A Bayesian Analysis of God's Existence

God loves me this I know, for the Bayesian analysis tells me so

1 Introduction

Bayes theorem is the mathematical theorem used to revise probabilities in light of new evidence. For example, suppose that you think that some hypothesis has a 50% probability. Then you learn some fact that is twice as likely if the hypothesis is true as if it’s false. Now you should think that the theory has 2/3 probability.

The simplest version of Bayes theorem uses odds ratios. Odds ratios tell you the ratio of the likelihood of one theory being true to the likelihood of it being false. For instance, a 2:1 odds ratio tells you that a theory is twice as likely to be true as false; a 3:2 odds ratio tells you that a theory has a 60% likelihood. If you learn new evidence, to figure out the probability of the theory after accounting for the evidence, you simply multiply the odds ratios of the new evidence. If a theory starts out with 60% probability (3:2 odds ratio) and then you learn something twice as likely if it’s true than false (2:1 odds ratio), then you multiply 3:2 by 2:1, to get 6:2 or 3:1.

Bayes theorem is the mathematically-proven way of changing probabilities in light of new evidence. It can also be used to decide upon beliefs. For instance, if you currently think something has a 50% chance of being true, and then learn some fact that’s three times likelier if it’s true than if it is false, you should then think there’s a 75% chance it’s true.

I thought I’d apply the methodology to the existence of God.

Now, this is hard. The numbers are a bit made up at times. But reasoning with made up numbers is often better than reasoning with no numbers at all. Human intuition isn’t good at figuring out probability, so it can often be improved by Bayesian analysis. Even though it often won’t be clear whether the odds ratio is 5:1 or 10:1, you can usually have a rough order of magnitude estimate.

So while you shouldn’t take the results of a Bayesian analysis too literally, and you should feel free to slot in your own numbers, using Bayes’ theorem can be a helpful exercise for evaluating hypotheses when there’s lots of evidence on both sides. For this reason, I thought I’d do this with theism. I find that when I do this exercise, almost no matter what numbers I use, theism seems to win out by quite a bit. Even if one uses pretty conservative numbers, theism ends up more probable.

So…without further ado…here is a Bayesian analysis of God’s existence. Much of the theistic evidence I’ll talk about has already been listed here. Also, it’s hard to compare theism to various other exotic hypotheses like axiarchism and natural teleology, so just for simplicity I’ll mostly be comparing theism to default atheism which doesn’t include these exotic hypotheses that are kind of like theism. I also use the term God to refer to an all-knowing, all-powerful, all-good being.

Lastly, given the sheer number of considerations, I won’t be able to assess them much in detail. I’ll mostly just link to other places I’ve discussed them, for this reason. If you want a fuller explanation of the arguments I give here, see this essay.

2 Prior

The prior probability of a hypothesis is how likely the hypothesis is before considering any facts about the world. So in deciding upon the prior you should ask: if you were in a dark room with no access to the world, and hadn’t considered your own existence (perhaps you were tripping on a powerful drug that eliminates your sense of self), how likely would you think it is that an omnipotent, omniscient, and omnibenevolent God exists.

I think a pretty conservative odds ratio is 1:1,000, meaning theism starts with a prior of 1/1,001.

Why this number? Well first of all, on some theories, theism is the very simplest way that the world could be. God might be the very simplest entity: a mind or agent totally without limit. Very simple theories tend to take up a very large portion of probability space; the theory that the universe has uniform laws that hold evenly throughout the entire cosmos beats all the theories of how the laws might not be uniform (even though there are infinite ways the laws could not be uniform).

There are also various accounts on which God grounds abstract objects like mathematical facts. I don’t find these super plausible, but I don’t think that they can be ruled out with certainty. Thus, just as one’s prior in nominalism and Platonism should be non-trivial, one’s prior in theism should be non-trivial if it’s considered as a new way to ground abstract objects.

Lastly, there are various ontological arguments that purport to explain why God’s existence is necessary as a matter of logic. I don’t think any of these work, but I’m not highly certain of this fact. Conditional on them working, God must exist, and this is knowable a priori. So one’s prior in God existing should be non-zero.

