      I don’t have exactly the right vocabulary for this, epistemology being a major gap in my education, but I think it’s worth making the distinction clear between the nature of truth on the one hand, and our ability to know it on the other. As for the first, in Mormonism there is an objective truth (as well as a Right and a Wrong). Like you, for us either God exists or he doesn’t–there is no “he exists for me, but maybe not for you.” We also put fairly high stakes on finding out what this objective truth is. Part of our morality is dedicating ourselves to finding out (to the best of our ability) what the truth is. Complacent agnosticism is not OK in Mormonism. Now to what we mean by relying on our own experience to find out the truth. I’m using experience more broadly than simply “perception.” Experience means applying all of our faculties–physical senses, logical reasoning, intuition, emotions–to interpreting what we encounter and for actively investigating hypotheses (e.g. does God exist). Your optical illusion is a great example. The most simplistic conclusion is there are two different colors, but by closer examination (but still using only our own faculties) we are able to get to the truth. We believe the same thing holds for more important questions like “does God exist.” A simplistic conclusion might be “I’ve never seen him, so he must not exist” or “my parents always told me he exists, so he must.” We believe that by more thorough investigation, including examining the sources of our initial simplistic conclusion, we can and will get to the truth if we earnestly seek it.

      However, in the end, what we believe is still a product of our own evaluations and judgment, but I think this is true for everyone everywhere and we have to live with that.

  4. Jason says:

    Teppo,

    I actually have a very weak background in physics but I try to keep tabs on what physicists think about physics (which is probably what “metaphysics” should mean!). I think our disagreement may stem partly from the ambiguity in the definition of “fine-tuning” – I agree with you that there are many puzzles in fundamental physics. There are many parameters whose value is not well-understood and seems to require an explanation. But I don’t think there are good examples of parameters whose values are well-explained by the hypothesis that they are fine-tuned for life.

    Let me clarify my remarks a little bit and say what I can given my limited understanding of some of the issues involved. There is one parameter which I agree is often considered “fine-tuned” and for which this is arguably the case – this is the cosmological constant. Steven Weinberg famously used anthropic reasoning to predict that if the cosmological constant is non-zero, it must be extremely small, and this prediction was ultimately borne out by the data. I think many physicists still hold out hope that the value of the cosmological constant will ultimately be explained by a more fundamental theory which does not invoke anthropic reasoning (and there is ample support for this view from the history of physics), but I think most physicists would also admit that the anthropic explanation might be right. This is not an example which favors theism however, because there is a theoretical value that is conducive to life and even more parsimonious – a cosmological constant of 0. Why should there be an extra unwanted term in the Einstein equations? The puzzle for physicists is why the value of the cosmological constant is BOTH non-zero AND small. Anthropic reasoning potentially explains this but I agree that absent more evidence of a multiverse, this explanation is not wholly satisfying. But theism does not solve this puzzle either (since it has nothing to say about why the value is non-zero). It’s just something we don’t know the answer to. Steven Weinberg – the one man who has arguably made a successful prediction using anthropic reasoning – explains why he is not “impressed with supposed instances of fine-tuning” here: http://www.physlink.com/education/essay_weinberg.cfm.

    I’m actally not sure which fundamental constant you’re referring to in your claims about galaxy formation. Re: the relative strength of the forces, it’s true that gravity is much weaker than the other three forces and that if gravity had a similar magnitude to the other forces nothing even remotely resembling life as we know it would be possible. So there is a legitimate mystery as to why gravity is weaker – this is the so-called hierarchy problem and I think almost all physicists would admit that it is a problem in need of a solution. HOWEVER, I think very few physicists regard it as evidence of fine-tuning or would invoke anthropic reasoning to explain it. Instead, they seek to formulate deeper theories which explain why gravity is so much weaker. Supersymmetry is one such theory. So I agree completely with your point that the investigation of theories like supersymmetry is motivated partly by the fact that certain fundamental parameters have values which are difficult to explain. But it is precisely because of the apparent theoretical success of frameworks like supersymmetry in resolving the hierarchy problem that the vast majority of physicists would not argue that the magnitude of the gravitational force is well-explained by the fact that only some values would permit life.

