The probability that God exists: A Bayesian perspective

by Alan Cohen

Many people are very certain either that God exists or that (s)he does not. This post is based on the premise that, from an empirical perspective, such certainty either way is almost certainly unwarranted, but that a combination of statistical theory and the current state of scientific knowledge can shed substantial light on the probability. After all, there is a rather big difference between an agnostic who thinks there is a 1% chance God exists and one who thinks there is a 99% chance. Obviously, there is no way to calculate a precise probability, but there certainly are ways to establish a likely range. In a nutshell, I will show that, while all the scientific advances in the world cannot disprove God’s existence, the accumulating capacity to explain the world through science substantially diminishes the probability that God exists.

First, let’s lay the groundwork.

What do I mean by “God”?

Or god. Or Gods or gods. Male or female. I want to keep the discussion broad but meaningful, so let’s refer to sentient, supernatural beings more powerful than humans. “Supernatural” is tricky, but let’s say it means outside the framework of modern science. Aliens from another galaxy that evolved through natural selection on their planet? Within the framework of modern science. Zeus, Vishnu, Yahweh, Christ, and Allah? Outside it.

Why sentient? Well, definitions of God such as “the universe in all its grandeur” are perhaps spiritually or therologically interesting, but from an empirical perspective they are indistinguishable from a purely materialistic (i.e., science and nothing else) view of the world, so they don’t give us much to go on.

Are God and science mutually exclusive?


God and science venn

Some definitions of God are clearly incompatible with modern science, but many are not. (For example, imagine a sentient being that created everything we might ever observe as a fun experiment, and then sat back to watch for billions of years.) This is obvious, but it will be a key point later because it means that the sum of the probability that God exists and the probability that science is accurate is greater than one. We therefore cannot take evidence for science as direct evidence against God.


  1. Science has largely succeeded in providing a plausible explanation for most things we observe in our world.
  2. It is meaningful to evaluate the probability a deity exists. It seems reasonable to suppose that it exists or does not, regardless of our evaluation; nonetheless, given our ignorance on the subject it is reasonable for us to try to establish probabilities, in the same way that scientists debated the probability that birds were descendants of dinosaurs (now considered near 100%).
  3. It is meaningful to lump many diverse definitions and conceptualizations of God into a single framework.
  4. I am not missing any other important hidden assumptions in my worldview.

Statistical framework

The core of the approach is Bayesian statstics, based on Bayes Theorem

P(A|B) = \frac{P(A)\, P(B | A)}{P(B)},

dating back to the 18th century. Basically it says that we can calculate the probability of some phenomenon of interest (A) given some knowledge or information about the world (B) based on the combination of the probability of A independent of this knowledge, the probability of the knowledge occurring were A to be true, and the probability of the knowledge occurring regardless of whether or not A is true.

As an intuitive example, suppose I wonder if my wife has been cheating on me given that she has been coming home from work late a lot recently. The things I need to consider are: (1) How likely was it she was cheating anyway, regardless of her lateness, P(A)? For example, have we had problems? Has she cheated before? Is there someone I suspect she was getting closer to? (2) If she were cheating, how likely is it that she would be coming home late, P(B|A)? Maybe she would think coming home late was too obvious, and hide it better to avoid questions. Or maybe that is the only time in her schedule that might be available for an affair. Crucial info to consider. (3) How likely is it she would be coming home from work late anyway, P(B)? Maybe her job has a lot of projects that require her to stay late often, maybe it doesn’t. Once we calculate these three things, we can arrive at a reasonable estimate of what may be going on given her recent lateness.

Traditional frequentist statistics generally ignore P(A). The best explanation of why this can be a bad idea is given intuitively by Randall Munroe in this excellent XKCD comic:frequentists_vs_bayesians

Putting it all together

OK, so let’s put these pieces together. We want to evaluate the probability that God exists, P(A), given what we know about science, B. A simplistic, non-statistical view is that, because God and science are not mutually exclusive, scientific advances do not provide evidence against the existence of God. We will see that this is not true, though the evidence is probabilistic rather than black and white.

