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December 02, 2005
A Little Hebrew Bible, a Little Archaeology, and a Little Math
One of the most interesting and dynamic discussions in ancient Near Eastern studies involves the short cuneiform alphabet how best to evaluate various discoveries that some say support the historicity of the Hebrew Bible. You can read about this discussion at several places on the web. I thought it might be interesting to try to apply Bayes' method to a couple of these discoveries. I offer the following discussion by way of illustration. One could, and many would, assign other probabilities to the truth-values of the various of the statements I will make below.
The first problem one faces is how best to state the proposition or hypothesis that one seeks to evaluate. For the purposes of this illustration, I will try to apply Bayes method to the following proposition.
There existed in the early10th century BCE an organized, self-conscious group of people in some part of the general area the Hebrew Bible calls Judah and/or Israel that was headed and/or founded by a person whose name was David (דוד or דויד in Hebrew).
I think there are only three subordinate propositions is this statement.
- There was an organized, self-conscious group of people in the 10th century BCE
- That group of people inhabited some physical space (without making any claim as to how much space or exactly where) in Palestine
- That group saw itself as founded or lead by a certain individual who was called David by his contemporaries.
I hope there are no other subordinate propositions because if there are they might screw up the works. Notice that I have sidestepped the question of whether or not the group was a kingdom. That issue adds a whole level of complexity that I want to avoid in this post.
This is a fairly modest proposition to which I will give an initial Bayesian probability of 50%. I will call this proposition D1 and the probability of D1 being true P(D1). Again, remember, this is an illustration; some may think that my value is excessively low, while others may think it is excessively high. However, anyone who would give the proposition a probability of 0% or 100% is not willing to play by Bayesian rules.
The first discovery that I would like to use in evaluating my proposition is Tell Dan stele. This stele famously contains the words בית דוד in line 9 which are literally translated 'house of David," While some have worried about this translation, I offer it as the most parsimonious but will deduct a percentage point or two for whatever uncertainty there may be. In addition it contains the phrase "king of Israel" in line 8 and apparently gives the names of a couple kings of Israel that André Lemair reconstructs, "Jo]ram son of [Ahab" and " [Achaz]yahu son of [Joram king]." You can read Lemair's complete translation in Wikipedia. The inscription has been dated between the 7th and 9th centuries BCE. This is a fairly large range. I discount something for every year it is more recent that the 10th century BCE. Again, we need to state the claim clearly. Here is my effort, "The Dan Stele gives confirming evidence to proposition D1." Even, with all the subjectivity that is involved in this, I would give it a reasonably high confirming probability, so I give it a probability of over 50%, say 75%. Let's call the evidence S and the probability that the evidence is confirming of D1 P(S given D1). Remembering that 0% and 100% is not allowed, you can give it your own confirming probability. If one were being completely rigorous, one would attempt to apply Bayes' method to each of the contingencies and derive an overall confirming probability to the stele, S. The problem with this approach is that with this kind of material we are deluding ourselves if we think we can judge such probabilities with greater than say 25-percentage point accuracy. Trying to be too rigorous could lead us to the wrong conclusion with six decimal points of accuracy.
Now we are ready to take a shoot at a Bayesian analysis. What we want to discover is the new probability of our proposition (D1) being true given the Dan Stele (S). In other words we want to calculate P(D1 given S).
Here's the formula:
P(D1 given S) = (P(D1) x P(S given D1)) / ((P(D1) x P(S given D1) + (P(D2) x P(S given D2)))
Oops, what is that P(D2) and S given D2 in the divisor? For our purposes, we can take D2 to be the negation of D1. There is a more rigorous definition but it is of little help here. So P(D2) is also 50%. But if P(D1) had been, for example, 60% then P(D2) would be 40%. We can take P(S given D2) to be the inverse of P(S given D1) or, in our case 25%. Again, this is not completely rigorous but good enough for our purposes.
So when we plug in the numbers we get a probability for (D1 given S) of 75%.
If you want to do the math yourself, you can download my simplified Bayes "calculator" in Excel format here.
It turns out that if one starts with a 50% probability with the assumptions and simplifications I have used, the answer will always be P(S given D1). But what if we now use our new probability of 75% that our proposition is true and consider some additional evidence. In this case, let's add the evidence from Eilat Mazar's 10th-9th century BCE monumental building in Jerusalem. Given the current state of our knowledge of the archeology of this evidence what is the probability that the statement, "The monumental building found by Mazar in Jerusalem is confirming evidence for our (D1 given S)" is true. Well for the sake of argument, I will assign a slightly confirming number of 55% to this statement. I do this in part because 50% doesn't give a very interesting answer. Let's call this evidence M. Now we need to crank the new numbers to calculate the probability of D1 given S given M. And we get a 79% probability that D1 is true given S and M. If we had assigned 45% to P(M given D1 given S) we would have gotten a probability that D1 is true given S and M of 71%.
Of course, to be completely rigorous we would need to go step by step through all the proposed evidence before we could settle on the probability that the statement,
There existed in the early 10th century BCE an organized, self-conscious group of people in some part of the general area the Hebrew Bible calls Judah and/or Israel that was headed and/or founded by a person whose name was David (דוד or דויד in Hebrew)
is true.
Now am I really proposing that scholars "do the math?" No, in most cases, and the examples I have used here are no exceptions, the evidence does not lend itself to mathematical rigor. However, the thought process that underlay the math can be applied to the types of evidence I have used in this post. One thing that becomes clear is that nothing is 100% confirming or disconfirming of any well formed proposition. Second, one needs to be very careful how one formulates the propositions (hypotheses) one seeks to test. In fact, I'm not sure that the proposition D1 I used in this post was as well formulated as it could have been.
If you want to learn more about Bayes' method and learn how I have somewhat perverted it, for the sake of simplicity in this post check out the references below. Those who want even more can follow the references in these works. Note: Bayes method was developed for statistically significant sets of data and I have greatly adapted the methodology to fit the kind of evidence under discussion in this post.
References:
Salmon, Merrilee, Philosophy and Archaeology, New York: Accademic Press, 1982, pp. 51-55
Posted by DuaneSmith at December 2, 2005 02:50 PM | Read more on Archaeology |
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