    Finally, the only other point on which we disagree is when you say that belief in the multiverse requires “faith”. My guess is you’re just playing on the semantic ambiguity in the meaning of the word “faith”, but this is a very misleading statement. You seem to agree that if string theory is true there is good evidence of a multiverse. Once it is more fully understood, string theory could be tested just like any other physical theory. Until then, I agree there is a great deal of uncertainty about whether it is correct, but those who believe it is correct do so on the basis of its surprising mathematical consistency (that is, time and again an apparent contradiction in the theory has been resolved on deeper inspection) and the naturalness with which it coheres with existing frameworks (e.g. postdicting gravity). So it is misleading to say that belief in string theory requires “so much faith” any more than Einstein’s belief in relativity prior to 1919 required “faith” (the analogy is not meant to imply that string theory will ultimately be right – who knows – but that sometimes theoretical reasons justify belief in a physical theory).

  5. Christine says:

    “I don’t know if this is something that distinguishes Mormons from other, say, Christian faiths, but primacy of our own experiences is definitely a central tenet of Mormonism.”

    I think that is a big distinguishing aspect of our faith, each person has to find out for themselves. We believe in the faith-strengthening power of sharing testimony and experience, but at the base of it all we say, “Find out for yourself.” No one stands between you and God, and he will answer your prayer if it is a sincere one.

    In fact, that’s how the modern organization of the church began. Joseph Smith had an amazing vision of God the Father and Jesus Christ. He told people about it, testified he was telling the truth. When they’d ask how they could possibly believe such an incredible story he would respond that they shouldn’t take his word for it, they should go and ask for themselves.

    Joseph Smith said that we live far below our spiritual privilege or opportunity. I think he’s right. I think the whole design of our life here on Earth was to teach us about God and our eternal destiny in the most effective way possible- through our own experience. I also believe that God is willing and wanting to be a bigger part of that experience than we allow him to be, even for believers.

  6. Jason says:

    I don’t disagree with the fundamental premise of what you are saying. The basic point is that on any given question, believers and atheists could look at the same evidence, evaluate it correctly and come away with different conclusions about the overall probability that God exists given that new evidence and everything else they believe. I think we can equivalently restate this as: your judgment about God’s existence depends on the totality of the evidence, not just on any one issue. And in fact, it sounds like you don’t even believe that cosmological observations make P(God | Data) > P(God), so we don’t disagree about that either (sorry, I previously missed comment #6 so this is the main point I was arguing for). So our only real disagreement on this point is that I think you are too quick to dismiss the multiverse as having no observable implications (we’ll know better once we have a consistent theory of quantum gravity).

    I’m actually most intrigued by your statement about prayer and personal experience. I’m curious – what do you think of people who believe they have been abducted by aliens and vividly recall details of that experience? What is the relevant disanalogy between that and your own faith? (the difference could be in the “prior”, but then you need to have some independent source of evidence beyond personal experience which makes the prior probability of God greater than that of aliens visiting earth).

    1. Brigham says:

      Are we taking as given that those alien abductions can’t really have happened? :) The biggest difference between that and my own faith is that I trust myself and my ability to parse my experiences more than I trust people who say they have been abducted by aliens. I’m sure people much smarter than I have done some thinking about whether one’s own perception deserves special primacy (to oneself, at least). Maybe I’m like one of Ed Leamer’s econometricians who believes no one’s analysis but his own. Of course this reasoning doesn’t and shouldn’t do much for you. You’ve got to trust your own experiences. I don’t know if this is something that distinguishes Mormons from other, say, Christian faiths, but primacy of our own experiences is definitely a central tenet of Mormonism.

  7. Jason says:

    A clarifying comment now that I’ve been led here by Brigham’s facebook posts:

    I think both Brigham and Roberto misunderstood why the vast majority of theoretical physicists do not consider “fine-tuning” to be evidence for God (and indeed, only 7% of physicists in the National Academy of Sciences say they believe in God: http://bit.ly/PQPm9).

    Firstly, the supposed fine-tuning is not a recognized scientific fact, but a claim made by religious apologists. The truth is that we have little idea what the preconditions for intelligent life are (we only observe one example of life evolving!), and little basis for concluding that intelligent life would not be possible if the fundamental constants were quite different than they are. In addition, we do not know how to define a probability metric over the possible value of fundamental constants of physics, and we don’t know whether a deeper theory would imply that they must have the value that they do for purely physical reasons. What we do know is that the physical laws are such that we exist – we do not know how to appropriately define a landscape of physical laws and we do not know whether intelligent life is a rarity in that landscape or something commonplace.

