The Probability of GodThe Probability of God: A Bayesian Analysisby Glen Davis, Chi Alpha @ Stanford, http://www.xastanford.org/inspired by Stephen Unwin's book The Probability of GodPrior Probability of God's existence50%Evidentiary AreaD FactorLikelihood of GodMorality150%Evil150%Suffering150%Answered/Unanswered Prayer150%Reports of Miracles150%Religious Experiences150%Origin/Nature of the Universe150%Logical Necessity/Impossibility150%Other Category150%Other Category150%Overall Probability of God's Existence:50%

This cell's formula is slightly different from the ones below it because it's prior is several rows above. Just keep that in mind if you start moving cells around.This is the maximum entropy assumption--in other words, we feign ignorance going in. Anything that would sway this initial number can be put down in the evidence. To strongly bias the initial value is really a way of sneaking your arguments into your assumptions.We have a sense that we ought to do certain things and refrain from certain others. Is this more likely in a universe with a God or one without?This is the probability that this evidence would arise in a godly universe rather than a godless one. P(E|G)/P(E|G')

In other words, how much more likely or less likely is this evidence to exist in a world with God than in a world without?Some of the bad things in this world can be accounted for by personal choice. If I torture someone, for example, that would be an example of moral evil. Is the extent of moral evil that we see around us more likely a result of a godly universe or a godless universe?This analysis is based on Bayes' Theorem, which can be used to determine what you ought to believe about something based on what else you already know or believe. See the other sheets for more info.Sometimes horrible things happen and no person is to blame. Someone gets cancer, for example, or they are burned in a fire caused by lightning. Are these events more likely in a universe with a God or a universe without?This category is not for flat-out miracles, but for those answers to prayer could just be coincidence. I pray for someone to be healed, and they get better. I pray for provision, and I have some money come my way. Given the level of reporting of these events (and your own experience of them), is it more plausible that these reports and your experiences are a byproduct of living in a divinely-governed universe or that they are a byproduct of living in a godless universe? This category should also include evidence from unanswered prayers.Some reported miracles cannot be explained by any natural mechanism. Someone being raised from the dead, for example, or someone teleporting around. Assuming this has not happened to you personally, are the reports of these miracles more likely in a godly universe or a godless universe?

Obviously, if one has happened to you you're no longer dealing with reports of miracles but with the experience of them. Your question is: is my experience more likely a clue that there is a God or a clue that there is not one?These aren't so much miraculous answers to prayer as they are encounters with the transcendent. People from all cultures throughout history have reported such experiences. Are they more likely in a universe with a god or a universe without a god?Is the universe so well-designed that it requires a designer? Or are the laws of physics so tight that they leave no room for the God hypothesis? This is where you could express your feelings about the anthropic principle, evolution, intelligent design, creationism, or any arguments related to the nature or origin of the universe itself.Some people have contended that God is a logical necessity (Anselm's ontological argument) and others that the term God is incoherent (logical positivism). If you've been impressed by an argument that God is a logical necessity or impossibility, this is the place to factor it in.

Obviously, it's only meaningful to do so if you have some doubts about the argument. If you're sure either way, the Bayesian analysis is a waste of time.INSTRUCTIONSEach evidentiary area subsumes many arguments for and against God's existence. Hover your mouse over the evidence category to see a description of it. Think about it, and then decide whether or not the evidence in that category is more likely a result of God existing or of a universe without God.

These are recommended value increments, but you can put down whatever you like.10: Much more likely if there is a God.2: More likely if there is a God.1: Equally likely regardless of God (put this if unsure).1/2: More likely if there is no God.1/10: Much more likely if there is no God.

Place your conclusion in the cell under D Factor.

Cells with red corners have comments--hover your mouse to learn more!The last number is how likely you ought to think it is that God exists given what else you believe about the universe.

If this approach intrigues you, I recommend Unwin's book as a great place to begin digging deeper.Did I overlook something? Create your own evidentiary area!Did I overlook something? Create your own evidentiary area!In case it's not obvious, what we're doing is chaining Bayes' theorem, using each output as the input (prior probability) for a new application of Bayes' with a different evidentiary area.

The order of evidentiary areas is irrelevant to the final number, which is the only one that really matters.Our goal is to have these areas be mutually exclusive and collectively exhaustive. In other words, cover everything with no overlap.

It's hard to do with a fuzzy subject like this. If you don't like my breakdown substitute your own.

About Bayes' TheoremUsing Bayes' theorem, we can show exactly how likely it is. Here is the problem expressed in terms of elements of Bayes' theorem.P(A given B)?What is the probability that I took a marble from the mostly blue bowl if the marble I picked is blue?P(B given A)80%What is the probability that I picked a blue marble if I picked from the mostly blue bowl?P(A)50%What is the probability that I picked from the mostly blue bowl?P(B)45%What is the probability that the marble I picked is blue?So here's how we would figure things out:P(A given B)89%

In the mostly blue bowl there are 8 blue marbles and two red marbles, so if I pick from that bowl there's a 8 in 10 chance of pulling a blue marble.There are two bowls, so there's an equal chance of getting either bowl.Overall there are 20 marbles, 11 of which are red and 9 of which are blue. 9/20 = 45/100 = 45%This is what we're trying to figure out.And this is the magic of Bayes' theorem: P(A|B)=P(B|A)P(A)/P(B)

Or in this Excel spreadsheet, =B27*B28/B29Bayes Theorem is an identity stating that the probability of A given that B has occurred is equal to the probability of B given that A has occurred times the probability of A occurring divided by the probability of B occurring.Confused?Think of it this way: suppose you have a bowl of ten marbles. Nine are red and one is blue.