All these make a 1/1,001 prior in God existing look pretty reasonable. Maybe something like one in ten thousand is more reasonable or 1%, but I don’t think one in a billion is reasonable. You shouldn’t be nearly certain in each of the following:

1. God doesn’t ground abstract objects.

2. God isn’t very simple.

3. All ontological arguments fail.

So for this reason, I’m sticking with 1:1,000.

3 Physical universe + laws + nomological harmony

How likely is it that there would be a physical universe given theism vs atheism? I think a physical universe is likelier given theism than atheism for a few reasons:

1. A physical world is contingent—it could have been otherwise! Contingent stuff existing at all is somewhat puzzling on atheism. Absent a necessary being explaining the existence of contingency, contingent stuff existing at all is surprising.

2. It would be much simpler for nothing to exist at all. On atheism, it’s not super expected that there would be a physical universe.

Now admittedly it’s not guaranteed on theism either—God could have made a world of merely mental entities or created nothing other than himself. But the odds seem a bit lower on atheism. I conservatively give this a 2:1 odds ratio in favor of theism, but I think anywhere between 1:10 and 10:1 wouldn’t be crazy.

One common objection is that the probability of a physical universe is 1: if there was no physical universe, we wouldn’t be here. But this is confused. The probabilities in question are conditional—how likely a physical universe is given theism vs. given atheism. Of course we know there is a physical universe, but we’re trying to see how likely that is given theism vs given atheism.

As an analogy, if you see someone get ten royal flushes in a row in poker, you now know that they got ten royal flushes. But still, it’s evidence that they’re cheating, because on the mere hypothesis they didn’t cheat, the odds they’d get ten royal flushes in a row are low.

Now, as for the notion that if we hadn’t existed we wouldn’t have been around to calculate probabilities, that’s true but irrelevant. Your existence gives you evidence that your parents didn’t use extremely effective contraception. This is true even though if they had, probably you wouldn’t have existed. (For more detail, see the section in this essay labeled “Anthropic principle.”) Whether some evidence is necessary for your existence is totally irrelevant to its conditional probability.

A next bit of evidence: the physical stuff that exists in the universe does stuff. Once again, this seems pretty surprising. It would be a lot simpler for all particles to sit dormant and effete, never doing anything. The fact that laws exist at all seems a lot likelier given theism than atheism.

Taking into account the existence of a physical universe, the odds of such a thing are very near one. After all, God has no reason to create particles that don’t do anything—the reason to make particles is to create agents and perhaps other things of value. So if God is going to make a physical universe, it’s almost guaranteed that he’d go through whichever steps are needed to produce agents. In contrasts, on naturalism the odds are a lot lower, because laws are brute and unexplained.

I give this an 8:1 update in favor of theism. This seems pretty conservative, but maybe if you’re really conservative you could go a bit lower and nearer to 2:1.

A third consideration is nomological harmony. Laws apply to the things that exist. This is surprising on each of the views of what laws are:

1. The first view of laws is called a governing view. This says that laws exist over and above the physical things and make them behave in certain ways. Laws function almost as forces that make things behave in some way—the law of gravity makes things with masses attract. But on this view, it’s surprising that the laws apply to the things that exist. There could be laws governing mass but nothing with mass—or laws that say “all Xs do A,” but no Xs to do A. As an analogy, finding that the laws are applicable is a bit like finding a random collection of game pieces, a rulebook next to them, and noticing that the rulebook lines up with the game pieces. Clear evidence of design!

2. The second view of laws is called the powers view. This says that laws simply describe the powers things have—the things they can do. The law of gravity just means everything has the power to pull on everything else proportional to its quantity of mass. But the powers things have are only operant under certain conditions—for instance, things might have the power to attract Xs, but if there are no Xs, they’ll simply lie dormant. Thus, it’s very surprising that the powers things have line up with the initial conditions that exist.

3. The third view of powers is called Humeanism. This holds that laws are just abbreviated descriptions of what things do. To say there is a law of gravity is simply to say that objects with mass tend to attract. However, things only behave in various ways under certain conditions, so on this view, it’s surprising that the conditions are right for the things in the universe to behave as they do.