    Secondly, those physicists who endorse a “multiverse” (something of a misnomer since it’s really just a big universe with different laws in different places…) do NOT do so because they believe it is the best explanation for a universe supposedly fine-tuned for intelligent life. The multiverse is a *prediction* which comes out of physical theories with an entirely different motivation. In particular, the multiverse is a prediction which emerges if we combine inflationary cosmology and string theory.

    So the situation is not, “The universe is fine-tuned. Religious people say God did it. Atheists assume God can’t exist so they are forced to postulate an infinite or very large number of universes so improbable things can happen by chance.” The situation is: “We don’t know if the universe is fine-tuned for life or not and we have no idea if the emergence of intelligent life is probable or improbable. We do have some independent reason to believe in a multiverse (although it is far from an established fact), and if further evidence confirms this hypothesis, then anthropic reasoning would be a well-substantiated explanation for any apparent fine-tuning we might discover.” Given *independent* evidence for the multiverse, any fine-tuning would be no more mysterious than the fact that you were born on Earth rather than on the moon. “God did it” would NOT be on equal footing.

    If the supposed fine-tuning of the physical universe provided a good reason for belief in God, isn’t it curious that the physicists who understand this fine-tuning best skew in exactly the opposite direction relative to the general public? I agree that there could be selection here – perhaps the smartest physicists in the world are predisposed to be atheists for reasons other than their knowledge of the physical world. Ideally we would want an experiment where we tested the impact of acquiring knowledge of physics on religious beliefs (perhaps an RD based on admission to grad school?). But I’d be willing to bet that the fact that 7% of physicists disbelieve in God relative to the 92% of the general populace is not wholly due to selection. Cosmological arguments might fool laymen, but they are completely unpersuasive to those best placed to understand them, so acquiring expertise in cosmology makes people less likely to believe in God.

    1. Brigham says:

      I think you and I might be making the same point, Jason, but I don’t think it really matters what most physicists’ motivation is for believing a “multiverse” to be plausible (and I didn’t use “believe” by accident–as far as I can tell there are no observable implications of a “multiverse” so don’t hold your breath for “independent evidence”) or even if it’s “settled science” that the universe is finely tuned. The main point is this: anybody who believes a multiverse with different realizations of physical constants to be plausible will not have their priors about God moved at all if presented with data that the universe were really finely tuned for life as we know it (see my comment #6 above). This is true no matter what your prior is about God. I think this is what you meant by “cosmological arguments are completely unpersuasive.” However, someone who does not believe a “multiverse” to be plausible may have their prior moved towards God existing when presented with evidence of a finely tuned universe.

      As far as the causal effect of acquiring knowledge of physics on religious belief, it sounds like you have some way of bounding selection effects–if so, I want in! But that aside, it wouldn’t surprise me at all if acquiring physics knowledge had a net negative effect on believing in God, but that would hardly be damning (no pun intended). People believe in God for a variety of reasons–superstition, because they were raised that way, because they have had personal experiences that have convinced them, etc. So even if studying physics only switched the beliefs of, say, those who believed because of superstition, and did nothing else, then it would have a negative effect. If your only point was “a lot of people believe in God for reasons that would be found invalid if they only knew more physics” then you’re right.

      In the end, I think you and my friend Roberto are right (and this is the main point of the original post)–cosmological arguments are not likely to make believers out of atheists. At most they will strengthen the faith of someone who already believes. The real question is where does that initial belief come from, and at least for me it doesn’t come from observations about a finely tuned (or otherwise) universe, but from personal experiences with prayer and other settings that try as I might I can only attribute to a loving God out there somewhere. Is this belief at least consistent with a rational view of the universe? 7% of physicists at NAS are evidence that it is!

