The strength of the evidence for a proposition is best measured in terms of the ratio of two probabilities, P(E|H) and P(E|~H) — that is, the probability of the evidence (E) given that the hypothesis (H) is true, and the probability of E given that H is false. That ratio may be top heavy (in which case E favors H), bottom heavy, or neither (in which case E favors neither hypothesis, and we would not call it evidence for or against H). Bayes’ Theorem is a mathematical tool for modelling our evaluation of evidences to appropriately apportion the confidence in our conclusions to the strength of the evidence.
To take an example, suppose that P(E1|H) = 0.2, but P(E1|~H) = 0.04. Then the ratio P(E1|H)/P(E1|~H) has the value of 5 to 1, or just 5. If there are multiple pieces of independent evidence of the same sort, their power accumulates exponentially. Five such pieces would yield a cumulative ratio of 3125 to 1. If the initial ratio were 2 to 1, ten pieces of independent evidence would have a cumulative power of more than 1000 to 1. By expressing it in mathematical terms like this, hopefully you can see how small pieces of evidence, no single piece by itself of very great weight, can combine to create a massive cumulative case.
The equation given below represents the odds form of Bayes theorem, which is used in developing cumulative cases. Translated, it states that the posterior probability of your hypothesis (H) given the available evidence (E) is equal to the prior probability (defined as intrinsic plausibility) of the hypothesis being true (expressed as a ratio) multiplied by the ratio of the evidence given the hypothesis against the probability of the evidence given the antithesis.
Dividing the probability of the evidence given the hypothesis by the probability of the evidence given the antithesis gives you what is referred to in probability theory as the Bayes Factor. The Bayes Factor is a measure of the strength of the evidence, and indicates how many times more likely it is that you will observe this evidence given that your hypothesis is true than if it were false. For instance, a Bayes Factor of one hundred indicates that your evidence is one hundred times more likely if your hypothesis is true than if it were false.
This form of reasoning is used routinely in the discipline of forensic science. For instance, the presence of a defendant’s finger prints on a murder weapon may be taken as evidence for the hypothesis of guilt over the hypothesis of non-guilt because the probability of the defendant’s finger prints being on the murder weapon is much higher on the hypothesis that the defendant is guilty than on the hypothesis that he is not guilty.
How might we make a powerful case for the existence of God based on what we have just learned about Bayes Theorem? We can begin by giving an estimate of the probability of the evidence given theism and the probability of the evidence given atheism, in order to calculate the Bayes Factor.
The Moral Choice Arena Evidence for God
One way to frame the argument for the existence of God – the approach I will take in this article – is to consider the evidence that we self-evidently live in what I call a moral choice arena. What is a moral choice arena? A moral choice arena is simply a community of persons, not necessarily humans, but persons in circumstances where they can engage in what we at least call moral decision-making, where they interact with other agents and grow their character in what gets called morally significant ways.
On the hypothesis of theism, a moral choice arena is something that God could be plausibly expected to bring about. Why? Intrinsic to God’s very character is the quality of moral goodness, and because of this it is not unlikely for an omnipotent and omnibenevolent entity to bring about the greatest goods. Since the greatest goods require a community of embodied moral agents in a moral choice arena, this is something that it might be plausibly expected for God to bring about. One might of course ask at this point why God would choose to bring about embodied agents. After all, couldn’t He have created spiritual agents that are not embodied? However, it is being embodied that amplifies our ability as agents to affect the world and each other. A world of physical pushes and pulls greatly increases the number of opportunities for free agents to morally flourish, mold their character, and co-operate with one another.
In order to progress with our argument, we need an estimate of the probability of a moral choice arena existing on theism, and of the probability of a moral choice arena existing given atheism. The probability of a moral choice arena existing given the hypothesis of theism is difficult to estimate. However, I think most of us would say that the probability is not likely to be lower than 1%. Thus, for the purpose of argument, let us use this conservative figure of 1% as the probability of a moral choice arena existing given the hypothesis of theism.
What, then, is the probability of a moral choice arena existing given atheism? We could just guess. But our estimate is going to be more informed if we break it up. For example, here is a selection of things that you need as preconditions for the sort of moral choice arena that I am describing:
- The origins of a Universe governed by physical laws such as gravity.
- The fine-tuning of initial constants and laws.
- The origins of life.
- The origins of molecular machines.
- The origins of animal body plans.
- The origins of consciousness.
- The existence of moral sensibilities.
To be generous, let us make the unreasonably charitable assumption that these things are all that we require as preconditions for a moral choice arena.