Now, I’m pretty sure this works on powers views or governing views. On Humeanism, it’s not as clear: philosophy wunderkind Joe Schmid has tried to talk me out of this, but I confess I find it hard to make sense of why it doesn’t work. I’m also pretty sure Humeanism is wrong, largely because it implies that if there are two different laws that both equally describe reality and have equal simplicity, there’s no fact of the matter about which is right. So if law A says “particle accelerators make atoms zig right,” and law B says “particle accelerators make atoms zig right,” then so long as no one builds a particle accelerator, there really will be no fact of the matter about what would have happened if someone had built a particle accelerator. Nuts! Humeanism also has other problems.

For this reason, I give the argument a conservative Bayes factor of 10:1. I think Humeanism is probably wrong and the argument still works if Humeanism is right, but I’m not super certain in either of those, so hard to go vastly above 10:1.

4 Laws and constants

Even once laws exist, most of them would produce nothing of value. The simplest laws don’t produce any value—it would be vastly simpler for laws to simply result in every particle spinning in a circle or moving in a line. Our laws are in an unlikely goldilocks zone between simplicity and complexity; they give rise to complex structures without devolving to random chaos. This is very surprising especially when you realize that for such a thing to happen, they need to exhibit the following features:

1. Even if things are broadly in the business of interacting with each other, there are certain conditions that have to apply. But most ways reality could be, they wouldn’t apply. Things only interact under certain conditions, but that means there are an infinite number of conditions under which they don’t apply. For example, you could have all the physical stuff really far apart so that it doesn’t interact. And that doesn’t seem much less intrinsically probable—it doesn’t compromise simplicity. Similarly, you could have there be other stuff that interferes with the interactions—potentially an infinite number of things could do that.

2. Even if the stuff that exists is broadly in the business of interacting, it has to be in the same spacetime region. That’s not guaranteed—you could have things occur in their own isolated spacetimes—distinct from each other the way David Lewis’s worlds are distinct. Them being in the same world is no simpler and so it’s pretty unlikely.

3. For stuff to interact in any way which brings about anything interesting there has to be time. Otherwise, everything would be static. Furthermore, things have to have interactions that play out across time.

4. The stuff has to be all of a similar kind to interact. You might have, for instance, disembodied minds, weird mathematical laws that do stuff, and then also have physical composites. One of the worries of the many worlds interpretation is that something as abstruse and mathematical as a wave function can’t explain the real physical stuff of our experience—you could have a world like that where there are abstruse mathematical laws but they don’t build up to anything concrete.

5. The stuff all has to be capable of interacting with other things. But surely it’s simpler for it to just follow its own plan—for instance, particles that move around in a circle whatever the other particles do.

6. Then, even when the stuff interacts, it has to be able to bind together in stable ways. If particles just bounced off each other but didn’t form more complicated composites, nothing interesting would happen.

7. For the stuff to interact, there also has to be space. Now, space might be necessary, but if not, it’s another thing needed for reality to hang together.

And this is just scratching the surface of the surprisingness. I think this is good for a Bayes factor of over 1,000, but I’ll be conservative and give it a Bayes factor of 100.

Then, on top of this, the constants of the universe are finely-tuned. Many fall in an infinitesimal range needed to give rise to any complex structures. If the cosmological constant didn’t fall in a tiny range, roughly 1/10^120 of the values it could have taken on, either the universe would have immediately collapsed or fallen apart. Conservatively I give this another Bayes factor of 100:1 in favor of theism.

5 Consciousness

One of the most surprising facts about our world is: experience exists. When you clump together matter in various ways, it produces consciousness. This isn’t surprising given theism—consciousness is the source of most value, so conditional on a perfect God bothering to make a finely-tuned universe, it would probably produce conscious minds

But it is surprising on naturalism. A naturalist would not guess that random arrangements of matter would give rise to mind. Even if physicalism is true, and physical stuff giving rise to consciousness is necessary, it’s still surprising. Merely stipulating that a thing which you wouldn’t expect in advance is necessary doesn’t blunt the improbability. If someone gets ten royal flushes in a row, it wouldn’t do for them to say “I’m a determinist, so I think my lucky streak was necessary.” It’s surprising that is necessary rather than something else. Bayesian reasoning is about epistemic probability—how unlikely you’d find something to be—rather than objective probability.