    2. Teppo says:

      Jason
      _
      You make very interesting comments and I think your reasoning is sound! However, I disagree with your starting point and hence with the final conclusions. Just to warn other possible readers, the text below is quite geeky! I think it should be understandable to many but I make no guarantees about how interesting you will find it :). Jason, it seems that you have a strong background in physics but I will include some wikipedia links for other readers.
      _
      You state that the fine-tuning of the universe is not a recognized scientific fact. I work daily with cosmologists and particle physicists at a recognized research university. I have no exact data but most of us take fine-tuning to be an established fact of fundamental physics. Granted, what we mean by fine-tuning is a bit more general. But I would argue that the fine-tuning that allows the formation of intelligent life is equally or more difficult to explain than the variety that we usually talk about. In our weekly seminars, a large fraction of the speakers motivate their work on things like supersymmetry by stating that it explains the fine-tuning of some natural constants. Hence, fine-tuning is considered to be enough of a problem to motivate top researchers around the country to spend months and years on building better models of particle physics and cosmology.
      _
      If I understood you correctly, a big part of your argument is that we do not know all the possible mechanism for intelligent life to evolve. I agree with you that there might be a myriad of ways for this to happen. However, there are some preconditions that are very robust and independent of the details. Not surprisingly, they do not deal with biology, which is much more complicated and less well understood than fundamental physics. Ignoring such details, it is clear that any kind of life will depend on very complicated structures. They need not be DNA and cells but there must be some physical objects that do more than bounce back and forth. We must have stable and complex building blocks for life, such as atoms and molecules.
      _
      The need for complex structure demands fine-tuning in at least two ways. The easier of the two has to do with Einstein’s theory of general relativity, which describes gravity. As far as we know, there is nothing stopping the universe from having a vastly different composition in terms of dark energy, dark matter, and ordinary matter (see http://en.wikipedia.org/wiki/Dark_matter for some introduction). However, most combinations of those ingredients would give a universe that was either too dense or too tenuous. A too dense universe would collapse into a massive black hole immediately after the big bang. A too tenuous cosmos would always expand as a gaseous cloud and never clump enough to form galaxies, stars, and planets. If we gave all values of initial densities an equal probability, the fact that galaxies did form points to an immense amount of fine tuning.
      _
      The second instance of fine-tuning is a bit more nuanced. Gravity is a very simple force compared to the standard model of particle physics that includes electricity and nuclear forces (see http://en.wikipedia.org/wiki/Standard_Model). The standard model would be a fine theory with most combinations of masses and electric charges for the various particles. However, if the charges or masses were too small, the forces that bind atoms together would not be strong enough to create stable molecules and cells. If we go in the other direction, namely much stronger binding forces, I agree with your point that we do not know exactly how things would go since modern physics is not very good at dealing with strongly interacting systems. However, the most likely outcome would be something like a neutron star – a dense clump of tightly bound matter with almost no interesting structure and definitely no potential for formation of life.
      _
      Since fine-tuning is a well-established part of fundamental physics, there are also lots of attempts to explain it. The most common is the anthropic principle (http://en.wikipedia.org/wiki/Anthropic_principle) which states that the universe is friendly to intelligent life because otherwise we would not be here to observe it. If this seems like circular logic to you, you are not the only one – many great scientists have opposed this line of reasoning passionately. However, if you postulate a multiverse where all the possible values for fundamental constants are realized infinitely many times, the anthropic principle makes a lot more sense. As you say, the concept of a multiverse was not developed to solve this problem – it is a natural prediction of inflation and string theory. However, I would argue that many people like the idea because it gives more credibility to anthropic reasoning. It is possible that this is how our universe really is but currently there seems to be little hope of ever testing the claim. We are confined to our own universe with no apparent way to interact with the rest of the postulated multiverse. There could of course be a way around this but I don’t know of even a highly speculative way of observing the multiverse. It is interesting that a large part of the physics community has embraced a solution that requires so much faith to deal with the problem of fine-tuning. To me it tells that there is something quite mysterious to be explained!
      _
      In closing, I agree with Brigham that cosmological arguments will not and should not convince people of the existence of God. I know that God lives because He has answered my prayers and influenced my life in ways that were not confided to just feelings in my heart. However, I would be happy if the fascinating problem of fine-tuning puzzled people enough to help cast doubt on their doubts and encouraged them to seek a personal relationship with God. I know from personal experience that such a relationship with a loving Heavenly Father will be a fulfilling and joyous foundation on which to build the rest of our lives.

  8. Anonymous says:

    I certainly agree with you that there are worse things parents can do than raise a child in a false religion, and my argument is clearly a stronger critique of religions that assign negative consequences to false beliefs. I won’t pretend to know enough to say whether your religion does this or not.

    As a practical matter, the fact that most people’s beliefs are determined in large part by accidents of birth makes it difficult for me to choose which religion to follow. The supernatural claims of every major religion seem equally implausible to me, and no one seems to have an argument for why theirs is preferable — instead they just stick with whatever they were brought up with, and for whatever reason I do not accept my parents’ religion. Most religious people tell me that I will know the truth because it will provoke in me some kind of subjective euphoria, but they all describe very similar euphoric experiences despite having radically different belief systems.