Our starting point is someone living before science (say, before Aristotle). Ignoring the problem that the distinction between materialistic and supernatural might be meaningless to such a person, we can suppose (s)he might assign a probability to God existing. There is very little empirical information available to arrive at this probability, so it probably should not be far from 0.5, but it really doesn’t matter. This is P(A), and we don’t know what it is and can’t ever hope to. So let’s rearrange our equation:

P(A|B)/P(A) = P(B|A)/P(B)

The left side now is a measure of how many more times likely God’s existence is, given what we know about science, than it was absent that knowledge. And this is equivelent to how much more likely what we know about science is, were God to exist, compared to how likely it would be with no knowledge of whether God exists.

None of this can be quantified precisely, but we can work through what it means for certain specific scientific findings. For example, consider Planck’s constant. I am not a physicist, but I am told that if it differed by even a fraction of a fraction of a decimal, the universe as we know it could not exist, and there would likely be no life. This leaves four possibilites: (1) We got really, really lucky. But the probability of this is so low that, in this statistical exercise, we must ignore it as essentially impossible. (2) God exists and made the constant just right. (3) There are an essentially infinite number of universes, and we happen to be  in the one with the right constant and are thus the ones asking the question. (4) Planck’s constant is constrained to that value by links with other aspects of the universe in complex ways we are far from understanding. Possibilities (3) and (4) are more “scientific” than (2) only in the sense that they share a worldview, not because there is any more evidence for them. In this case, science is not informative, and the ratios on both sides of the equation are equal to 1.

But now consider evolution through natural selection. Whether or not one believes in evolution, there is a plausible materialistic explanation that unifies an incredibly large number of facts we observe about our world: (1) Geographic distributions of millions of species, relative to each other; (2) genetic divergence patterns among those species; (3) which rock strata different fossils are found in; (4) why organisms are uniquely adapted to their environments; (5) why organisms are imperfectly adapted to their environments; (6) how organisms develop, and how these developmental patterns vary across species; etc.

Clearly, none of this proves that God does not exist. But, compared to our state of knowledge before this, it means that there is a plausible materialistic explanation for something for which, previously, such an explanation was not obvious. Referring to the Venn diagram above, the presence of a plausible explanation for such a complex set of facts dramatically increases the probability of the blue ellipse, without providing any direct information on the probability of the orange ellipse.

Plugging evolution into our equation as B, we would have to conclude that evidence consistent with evolution would be substantially less likely if God existed, since many of the scenarios involving God would not have resulted in a world in which it appears as if evolution had happened. Therefore P(B|A)/P(B) < 1, and in consequence P(A|B)/P(A) < 1.

Similar things could be said for meteorological explanations of the weather, neurobiological explanations of consciousness, astronomical explanations of the stars and seasons, geological explanations of continental drift, volcanoes, etc. There are still major outstanding scientific questions, but for all or almost all of these questions, it is easy to conceive of how they might be explained within the context of science. For example, we don’t yet know exactly what aging is and why some species don’t age, but the answer seems unlikely to be supernatural.

Taken together, scientific findings make P(A|B)/P(A) substantially less than 1. They provide a plausible materialistic interpretation for almost everything we observe about our world, and as such increase the probability that such an explanation is the true one, relative to what that probability was without knowing that a plausible explanation existed. However, it is important to remember that we cannot measure or estimate P(A) and P(B). This means that, while scientific findings make God much less likely than in the absence of those findings, we cannot claim that the absolute probability is low.

Lastly, an important caveat is that science explores questions it can tackle. Many people believe in ghosts and contact with dead friends and relatives, but in the modern world most people do not discuss it much because it is considered crazy or unscientific. Scientists do not study this. Whether or not there is any truth to such beliefs, we must be wary that science may be carried out in such a way as to confirm its own importance and ability to explain the world, inflating our estimation of its true explanatory power.