If dualism is true, things are even more surprising. Why would there be laws that give rise to minds, of a character utterly unlike all the things in the universe? Why would there be random extra laws, that apply at the level of brains, that make them produce experience?

Then, even if there are laws of the form “such and such physical arrangements give rise to consciousness,” it’s surprising that such and such physical arrangements ever arise. It could be that brains produce consciousness but there simply aren’t any brains that ever get created!

I conservatively give this a Bayes factor of 200:1. I think the fact that some arrangements of matter give rise to consciousness is good for a Bayes factor of at least 100 (a theist would be at least a hundred times less surprised by that development than a naturalist). The fact those arrangements of matter exist is good for a Bayes factor of at least 2.

6 Psychophysical harmony

The argument from psychophysical harmony is one of the best arguments for God’s existence. It begins by noting that there’s a harmonious pairing between the mental and the physical. Often, people have experiences that feel bad. When this happens, they behave to avoid those experiences. When you put your hand in hot water and feel pain, you pull your hand away. Similarly, we often talk about our mental states, and our talk is accurate. For example, when people say “I’m sad,” or “I see a brown desk,” most of the time what they say is true.

What happens physically lines up with what’s going on mentally. The way we talk and behave matches up harmoniously with what we’re experiencing. When you have a desire for your limbs to move in a particular way, they move in that way.

This doesn’t sound that surprising, but it is when one takes the time to really think about it. First, however, we’ll have to get clear on the notion of psychophysical laws. The psychophysical laws simply govern the relationship between the mental and the physical. They determine which mental states cause which physical states and which physical states cause which mental states. Specifically, our psychophysical laws result in a few kinds of harmony:

1. Normative harmony: when we have experiences that make it fitting for us to behave in certain ways, we often behave in those ways. For example, pain makes it worth pulling your hand away, and when you’re in pain, you pull your hand away.

2. Semantic harmony: our talk about consciousness lines up with what’s going on consciously. For instance, when people say “I have a blue square in my visual field,” they usually do.

3. Behavioral harmony: our behavior lines up with our conscious desires. For instance, when you want to move your limbs, your limbs move. When you want to write a poem, tilt your head, or do anything else, your body cooperates.

None of this sounds strange. It’s all so natural as to be banal. But when you seriously reflect on what’s involved it starts to look really weird. Most ways the mental and the physical could have paired would have produced nothing of the sort. For instance, you could have had the following pairings:

1. We could behaviorally disposed to do the opposite of what we want. So, for instance, you’d bang your finger, feel pleasure, think “I want more of that,” and then that would cause you to pull your hand away. Note: it’s not that you’d want to pull your hand away, but instead that you’d be behaviorally disposed to do the exact opposite of what you want.

2. Every creature with a brain processing information could simply have the experience of seeing a red wall. This would be a very simple law—far simpler than our rich and integrated set of mental-physical pairings—and yet wouldn’t produce harmony, or anything of significant value. In such a world, creatures would evolve to have all sorts of complex behavioral dispositions, but would have a rudimentary mental life.

3. There could be some other set of causal links between the mental and the physical. Thus, perhaps the experience that would precede you moving your right arm would be having an image of a yellow star. Evolution would select for your brain being disposed to see a yellow star when it would be beneficial to move your arm.

If any of these were the laws, very different sorts of creatures would evolve. They would, in many cases, be behaviorally like us, but would have radical disharmony. Thus, the fact that there’s harmony is good evidence for theism.

I’ve only briefly begun to scratch the surface.

There are plenty of helpful resources for getting the argument. There’s a law of nature that everyone who finds the argument persuasive gets so irritated by people misunderstanding it that they write a brief explainer on it. My friend Amos has a several-part series on it; my other friend Apologetics Squared (not the name his parents gave him) has an incredible series on YouTube of 38 very short videos, roughly 1 minute in length, explaining and defending the argument, as well as a pretty short and comprehensive video on it; Dustin Crummett and Brian Cutter have a paper on the argument (I’m starting to think I’m friends with everyone in the world who likes the argument); and I have an article on it.