    I would also dispute your claim that “God has provided a way for us all to be able to fulfill our purpose and find true happiness.” Many people lead lives that do not turn out to be happy for one reason or another, and I would hesitate to say that this is because they’ve all turned away from some path that God has laid for them. You might argue that everyone who lives a good life will be happy in the afterlife, but of course as a non-believer I don’t see much reason to think this is true.

    I apologize if I’ve taken this thread off track; don’t worry about responding.

  9. Anonymous says:

    I think Bayesian learning is a really useful framework for thinking about religious belief. It can also be applied to choice of religion (rather than just belief/non-belief).

    It’s an empirical fact that most people come to believe the tenants of the dominant religion in the society in which they are born. It is unlikely that you would have the religion that you do if you’d been born in India, for example. This says to me that the choice of religion is in general dominated by the prior — beliefs in a particular religious system are instilled in people when they are young, and these priors are sufficiently strong that there is little convergence in belief across people who start with different priors as they age, become more rational, and gather more evidence.

    My question is this: Why would a benevolent, fair god create a world in which most people’s religious beliefs are determined by accidents of birth? Couldn’t he have designed us to believe less strongly in whatever priors our parents give us? If only one religion is actually true, doesn’t it seem hopelessly unfair that many people are unlucky enough to have strong false priors instilled in them before they can think, while others get the truth? To put it another way, I think the probability of observing a world in which most people unreflectively believe in their parents’ religion is pretty low if there is actually a benevolent and fair god.

    1. Brigham says:

      I think there are much worse things parents can do to their kids than raise them in a religion that doesn’t happen to be the one that has the most complete truth. I am sure I would agree with you if I believed that being raised in some other religion somehow resulted in damnation or not getting to go to heaven, but I don’t. From God’s point of view, we are able to fulfill our purpose in this life whatever circumstances we’re born into, and we all have potential to find true happiness, whatever beliefs happened to be instilled in us. Being raised in different religions is just one of the countless ways each of our individual circumstances differ, for better and for worse, but the fact that despite the vast range of human experience, God has provided a way for us all to be able to fulfill our purpose and find true happiness is for me evidence *for* his benevolence and fairness, not against.

  10. Mark Pickering says:

    “True, we may start out with different prior probabilities that are functions of preferences, choices, upbringing, etc., but once we start observing data and updating our beliefs, as we incorporate more and more evidence into our beliefs, the importance of the initial prior belief fades away.”

    I don’t think that’s true. Any evidence can be accounted for by any theory, and “updating” our beliefs in response to it is no straightforward matter. How or whether new data should lead one to revise one’s beliefs depends on one’s prior beliefs, as well. There can be no evidence for the standards one should use for interpreting evidence.

  11. This discussion is so far above my head, I hesitate making any response lest I reveal my complete ignorance… But I will just say that as much as I love the idea of being able somehow to reduce faith to a mathematical formula, something quantifiable and universally replicable, I’m not sure it will ever be that systematic, nor do I think it should be. Faith as the SUBSTANCE of things hoped for and EVIDENCE of things not seen, notwithstanding, faith will never be an intellectual pursuit or a matter of gathering data. It’s that initial predisposition to either believe or disbelieve that is the core of faith, and that just is not an intellectual decision, it doesn’t seem to me. It seems to me that is “desire,” something seated in the heart or soul rather than the mind. That is what makes faith something beyond data-gathering, but instead an offering, a gesture of love and devotion, an act of worship, if you will. Still, I loved this discussion! For the record, my favorite theorem is Heisenberg’s Principle of Uncertainty :) I would LOVE to see someone explore the spiritual ramifications of that one!

  12. LH says:

    Very interesting article — although I think there’s a slight gap in the reasoning, specifically when you say, “the higher somebody’s prior probability that God exists, the higher the probability they will assign to God existing even after observing the data, and vice versa”, you’re forgetting that Pr(data given God exists) is related in some unpredictable way to Pr(God exists). If your “belief” Pr(God exists) “increases” by a factor of 2, Pr(data given God exists) could actually decrease by a factor of more than 2, depending on the exact relationship between that belief and the data (i.e. depending on how it looks on a Venn diagram). Saying “belief” for Pr(God exists) could be a little misleading, and saying the prior “increases” or “decreases” feels totally screwy in a way that I can’t put my finger on. In reality in this case, you really have to use the Laplacian prior of Pr(God exists) = 0.5 no matter whether you’re an atheist or a believer, since, in the absence of further information that can provide strong objective reasons to believe either way, that is the “only reasonable” starting assumption.

