Saturday, January 25, 2014

The Essential Equation of Theology

Most theological questions cannot be answered by direct observation. Some might be unknowable. Many require understanding an ancient language and culture that nobody fully understands. Nearly all involve uncertainty. They are matters of belief rather than knowledge. In other words, most theological questions require probabilistic answers. Unfortunately, the human brain is not very good at processing probability and uncertainty.

Fortunately, there is a practical solution: Bayes' theorem. It is a mathematically valid method to calculate probability when there is uncertainty in the data. The formula (color-coded to help you keep track of the terms) is:

Where:
• P(H|E) is the probability that hypothesis H is true, given evidence E
• P(E|H) is the probability that E would be observed if H is true
• P(H) is the prior probability that H is true, without considering E
• P(E|¬H) is the probability that E would be observed if H is not true

Bayes' theorem has important implications for theology. It suggests we should adjust our beliefs whenever we learn new evidence. It also implies that the way many of us interpret the data is wrong. Instead of asking "What (if anything) does the evidence prove?", which does not account for uncertainty and can lead to erroneous conclusions, Bayes' theorem implies that we should instead ask 3 questions:
1. P(H): What is the probability that our belief is true without considering the new evidence?
2. P(E|H): What is the probability that the new evidence would be what it is if our belief is true?
3. P(E|¬H): What is the probability that the new evidence would be what it is if our belief is not true?
The probability that the belief is true is adjusted whenever new evidence is considered. It increases if the answer to #2 is larger than the answer to #3 and decreases if #3 is larger than #2. If #2 and #3 are the same, the data (not really "evidence" in that case) does not move the original probability (#1).

Let's apply Bayes' theorem to a basic theological belief: "God exists". For this example, I'll define "God" only as a personal being who created the universe.

The first evidence to consider is that all relevant observations indicate that our universe had a beginning (which is the scientific consensus). To answer the 3 Bayesian questions:
1. P(H): A prior probability of 0% or 100% would be circular and would neglect uncertainty. 50% seems too high when no specific evidence has been considered yet, so I'll use 10%. It's somewhat arbitrary, but if enough evidence is considered, what we use for P(H) shouldn't matter.
2. P(E|H): If God is the creator of the universe, the probability is very high that the observations would indicate the universe had a beginning. I don't trust my mind enough to use probabilities above 95%, so I'll call it 95%.
3. P(E|¬H): If there is no God, this is a more difficult question with a high level of uncertainty. I can't go too high because it would seem to defy the First Law of Thermodynamics. However, I've heard some interesting theories that don't seem entirely implausible. I'll go with 25%.
Plugging into the equation, P(H|E) = 0.95*0.10/(0.95*0.10 + 0.25*(1-0.10)) = 0.297, the probability that God exists becomes approximately 29.7%.

Now let's consider negative evidence: the current lack of any direct observations of a God. The prior probability is now the previous result: 29.7%. If there is a God, a lack of any direct observations of him may or may not be probable, depending on what kind of God it is. I'll say 50%. If God does not exist, a lack of direct observation is almost certain. I'll again use my 95% rule. The result, P(H|E) = 0.50*0.297/(0.50*0.297 + 0.95*(1-0.297)) = 0.182, is an updated probability of 18.2%.

Finally, let's consider neutral evidence: religious writings contain apparent errors and contradictions. If God exists, it's still highly probable that religious writings would contain apparent errors and contradictions, whether real or perceived. P(E|H) = 90%. The same would be true if there is no God. P(E|¬H) = 90%. The result, P(H|E) = 0.90*0.182/(0.90*0.182 + 0.90*(1-0.182)) = 0.182, 18.2%, no change.

This process should be repeated until all data is considered.

There is much more (and in my opinion, much better) evidence to consider, but my point here is the thought process, not the numbers. We can disagree about what the numbers should be, but if that's what we're debating, we've come a long way. It would mean we're asking the right questions and analyzing the data in a way that properly accounts for uncertainty.