I conservatively give the argument a Bayes factor of 20:1 (it would be higher but we already have a lot in the background and these things aren’t independent).

7 Miscellaneous

You exist! Out of the uncountably infinite possible people who could exist, you happen to. On atheism, this is very surprising—even if someone exists, it’s very unlikely it would be you. On theism, however, God is likely to make a giant infinity worth of people, which makes your existence vastly likelier. Your existence is likelier if more people exist; God’s existence predicts the existence of vastly more people than atheism, so your existence is very strong evidence for God’s existence.

This is called the anthropic argument. It’s a hard one to get your head around, but I think when you get it, it’s one of the most persuasive theistic arguments. Here’s a brief article where I’ve explained it in more detail and here’s a comprehensive post addressing every objection. I think it’s good for a Bayes factor of at least 100, because SIA (the idea that your existence is likelier if there are more people) is almost certainly true, and a reworked version of the argument works on every view of anthropics. This might be controversial—smaller minds can’t fathom the ways of anthropics or the high Bayes factors therein, but I felt constitutionally incapable of giving the argument a Bayes factor below 100.

Then, there is the fact that we are capable of grasping morality, modality, and various truths about metaphysics. If our brains were purely the result of unguided natural selection, it would be a vast coincidence if we knew anything about morality—if our beliefs bore any relationship to what was morally right. But we do know things about morality; we know, for instance, that it’s wrong to torture babies. This is evidence for theism. And as I explain in the article, even if you don’t believe in objective morality, similar puzzles arise in other domains, like our knowledge of induction, so the argument still has force.

I think this is good for a Bayes factor of 20:1. Even putting aside my in principle arguments for why naturalism wouldn’t produce moral and inductive knowledge, it’s pretty surprising on naturalism that we’re the sorts of creatures that can discover the universe! A priori you wouldn’t expect monkeys in suits to be able to discover the laws of physics.

Another argument good for a sizeable Bayes factor: we’re not in a skeptical scenario. There are many different ways we could be in a skeptical scenario given naturalism. These include:

• One of the many infinite scenarios where probabilities break.

• Us being Boltzmann brains.

• Modal realism.

• Us being brains in vats.

And more. The odds that none would obtain are low on naturalism but high on theism. So I’ll give this a Bayes factor of 5:1.

8 Evil

Here’s a big fact that favors atheism: the world is filled with blind, indifferent natural laws that churn out vast quantities of evil. Animals have been suffering and dying for millions of years. The laws operate with no regard for value. Now, there are some theodicies that can explain this, but it’s still pretty surprising: let’s be generous and give it a Bayes factor of 10,000:1 in favor of atheism.

I think this is pretty generous. The disjunct of all theodicies that explain a broadly indifferent universe is non-trivial. Even if you think evil totally disqualifies theism, you shouldn’t be certain of that judgment, so you shouldn’t give it a Bayes factor much higher than this.

Then, there’s divine hiddenness. I don’t actually think this is good for a very high Bayes factor. Once we have evil in the background, then we’ve already admitted the presence of a world filled with hideous depravity: rape, cancer, torture, murder, disease, and predation. If there is an explanation of all that, it’s not that unlikely that a similar explanation would provide for our blindness to God’s existence. And while it is somewhat unlikely given theism that God would be unknown to so many, it’s also unlikely given atheism that God would be known to so many. Let’s be generous and give hiddenness a Bayes factor of 10:1, even conditioning on evil.

Taking into account the scale of the universe—the fact that it’s mostly inhospitable and empty—maybe at the high end, we’ll get another Bayes factor of 5. I don’t think this is hugely evidentially significant because it’s mostly just pointing to bad things about the universe (the fact it’s barren and we can’t live in most of it) but that’s mostly covered by the problem of evil. Maybe we’ll get another Bayes factor of 2 from religious confusion, but again, this is mostly covered by evil and hiddenness.