    However, overall, it’s a nice fairly simple illustrative way of explaining why atheists usually choose not to believe no matter what evidence is presented to them, and vice versa for people who have already chosen to believe something.

    1. Brigham says:

      Yes, LH, there are a few subtleties in the mechanics of the “updating” formula that I glossed over (remember, I’m trying to minimize nerdiness!), and I think you’re hitting on some of them. But it’s not quite as arbitrary and unpredictable as you might think. You can think of Bayes’ rule as consisting of an objective part multiplying a subjective part (which is the prior probability of God existing). The objective part is: Pr(data | God exists) / Pr(data). By objective, I mean reasonable people should be able to agree on what this number should be, or at least agree on what assumptions are necessary to pin down the number, independent of what each person’s subjective prior probability is. In Bayesian parlance, this number is called the “proper likelihood.” So in other words, if somebody has a prior that is, say 2x somebody else’s prior, they should still be using the same “likelihood” to update their priors, since typically the likelihood does not depend on the prior distribution.

      As far as feeling totally screwy, my hunch is that you are a deep-rooted frequentist chafing at the Bayesian notion of probability! (and I completely sympathize) To a frequentist, probability means the fraction of times some event would occur if a process were repeated many times. In this sense, thinking of your degree of belief in God as a probability is silly, since we’re not repeating (even hypothetically) whatever process generated reality a bunch of times, and seeing how often God shows up. But to a Bayesian, probability just denotes the degree of belief in some hypothesis, and thus can be subjective. Bayesian analysis will often start with a laplacian prior as some kind of agnostic benchmark, but it’s perfectly valid to start with any prior as long as it’s greater than zero and less than one.

  13. Teppo says:

    Brigham, this is awesome! I *love* the geekiness of it all!

    Coming from a physics background, I wanted to contribute my two cents to the debate on multiple universes. Yes, Stephen Hawking believes in it along with many other respected physicist. At the same time, many respected physicist find it to be utter nonsense. For me it is a lot more distasteful to explain the world with a hypothesis with no-experimental evidence and perhaps no possibility to test it (multiverse) rather than to use a hypothesis that is very subjective but testable by many (subjective) experiments (God).

    I totally agree with Roberto that believing in God does nothing to explain the existence of the universe. However, it can do a lot to explain our experiences in the universe. If one is willing to trust subjective experiences about the world (and everybody is as part of their everyday actions), then interactions with God can connect Him very strongly with objective reality.

    1. Brigham says:

      I like the distinction between a the untestable multiverse theory and the subjectively testable hypothesis that God exists. Along the same lines, our belief in a creator, together with our belief that we will eventually be in his plane of existence isn’t just inserting another turtle in the stack of turtles holding up the world, because it implies that at some point we *will* be able to understand the beginning from the end. We acknowledge that there are many things that we simply will not be able to comprehend in this life (does science acknowledge this?) but at least we put a definite timetable on when we will be able to understand it!

  14. Barbara says:

    Pretty Nerdy, Brigham, but well put! Thanks for this post!

  15. Tim says:

    This is a great piece Brigham. I feel like it ties together all of my posts under a simple mathematical framework. Bayes’ Theorem always has been my favorite theorem, now I know why.

    1. Brigham says:

      Thanks, Tim. Bayes’ theorem is one of my favorites, too, although i dont think i could ever be a true bayesian. Btw I think you have a .com instead of a .org on your name URL!

  16. Natalie says:

    Roberto, you crack me up. That’s all. Love reading your thoughts!

    1. Roberto says:

      I’ll take that as a compliment.

      1. Natalie says:

        Yes yes, you should. It was the “Big hug!” that got me.

  17. Roberto says:

    Off the mark for the exercise you propose, yes. But only because your exercise presuposes a marginal belief in God as a seed for the exercise. But even then, I do not see how we can discard Hawkins’ response. If the variables had not had those values in this universe, there would be no Brigham and Roberto to ponder the million dollar question. For me this question is fundamental, as much so as the question of the origin of anything. Consider the two following arguments.

