## Saturday, January 29, 2011

### Was Jesus Raised: Bayes' Theorem

The McGrews' approach to proving the resurrection is Bayesian; That is, it employs Bayes' Theorem.

In order to address their argument I realized that I had to become familiar with Bayes' Theorem. So I went to my local library and found a textbook called Introduction to Probability and Statistics which had an introduction to Bayes' Theorem.

According to the book, the theorem should be used when various theories are both mutually exclusive and exhaustive. In the case of the Resurrection, I think it's quite clear how we can do this; Logically there are only two possibilities: Either the resurrection occurred or it did not, and these are mutually exclusive.

Bayes' Theorem can be expressed this way:

Pr (h/e.b) = Pr (h.b) x Pr (e.h&b) / [Pr (h.b) x Pr (e.h&b)] + [Pr (not-h.b) x Pr (e.not-h&b)]

This looks scary as hell, I know. So let me explain what all this means Pr (h/e.b) describes what we're trying to figure out: What is the probability (Pr) of a hypothesis (h) given the evidence we have (e) and our "backround knowledge" (b) ? In other words, how likely is the hypothesis we are proposing, all things considered? That's what Bayes' Theorem helps us figure out.

According to Bayes, we can figure out how likely a hypothesis is if we know the following values:

1. What is the probability of our hypothesis being true independent of its explanatory power?
2. Assuming our hypothesis is true, and taking into account other facts we know about the world, how likely is it that we would have the evidence that we have?
3. What is the probability of our hypothesis not being true independent of its explanatory power?
4. Assuming our hypothesis is not true, and taking into account other facts we know about the world, how likely is it that we would have the evidence that we have?

The values of these four things can then be plugged into Bayes' Theorem and will answer how our question: All things considered, how likely is it that our hypothesis is correct?

A Test Case

The book that I checked out gives the following example on page 157 which I will paraphrase:

Medical case histories tell us that different illnesses may produce the same symptoms. Pretend that a particular set of symptoms (refered "H") occurs only when any of the three illnesses A, B, or C occur. For simplicity assume that these illnesses are mutually exclusive (they never occur in the same person).

Here's how often people in general get these illnesses:
A occurs in one percent of the population
B occurs in half of one percent of the population
C occurs in two percent of the population

When illness A is present, the probability of symptom H developing is .9 (or 90 percent). When illness B is present, the probability of symptom H developing is .95 When illness C is present, it's .75 .

Now, pretend we're doctors and we observe that our patient has symptom H. How likely is it that this person has illness A?

Bayes' Theorem allows us to figure that out from the above information. Remember those four questions I posed earlier? Let's take them one at a time:

1. What is the probability of our hypothesis being true independent of its explanatory power?

Our hypothesis, in this case, is that the patient has illness A. We know illness A occurs in one percent of the population, so the answer to this is .01

2. Assuming our hypothesis is true, and taking into account other facts we know about the world, how likely is it that we would have the evidence that we have?

The answer to this is in the information that we're given about the probability of symptom H occurring if illness A is present. The answer to that is .9

3. What is the probability of our hypothesis not being true independent of its explanatory power?

Things are a little bit trickier here, since there are two other hypotheses on the table (the hypothesis of illness B and the hypothesis of illness C). But we do know how likely they are, independent of their explanatory value, and that is contained in the statement given earlier:
Here's how often people in general get these illnesses:
B occurs in half of one percent of the population
C occurs in two percent of the population

4. Assuming our hypothesis is not true, and taking into account other facts we know about the world, how likely is it that we would have the evidence that we have?

The probability of the evidence that illnesses B and C is, of course, given earlier as .95 and .75, respectively.

These four questions give us every value we need to figure out the overall likelihood that the patient has illness A. Let's go back to the equation:

Pr (h.b) x Pr (e.h&b) / [Pr (h.b) x Pr (e.h&b)] + [Pr (not-h.b) x Pr (e.not-h&b)]

One at a time: The probability (Pr) of the hypothesis (h) given our backround knowledge (b) is answered by question 1, it is .01

We must multiply that number by the Probability (Pr) of the evidence (e) assuming our hypothesis is true (h) and given all our other medical knowledge (b). This value is given by question 2, it is .9

When we multiply .01 by .9, we get .009
We can put that in our equation:

.009 / [Pr (h.b) x Pr (e.h&b)] + [Pr (not-h.b) x Pr (e.not-h&b)]

Now, look at what's in that first set of brackets following the slash. The symbols there are precisely the same as those that were in front of the slash. Therefore that should be replaced with the same number:

.009 / [.009] + [Pr (not-h.b) x Pr (e.not-h&b)]

Not so scary anymore, huh?

The only thing left is to plug in the alternatives to our hypothesis. There are two alternatives to our hypothesis, not just one alternative which might be implied by the crude expression "not-h" or NOT our hypothesis. So our equation needs to take into account B and C:

.009 / [.009] + [Pr (B.b) x Pr (e.B&b)] + [Pr (C.b) x Pr (e.C&b]

For both of these bracketed equations we're doing something very similar to what we've done before. We need to find the probability of B given our backround knowledge, then find how likely the evidence is assuming B is true, then multiply those two figures together. The same goes for our next set of bracketed symbols: find the probability of C given our backround knowledge, then find how likely the evidence is assuming C is true, then multiply those two figures together.

If you've been reading my post and paying attention, you'll already know what numbers to put in:

.009 / [.009] + [.005 x .95] + [.02 x .75]

Now work that out:

.009 / [.009] + [.00475] + [.015]

Add the numbers in brackets together:

.009 / .02875

Now divide, of course, and you'll get a number of roughly .3130 or 31.3%

I checked the answer I got here against the answer given in the end of my textbook and it confirmed this answer.

The Relevance of This Post

Now that we've gotten some backround in Bayes' Theorem, how does this apply to the Resurrection? Just as we can estimate the intrinsic likelihood of a disease, so too we can estimate the intrinsic likelihood of a resurrection, as I will demonstrate in future posts. Just as we can look for symptoms of a disease, so too can we look for "symptoms" of a resurrection (i.e. the tradition of an empty tomb) and we can estimate how likely these symptoms are if the resurrection happened versus how likely these symptoms are if the resurrection did not happen. As I'll demonstrate in future posts, we have ways of estimating values for each of the four questions I presented earlier. And so we can make a definitive statement about whether the resurrection happened.

BlogHer said...

Hello, Nicholas. I've been reading your AIG Busted blog and am finding it to be excellent. The photo that you've provided on your 'Gill Slits' blog reveals someone very young. Is it a recent pic? I ask because you are quite accomplished for someone so young. And a book already!

AIGBusted said...

Hi BlogHer,

I'm 21. So yeah, I guess I'm young, although my younger sister tells me that anyone older than ten is ancient.

And thank you for the compliments. I'm glad you like the blog.

-Nick