1. Hi Matt,

Interesting stuff! I'm not sure I'm completely sold on using Bayesian arguments in this way regarding the *particular question* of the existence/nonexistence of God, but neither am I sold against it. But, you've definitely got me thinking more about it. One issue of course is what sort of prior probabilities you apply, and even if it makes sense to apply such a probability in this case. (You'll recall we discussed this at some length at lunch the other day). Another thought: where do ontological arguments for God's existence come in to this analysis? Would they be considered at level 1 above, or level 2? What about some philosophers' arguments that belief in God can be considered "properly basic" (i.e. Alvin Plantinga has argued this way). If this is true, wouldn't this pretty much bypass Bayes' theorem, at least for this question? (Obviously, it might still apply to questions about God's character, whether Jesus is God, whether God acts in a scientifically detectable way, and so on.)

Just some thoughts. I'm inclined toward a holistic view that God, and particularly the Christian God, makes the most sense out of everything I see in the natural world, human history, and my own experience. From this starting point, arguments like the ontological and moral arguments, while previously not holding much water on their own, seem much more reasonable to me, and I seek to understand them better within this paradigm (i.e. "faith seeking understanding"). Like we discussed, rather than a circular approach, I see it as more of a spiral staircase, where new information causes a re-evaluation of previously weakly-attested arguments. Perhaps what I'm doing is Bayesian, and I don't even realize it?

1. Thanks Dan! I think all of that fits nicely into a Bayesian framework (pretty sure any probabilistic argument can at least be re-written in Bayesian terms, even if the probabilities are 0 or 1).

I don't consider things like the ontological argument to be evidence per se, but it seems appropriate for estimating a prior probability. I don't think it would justify anything near 100% though, if you're talking about a personal and specific (e.g., Christian) God. 100% would only "bypass" Bayes' theorem in that it makes all evidence irrelevant.

I suppose you could use those arguments the other way, as "evidence" that updates a lower prior (they might have to be "translated" into the form of evidence), but I think it'd be a lot more difficult that way. It would be an interesting exercise though!

2. The following is a tangent, but I don't get to talk about this often. Dan mentioned moral arguments for God in his post. That line of argument has very little weight for me. I can easily conceive of a universe exactly like ours except without God and, therefore, without good and evil. People would still be compelled by their biologically and culturally evolved consciences to believe that willed actions have a moral dimension; they would simply be mistaken. I see nothing internally consistent about such a worldview. And, I don't think it makes much impact on the Bayesian analysis, since any species like ourselves would very likely have (possibly fictitious) moral inclinations, otherwise that species would very likely not have evolved (independently of whether God exists).

Of course, this view makes me a bit of an oddball, as the vast majority of theists and the majority of atheists take objective morality as a properly basic belief!

2. Matt...I think the Devil is in the details! Reference the following short lecture by Stephen Hawking at http://www.hawking.org.uk/the-beginning-of-time.html. This would seem to support your .95 for P(E|H) IF the Universe was created as described in the book of Genesis. However..."the Big Bang is a beginning that is required by the dynamical laws that govern the universe. It is therefore intrinsic to the universe, and is not imposed on it from outside." There have been many attempts to "explain away" a possible beginning of time...the breakdown of physics re: the Second Law of Thermodynamics. Stephen agrees that the Universe had a beginning ...but uses imaginary time, real time, and boundary conditions to satisfy the dictates of physics. The ultimate point is that there may be an even higher probability for P(E|¬H) and a lower probability (e.g., .01) for P(H). In fact...if you only lower P(H) to .01 the resulting answer is .0024. Anyway...I feel your initial assumptions need much more clarity if you want to make a serious case for the existence of God. And of course...your definition of God requires more clarity as well. You may find this interesting...The Catholic philosopher John Haught warns: "For even if scientists concluded that some intelligent being had tinkered with the initial conditions and cosmological constants, pointing them in the direction of life and mind, this "being" would still be an abstraction, and not the living God of religion. It would be a great empty plugger of gaps, and not the personal God of Abraham, Jesus and Muhammad. The [strong anthropic principle] is no more capable of confirming or deepening our religious life than are the old arguments of God's existence. The realms of science and religion are radically distinct. Once again, then, in the interest of maintaining the integrity of both religion and science, we refuse to derive any theological consequences or religious comfort from this spuriously popular "scientific" theory."