There are various other considerations—the biological role of pleasure and pain, for instance, as well as the suffering of moral patients—but these are already covered under evil, and more broadly, they’re covered by an explanation of a pitiless and indifferent world.

I’m somewhat left scratching my head at what the other atheist evidence is supposed to be. It’s mostly just stuff about the world that’s bad. But once you take into account blind and indifferent laws, you can explain those things.

Maybe you can get another Bayes factor of 4 or something from the absence of miracles. But probably this is already covered under indifferent laws, and there are some well-documented miracles. So I’ll generously give this a Bayes factor of 4 in favor of atheism, but I don’t really buy it.

9 Time to calculate

I didn’t include some other arguments:

1. The Kalam because it has negligible weight.

2. The moral argument for the same reason.

3. Arguments for theistic incoherence because those went into the priors.

4. Arguments from biological design because they have no force.

5. Any of Feser’s arguments because they suck.

6. Arguments from widespread religious experience which I actually think have decent force, because I was being generous and think most people don’t find this convincing (still, a world where people go around constantly having powerful religious experiences is surely at least somewhat likelier given theism than atheism).

7. Arguments from souls and libertarian free will, because even though I find them moving, most of you don’t believe in such things.

8. The argument from haecceities because it’s weird enough that even though it might work, it’s not good for any serious Bayes factor.

9. The argument from widespread theistic belief because I counted that as a mitigating factor under divine hiddenness.

But taking into account what I did include we get a Bayes factor of 64000000000000:2000000000. Or, to simplify, 32,000:1 in favor of theism. If you take these numbers completely literally, your credence in atheism would be one in 32,001.

Now, obviously you shouldn’t take these numbers too literally. It’s important to distinguish between confidence levels inside and outside of an argument. If an argument tells you with 99.99% certainty that some view is right, it’s likelier the argument is wrong than .001% odds. The odds I’ve gone wrong somewhere are a lot more than one in 32,000, which are the odds this tells us to assign to theism. And if you don’t buy some of the big theistic arguments, this could pretty easily flip (though you should also not buy the high Bayes factors I’ve given to atheist arguments then either).

But I did think my numbers were pretty conservative. I gave decent weight to atheistic arguments I thought had negative weight and assumed that the problem of evil was vastly more significant than any of the arguments for theism. Even then, theism won by quite a lot.

I think the core probabilistic argument for atheism is this: atheism really only has one very significant piece of evidence. That evidence is the suckiness and indifference of the world. Theism, in contrast, has many converging lines of evidence. So it’s pretty hard for the atheistic evidence to outweigh the many different converging lines of theistic evidence. Just like a single argument wouldn’t be enough to overturn the many converging lines of evidence favoring evolution, a single argument isn’t enough to overturn the many convergent lines of evidence favoring theism.

Now, maybe I should have given the arguments a lower Bayes factor. Perhaps I should think it’s likely enough that I’m wrong about psychophysical harmony and anthropics that it’s not good for the kind of Bayes factor I gave it. But then we’d also have to lower the problem of evil Bayes factor—lots of smart atheistic philosophers don’t think evil has any force. If we did that, and also made our priors less extreme, theism would win by even more.

So even though you shouldn’t take these numbers too literally or have them as your exact credence (my credence in theism is no more than 70%, for instance) it is suggestive that even when assigning conservative probability estimates, Bayesian analysis ends up favoring theism by a lot! This isn’t the only thing that should affect whether you believe in God, but it should at least be one of the factors you consider!

If you disagree with my results, plug in your own numbers! I think you might be shocked by what you find. An atheist friend of mine assigned pretty atheist friendly numbers (I think his prior in theism was one in a billion, for instance) and his credence in theism output by the model was about one in seven. Avoiding theism is surprisingly hard when you do the math!

Atheists often say that they would believe in God if they received a divine revelation. Well they have something almost as good: a Bayesian analysis! Surely this should be enough to convince everyone to love the lord with all their heart, soul, and might! If you would believe the good lord, resplendent in power and glory, if he announced his existence, then you should believe his good reverend—Thomas Bayes—when he tells you that God exists!