A Universe From Nothing
What is the probability, on the assumption of atheism, that there would exist a Universe governed by physical laws such as gravity? For the purpose of argument, let us make the generous assumption that the probability of a Universe governed by physical laws, on the assumption of atheism, is 0.1%.
A Finely-Tuned Universe
But you don’t just need any old Universe. As it turns out, the laws and constants of physics have to be finely tuned in order to be conducive to sentient life forms – or, for that matter, any life forms. As Geoff Brumfiel confesses in this Nature News article,
“If you believe the equations of the world’s leading cosmologists, the probability that the Universe would turn out this way by chance are infinitesimal — one in a very large number.”
In 2012, astrophysicist Dr. Luke Barnes (University of Sydney) published an extensive review paper, surveying more than 200 academic papers that document the fine-tuning of our Universe for life  (Barnes, 2012). He stated that he can only think of “a handful of physicists that oppose this conclusion, and piles and piles that support it.” Dr. Luke Barnes also co-authored a book on the subject in 2016 with Dr. Geraint Lewis, called A Fortunate Universe: Life in a Finely-Tuned Cosmos, which you can purchase at Amazon.
Just to give an idea of how finely-tuned our Universe is, consider the cosmological constant, which determines how rapidly the Universe expands. It is thought to be finely-tuned to 1 part in 10120. If you get it wrong, the Universe either expands so rapidly that you only ever get the two lightest elements, or it collapses within picoseconds of the big bang. In such circumstances, no life of any kind could arise. Another factor is the ratio of electrons to protons, thought to be finely-tuned to 1 part in 1037. If it was larger or smaller, chemical bonding would be insufficient for life chemistry.
That’s just two of many constants and laws that have to be delicately balanced in order to produce a life-permitting Universe. I will not here delve into the problems inherent with the usual responses to fine-tuning based on the multiverse hypothesis and weak-anthropic principle. That has been done extensively elsewhere. For the purpose of our argument here, I will generously assume that the level of fine-tuning of our Universe has been greatly over-estimated, and that the probability of getting a life-permitting Universe, on the assumption of atheism, is an absurdly high 0.1%. Given the ridiculously generous nature of the estimate, no atheist can sensibly contest that figure.
The Origins of Life
Next, given there is a life-permitting universe, how much should we expect that life would actually arise in it? I will not hash out the probabilistic arguments pertaining to the origins of life here. I would also direct less-acquainted readers to Stephen C. Meyer’s book, Signature in the Cell – DNA and the Evidence for Intelligent Design, or to William Dembski and Jonathan Wells’ book, How to be an Intellectually Fulfilled Atheist (or Not) for an accessible entry point. Needless to say, the probability of forming life from inorganic chemicals by naturalistic processes is very, very low. Nonetheless, for the purpose of this argument, I am going to assume that the likelihood is unreasonably high – at 0.1%. Again, no naturalist can sensibly contest such an estimate.
Molecular Machines in the Cell
What about the origins of the molecular machinery in the cell? Again, I will not hash out the various probabilistic arguments against the origins of the irreducibly complex machinery found inside the cell. For an accessible introduction, I would refer readers to Michael Behe’s three books, Darwin’s Black Box, The Edge of Evolution, and Darwin Devolves. Needless to say, again, the odds of forming by naturalistic processes the numerous machines of the cell with their many interacting protein parts is prohibitively small. Nonetheless, for the purpose of our argument we can give an absurdly high estimate of the odds of 0.1%.
The Origins of Multicellularity
What about the origins of multicellularity? Our bodies are made up not just of a single cell type but of many – including nerve cells, red blood cells, smooth muscle cells, fat (adipose) cells, intestinal epithelial cells, striated muscle cells, bone tissue with osteocytes and loose connective tissue with fibroblasts. Is this plausibly accounted for on a naturalistic scheme? It seems not. At the level of the single cell, cells that are more able to reproduce are selected for. However, if those cells reproduce in an uncontrolled fashion in a multicellular organism, it will cause serious harm, even death, to the organism. While a single cells seeks to reproduce more than its competitors, in the context of a multicellular organism, the reproduction of cells must be controlled so as to facilitate the needs of the whole organism. Indeed, as John Pepper and his colleagues state in a paper published in PLoS Computational Biology 
“Multicellular organisms could not emerge as functional entities before organism-level selection had led to the evolution of mechanisms to suppress cell-level selection.”
Furthermore, in multicellular organisms there is a need for a communication network between cells that controls the positioning and abundance of the various cell types within the organism. Fundamental to this is cellular differentiation, a process that takes place in all multicellular organisms. This level of organisation is inexplicable by the sum of the parts, cells, since the coordination requires a level of organisation above that which is present in individual cells.