1. Thanks Josh. I'm not really trying to make an argument for God here, just using an example to explain how to weigh evidence with uncertainty. You could be right that, in this example, P(E|¬H) should be higher and P(H) should be lower, although generally I'm not comfortable with probabilities below 5% or above 95% for such things (for reasons I'll eventually write about). What those probabilities should be is a very useful discussion though -- much more so, in my opinion, than the usual debates people have on this subject.

I'm not following your math. Using 0.01 as the prior instead of 0.10, P(H|E) becomes 0.037, unless you use a higher P(E|¬H). But it wouldn't go down to 0.0024 unless P(E|¬H) > P(E|H), which I think is a stretch, Hawking argument notwithstanding. I'll defer to him on the cosmology, but I think his argument (which I'll have to study more) is not so much an argument for a high P(E|¬H) as it is an explanation that fits the evidence, which is a bit different, though the fact that it does fit the evidence suggests P(E|¬H) shouldn't be too low.

I agree that what you define as "God" makes a huge difference for the prior probability, and that the more vague definitions of God are a long way from the personal God of Abraham, etc. I've been planning to discuss all of that (including a graph) in my next post.

2. Perhaps I got lost in a parenthesis...wouldn't be the first time...will recalculate later.

3. Hey Matt,

Very interesting stuff! Obviously this is just the tip of the iceberg, and I am curious how it develops. But just for the sake of throwing a wrench in the gears, there is also a probability that God may have written into natural law, physics and more, and even into our human capacity, that He will not be proven or disproven by natural means. Just a thought... :)

1. Hi David, thanks for the comment. I agree. I don't think it's possible (at least with our current level of knowledge and technology) to prove or disprove God, by natural or any other means. If we could, it wouldn't be faith. The "Does God exist?" question is only one of many for which the Bayesian framework is useful. It's also applicable to things like Bible interpretation. I'll be using it often on this blog, but I'll also write about a lot of other data analysis methods that are useful in other ways.

4. I'm very glad you wrote this post, Matt. I had been planning to start a blog in part addressing some of these issues, but kept putting it off. I think explication of the Bayesian method is a critical apologetic tool, largely because it highlights the importance of carefully considering the prior probability of God's existence. If one takes P(H) to be virtually zero, then one is largely dismissing the evidence from the start. This, I believe, is a common mistake of atheists. To assume that God is extremely improbable a priori is to make some very unwarranted assumptions. I find that many otherwise empirically minded people too quickly dismiss the evidence available to them based on their (often latent) metaphysical beliefs. It's a stark example of unfounded faith trumping evidence.

1. How...precisely!...you define GOD has a profound effect on everything else. Every culture in history has had a belief in the divine and a mythology to explain the world around them....no room for grey shades here.

2. This comment has been removed by the author.

3. I think both of you make some good points here. Corey, I totally agree that a lot of atheists have a very low prior (i.e., P(H)) for God (maybe even 0 in many cases), which means there's practically no amount of evidence that would be sufficient for belief. And Josh, I think you are right that many Christians have too high of a prior (maybe even 1 in many cases), which means they will always end up believing in God regardless of the evidence or lack thereof.

I also agree, Josh, that how you define God is of paramount importance. That's why my next post will be all about defining God, including some mathematical tools that are very helpful for doing so.

5. Sorry...thought I was editing...but deleted.

Your last sentence is very puzzling. You are not describing an atheist...you seem to be describing how you prefer to think of atheists.

Cosmology is a central branch of metaphysics...but metaphysics has increasingly become a non-empirical (since the scientific method) philosophical pursuit into the nature of existence. The scientific method derives provisional truth...and new evidence either supports or diminishes veracity.

Again...how you define GOD is profoundly important...is GOD an abstraction based on evolution...the GOD of the Old Testament...the GOD of a particular historical mythology?

1. Josh, I agree that one's starting definition of God is paramount. In my post, I was referring to "a personal First Cause", that is, a free-willed agent that created our universe. It's certainly not as precise as the Christian description of God, but I find that many atheists are reluctant to grant the plausibility of even so general a definition of God, making subsequent argument to the plausibility of the Christian God futile.

In my previous post, I was describing atheists who assign a vanishingly small prior probability to God (as defined above). Any defense of such a position immediately brings in esoteric metaphysical premises. In my experience, many atheists fall into this category. But I certainly wouldn't say this categorically; I readily acknowledge that atheists, like theists, span a wide range of worldviews.