Now pull a marble at random from the bowl--how likely is it to be blue?

Clearly 10%.Now imagine two bowls of ten marbles. One has nine red and one blue, and the other has eight blue and two red.

Shuffle the bowls around as if playing some magic trick, and then pick a marble at random.

You pick a blue marble.

Which bowl did you likely pick from--the one that's almost all red or the one that's almost all blue?

You probably got your marble from the bowl that's mostly blue. It doesn't take a genius to figure that out.So there's roughly a nine in ten chance that you pulled from the mostly blue bowl.

What's the significance of this? We inferred a reliable probability of something we did not directly measure by looking only at related data!

Surprisingly, this approach has proven quite useful in many non-mathematical applications (although not without controversy). It's possible to apply Bayes' Theorem to all sorts of relatively fuzzy subjects if you can quantify some of the related data, even if only in a vague and somewhat subjective way. See the next sheet for an example of that.

You can learn a lot more about Bayes (and some controversy about the proper application of his theorem) on the Wikipedia (www.wikipedia.org).Incidentally, the astute will have noticed that the formula on the first sheet is different from the one that I used above. They are equivalent to one another (I leave it to you to demonstrate this if you're really nervous about it). The different formula on the first page was occassioned by the need to integrate user data into the equation. Stephen Unwin chose to do that using the variable D, and made the appropriate transformations to Bayes' to make it work. I think it makes it much more accessible for non-math types.

Non-Mathematical ExampleA Non-Mathematical Example of Bayes' Theorem

Here's a non-mathematical example that gets across Bayes' theorem. While the other sheet is more helpful for the math, this may be more helpful for understanding the basic concept.

Question One: You see me walking down the street with tears in my eyes. How likely am I to be sad?

Well, pretty likely. Based on the information you have, thats a reasonable assumption. People cry when they are really sad or really happy (usually in places like wedding chapels and hospital delivery rooms) or when they have something in their eye, but sadness causes more tears than anything else.

But say you get to know me well, and realize that I have pesky contacts that keep causing me to tear up for no good reason at all. In fact, I cry far more often due to my contacts than due to sorrow or joy.

Question Two: You see me walking down the street with tears in my eyes. How likely am I to be sad?

This is the same as question one, but now we have more information that can help us give a better answer.

One way we can try to answer this question is using Bayes theorem. To do that, we have to try to quantify our knowledgeput numbers on it. This is going to be somewhat subjective (you might choose to assign different numbers than me), and yet oddly objective (Im making my assumptions plain and putting them out in the open for discussion and possible refutation).

Lets say a = me crying.Lets say b = me being sad.

What is P(b|a)? That's statistical notation for "the probability (P) of b given that a is true." In other words, how likely is it that Im sad (b) given that Im crying (a)?

This is where Bayes theorem becomes useful. It tells us that P(b|a)=P(a|b)P(b)/P(a)

In English: the probability that I am sad given that you see me crying is going to be equal to the probability that I would cry if I was sad times the probability that Im sad divided by the probability that Im crying.

Now we obviously don't have hard numbers that will help us make an inference. Nobody (to my knowledge) has ever followed me around and recorded how frequently I cry or regularly assessed my emotional state to develop a normative model of my personal psychology.

Does that mean that we're completely stuck?

No, not at all.

If we can get some round numbers we'll still be able to get some information. That information will be useful if our numbers are reasonably accurate.

So let's think for a minute. Knowing me, you'd say the following numbers are close:

* If I was really sad, theres probably a 75% chance that I would cry. [thats P(a|b)]* Im only really sad about 1% of the time, though. [thats P(b)]* And I have tears in my eyes about 5% of the time. [thats P(a)]

Now lets plug the data into Bayes theorem: .75*.01/.05=.15

If our numbers were perfectly accurate, that would mean that theres a 15% chance if you see me crying that Im actually sad. The rest of the time it's either bad contacts or rapturous joy (or some other unidentified cause).

The numbers are probably not perfectly accurate, so relying on the 15% figure as opposed to a 25% figure is a little silly. The conclusion, though, is solid: "If I see Glen crying, it's probably not because he's sad."

That's a somewhat silly example, but it gets the core concepts across.

You can use Bayesian analysis for all sorts of things IF you can come up with reasonable numbers for the related evidence. You can come up with reasonable numbers in all sorts of ways--experimentation, research, or interviewing experts.

To the extent that your numbers are externally justifiable your analysis is persuasive to others. To the extent that your numbers are subjective your analysis is persuasive only to you (and to those who mostly agree with you).

But here's the thing--you ought to believe the results of your analysis until you are persuaded to change some of your numbers.

What Bayes' Theorem is doing in that case is helping you be consistent: "Given that you believe X, Y, and Z, W is the most reasonable expectation for you to hold of the universe."

And that's what's happening on the first sheet. You're not necessarily creating an argument that will crush all in your path--you're discovering how much confidence you should place in the God hypothesis BASED ON WHAT YOU ALREADY BELIEVE ABOUT ALL THE RELEVANT EVIDENCE.

And so I encourage you to think deeply about the arguments for and against the existence of God. If you'd like to see a helpful discussion in the context of Bayes' theorem, I encourage you to go read Stephen Unwin's book The Probability of God.