A further requirement of multicellularity is genetic sameness. Developmental biologists Lewis Wolpert and Eörs Szathmáry explain ,
The first step in the development of a complex organism is the establishment of a pattern of cells with different states that can differentiate along different pathways. One mechanism for pattern formation is based on positional information: cells acquire a positional identity that is then converted into one of a variety of cellular behaviours, such as differentiating into specific cell types or undergoing a change in shape and so exerting the forces required for the formation of different structures. This and other patterning processes require signalling between and within cells, leading ultimately to gene activation or inactivation. Such a process can lead to reliable patterns of cell activities only if all the cells have the same set of genes and obey the same rules.
Much more could be said on this subject, but suffice it to say that on any reasonable estimate, the probability of evolving multicellularity by chance and necessity is less than 0.1%. So let’s take that as our estimate.
Animal Body Plans
Now that we have multicellular life, what is the likelihood of animal body plans emerging? Again, there is much that could be (and has been) said on this subject. For a more detailed discussion than what I offer here, I refer readers to the volume Theistic Evolution: A Scientific, Philosophical and Theological Critique, in particular chapter 7 (by Jonathan Wells) and chapter 9 (by Sheena Tyler). I would also recommend reading chapter 13 of Darwin’s Doubt by Stephen C. Meyer. Paul Nelson also has various articles on Evolution News and Science Today discussing the problem of ontogenetic depth, and his argument is summarised succinctly in this short video. Embryonic development takes place by a process of serial cell differentiation and specification. All animals have a programmed developmental trajectory going all the way from the fertilised egg (zygote) to the final form of the organism capable of reproduction. Only the final form capable of reproduction is ‘visible’ to natural selection. The developmental pathway must be put together by a process lacking foresight. Traversing half-way across the chasm that separates the zygote from the final form is no good – the organism is still non-viable. The chances of a developmental pathway being produced without a process with foresight, therefore, is infinitesimally small. Nonetheless, for the sake of our argument here, let’s just assume a probability of, yup you guessed it, 0.1%.
I will add a note at this point to say that I assume that the only way for a body plan to come about (by which I mean something which can house a mind whereby it can control its movements and so forth) is via cellular RNA or DNA body plans, or via some chance assembly of a computer. Of those options, the DNA-based body plans that we actually observe seem to be the easiest, and it is immensely improbable. Thus any other hypothesis is likely to be even more improbable.
The Origins of Consciousness
What about the origins of consciousness? There is nothing in known physics that would allow someone to look at the brain and conclude “hey, there’s someone in there; this thing has first person experiences.” Thus, we cannot predict consciousness by way of physics and examining the brain. You might well be tempted to think that only brains with subjective experiences would avoid pain and so forth, and thus we could predict the evolution of consciousness because its adaptive. However, this response is mistaken, since only people who believe in souls believe that the mind affects the brain like that. Most naturalists would say that your body would do what it does, even if no consciousness existed, since it is a physically closed machine. All of your neurons would fire just the same and move your body the same way without ‘you’. Evolutionary history presumably would be identical without subjective experience. What is the probability, on the hypothesis of atheism, that consciousness – personal first-person subjective experience – would arise out of matter? Let us again be generous and assume a probability of 0.1%.
The final ingredient we need for our moral choice arena is moral sensibilities – for conscious agents to recognize some behaviors as morally virtuous and others not. Again, for the purpose of our calculation, we can assign a probability here of 0.1%.
Let’s summarize the various ingredients we have looked at and the probabilities on atheism that we assigned to them:
- Pr(Universe [laws etc.] | Atheism) = .001
- Pr(Life-permitting Universe | Universe & Atheism) = .001
- Pr(Origin of life | Life-permitting Universe & Atheism) = .001
- Pr(Origin of life | Life-permitting Universe & Atheism) = .001
- Pr(Molecular machines | Origin of life etc. & Atheism) = .001
- Pr(Multicellularity | Molecular machines etc. & Atheism) = .001
- Pr(Body plans | Multicellularity etc. & Atheism) = .001
- Pr(Consciousness | Body plans etc. & Brains & Atheism) = .001
- Pr(Moral sensibilities | Consciousness etc. & Atheism) = .001
Remember, I am charitably assuming for the sake of argument that this is all that is required for a moral choice arena. I am also purposefully grossly over-estimating the probabilities of each individual step. In addition, if you recall, I grossly under-estimated the probability of a moral arena on the assumption of theism, which I gave as a mere 1%, or 0.01.