    1) Where does the universe come from? God made it. Who made God? Nobody: he has always existed. He’s eternal, self-sufficient.

    2) Where does the universe come from? It has always existed.

    Explaining the universe in terms of a God just puts one more turtle below the previous turtle. And who is supporting that turtle? Etc.

    In my view, both religion and science are equally inadequate at answering that question. All things considered, and lacking any experience to the contrary, I remain a skeptic. Big hug!

    1. Brigham says:

      You are right on, Roberto–my example isn’t for the anthropic principle. In fact, Bayes’ theorem vindicates the anthropic principle, because it says that the probability of observing a finely tuned universe universe–given that anybody is there to observe anything–is one! So the updating formula says divide the probability of the finely tuned universe given that there is a God (and I would say this is one) by the marginal probability of a finely tuned universe (which by the anthropic principle is one), so those cancel, and you’re left only with the original prior probability of God existing. In other words, if you buy the premises of the anthropic principle, then observing a finely tuned universe should not change your initial belief at all about whether or not God exists. And this is exactly Hawking’s point, I believe.

  18. Roberto says:

    So it seems we have circled back again to the same premise as before: experience has, at least for a given person, the last word. You have experienced ‘God’ (excuse the quotation marks), in the sense that you have experienced something that you identify as being God, and this experience, in turn, has made it palatable for you to: (a) include the assumption / knowledge / previous belief in god as part of your little bayesian experiment, and (b) to give more credence to a ‘finetuning god’ belief than to a ‘all possible universes’ speculation.

    Me, on the other hand, not having had the joy of religious experience (e.g. one that made me fee there was ‘god’ there) am neither inclined to start form the assumption of a finetuner god nor to give it more weight than to what Mr Hawkins proposes.

    So, I think this is – at least as far as I can see – a matter of faith and personal experience. Would you agree?

    1. Brigham says:

      Haha, yes, I think you are right. Although experiences are really just another form of data to feed our Bayesian updating process, and thus are subject to different interpretations given different priors. Let’s put these personal experiences in a Bayesian framework: let’s say the “data” is an intense emotional feeling during a church service. The believer would interpret this feeling as God communicating the reality of his existence and love, and therefore as additional evidence for God’s existence, and the updated belief would be stronger. The skeptic, however, would counter that emotional responses of that type occur in a variety of settings and for a variety of reasons, most of which have nothing to do with God, and more to do with priming, upbringing, etc, and should not be interpreted as evidence. But in the Bayesian framework the skeptic’s argument is completely off the mark: what matters for updating beliefs is Pr(data | God exists), while the skeptic is talking about Pr(data | God doesn’t exist), which has no direct role in the updating formula.

  19. Roberto says:

    So, let’s see. With the disclaimer that I’m not a statistician, I want to add a counter view to Brigham’s (and Richard Price’s, who made it in 1763).

    The way I interpret Bayes’s theorem on conditional probability, in simple terms is that, when you are given additional information, you can make a better guess. Say, in the Monty Hall problem, you are given the additional information that a given door does not have the price. So you can better guess which door does have the prize.

    In the case of a universe with ‘a finely tuned set of variables’, I’m not sure we can apply Bayes’s theorem to argue that our living in such a universe makes it more likely that god exists, because we do not have the prior probabilities of there being a god and of this set of variables occurring separately, and the prior probabilities are needed to use Bayes’s theorem.

    Furthermore, I would argue along the same lines as Stephen Hawkins, when he says that if these variables did not have the exact values for life, we would not be here asking the question. I find equally plausible that there are infinite universes, many with other values for these variables, or other variables altogether, in most of which there are no intelligent lifeforms asking the question.

    1. Brigham says:

      Aha, you have hit on the sometimes acrimonious debate between the “frequentists” and the “Bayesians” about what probability really means. You seem to come down on the frequentist side, in which probability means the fraction of times some event would occur given many repetitions. Bayesians, on the other hand, use probability to mean the degree of belief in a hypothesis. In this sense, a prior probability can be purely subjective, which is the sense I’m using. For the record, when wearing my econometrician hat, I prefer the frequentist interpretation.

      I have read and appreciate Hawking’s “anthropic principle” and it has kept me up more than a few nights, but in the end I find the notion of a continuum of potential universes with different realizations of physical constants much more far-fetched than the existence of God, with whom I have had (what I believe are) real personal experiences!

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