The next step is to multiply those numbers together to arrive at the probability of a moral choice arena given atheism. The calculation works out as 10-24. Since that figure is much, much lower than 0.01, we can conclude that the moral choice arena is very strong evidence for theism over atheism.
Factoring in the Prior Probability
Our calculation, however, is not complete until we have factored in the prior probability. Prior probability relates to the intrinsic plausibility of a proposition before the evidence is considered. Normally the prior probability will be somewhere between zero and one. A prior probability of one means that the conclusion is certain. For instance, the fact that two added to two is equal to four has a prior probability of one. It is definitionally true. A prior probability of zero, conversely, means that the hypothesis entails some sort of logical contradiction (such as the concept of a married bachelor) and thus cannot be overcome by any amount of evidence.
Priors can be established on the basis of past information. For example, suppose we want to know the odds that a particular individual won last week’s Mega Millions jackpot in the United States. The prior probability would be set at 1 in 302.6 million since those are the odds that any individual lottery participant, chosen at random, would win the Mega Millions jackpot. That is a low prior probability, but it could be overcome if the supposed winner were to subsequently quit his job and start routinely investing in private jets, sports cars, and expensive vacations. Perhaps he could even show us his bank statement, or the documentary evidence of his winnings. Those different pieces of evidence, taken together, would stack up to provide powerful confirmatory evidence sufficient to overcome a very small prior probability to yield a high posterior probability that the individual did indeed win the Mega Millions jackpot. In other situations, setting an objective prior is more tricky, and in those cases priors may be determined by a more subjective assessment.
In the case of the existence of God, estimating a prior probability is difficult. Since nobody has ever put forward a convincing case that the attributes of God entail some sort of logical contradiction, we can safely assume that the prior probability is non-zero. That means that the existence of God can in principle be demonstrated by evidence, provided there is enough of it relative to whatever the prior probability is. For a good discussion of the considerations involved in determining the intrinsic probability of theism, I refer readers to a paper published in 2018 by Oxford University philosopher Calum Miller. 
For the sake of argument, let us be extraordinarily conservative and say that the prior probability of God is 10-18. Again, it is very hard for an atheist to seriously contest the assignment of a prior that is that low.
Performing the Calculation
Given the above analysis, what posterior probability should we assign to the existence of God? Let’s look once more at the equation of the odds form of Bayes’ Theorem, given earlier in this article.
Thus, the posterior probability ratio is equal to the result of the following calculation:
(1022/1) x (1/1018)
The result is a ratio of 10,000 to 1 that the hypothesis of God is true. Converted to a decimal, the posterior probability of God is thus 0.9999.
I hope to have shown in this article the power of a cumulative case for God based upon Bayes Theorem. In particular, while assuming outrageously generous estimates for the probabilities of the various preconditions necessary for a moral choice arena, we have accumulated sufficient evidence for the existence of God to overcome even an astronomically small prior probability of 10-18. and still achieve posterior odds of 0.9999 for the existence of God. In view of how generous we have been with our assignments of the relevant probabilities, the actual posterior probability, based on the available evidence, is in fact much higher than that.
 Barnes, L.A. (2012) The Fine-Tuning of the Universe for Intelligent Life. Publications of the Astronomical Society of Australia 29(4):529-564.
 Pepper, J.W., Sprouffse, K. and Maley, C.C. (2007) Animal Cell Differentiation Patterns Suppress Somatic Evolution. PLoS Computational Biology 3(12):e250.
 Wolpert., L. and Szathmáry, E., Evolution and the egg, Nature 420:745, 2002.
 Miller, C. (2018) The Intrinsic Probability of Theism. Philosophy Compass 13(10):e-12523.
2 thoughts on “What is Bayes’ Theorem, and What Does It Have to Do with Arguments for God?”
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Similar reasoning has been employed to argue that the probability of the simulation hypothesis or the alien terraforming/cosmoforming hypothesis may be extremely high. Given that both of these possibilities could have a purely naturalistic explanation and generalized versions of these are also logically independent of theism (God could be running the universe as a sim on a divine supercomputer instead of “actually” creating it, for example), how do we figure out which specific claims are actually justified? Keep in mind that given the simulation hypothesis, all “miracles” suddenly have a trivial naturalistic explanation.
This may sound like a ridiculous Spaghetti Monster argument or a Descartes’ Demon-style skeptical scenario to you, but I assure you that I am absolutely serious and that I find both the simulation and terraforming/cosmoforming arguments have some plausibility; in fact, I would confidently assign nonzero probabilities for these while I’m not sure I can for a god.
Nick Bostrom’s Simulation Argument: