The question

Christoph Heilig raised an interesting question about our epistemological access to Pr(H|E), commonly called the posterior probability in a Bayesian analysis. I had argued that sometimes we have better epistemological access to Pr(H|E) than to the terms by which one calculates Pr(H|E) by Bayes’ formula. Here’s the standard formulation of of Bayes. 

Simply put, my position is that sometimes there are epistemologically more reliable ways to get Pr(H|E) than by using Bayes. In fact, sometimes Pr(H|E) is impossible to gain by using Bayes while it is possible via a different route, i.e., sometimes we have better epistemological access to the left-hand side of the formula than to the right-hand side.

Note that I am only saying that sometimes skipping Bayes is better, not that it is always better. To show how this works, I’ll expand on an example I gave to the Bayes and the Bible group. Then I’ll look at how this works with hypotheses. After that I’ll take what was discussed here and apply it to the issues in Biblical Studies that Christoph asked about.

Example 1: Hares

Suppose that you are wandering through the woods and come to the top of a tall hill. Gazing down at the surrounding countryside, you find a nearby hilltop, somewhat lower than your own, covered in grassy meadow. Upon that meadow are a number of creatures of several different kinds. Being a naturalist of a statistical bent you wonder what the probability that a creature on that hill is a hare assuming it is ecru; i.e., Pr(H|E). As I’m sure all woodland naturalists know, ecru is a beige-like, yellowy-brown color. I use it to avoid giving offense to the hares—beige seems like such a blasé colour.

Pr(H|E) Without Bayes

Looking down, we discover that it is hard to count all the creatures. It’s hard to tell which things at the far end of the meadow are creatures and which are leaves blowing in the wind—all are a similar blackish sort of color. But this makes it impossible for us to calculate Pr(H). In our case Pr(H) = the number of hares / total number of creatures and because of the darkness we don’t know how many hares there are and we don’t know how many total creatures there. This means we don’t know either the numerator or denominator. However, fortunately for us, all the Ecru colored creatures have gathered on the near side of the hill and the type of creature is easily recognizable. There are 10 Ecru colored creatures, 6 of which are bunnies. The chart describes this situation.

Since we don’t know how many ~Ecru (the ‘~’ means not) things there are we have a lot of question marks left. However, we have everything we need in order to calculate Pr(H|E). Simply put, out of the group of Ecru things, how many of them are Hares? Another way looking at this is in terms of a random drawing. The second part of the formula [here the E in Pr(H|E)] is telling us which bag we are drawing from. In this case, it is the bag full of Ecru things. The question then becomes, what are the odds of pulling a Hare out of the Ecru bag? Or, more magically, what’re the odds of pulling a rabbit from the brown hat?

In our case we know how big the hat is—there are 10 things. And we know how many hares are in the hat (6). The odds are then 6/10 and hence Pr(H|E) = .6. We found this all without needing to know H (I’ll discuss how this works with the definition of conditional probability as Pr(A|B) = P (A∩B) / P(B) more in the next post). Hence we have succeeded in learning Pr(H|E) where we would have failed when following standard Bayesian calculation. This is due to the lack of Pr(H). Actually, we are lacking all the terms on the right side of the equation, Pr(E) and Pr(E|H) as well as Pr(H).

Pr(H|E) With Bayes

Now it is important to note tha this situation does not always hold. Consider a variation on our example. Here the population stays the same, but our epistemological situation is different. We are still on the hill, this time at sunset. The light makes it easy to see the colors of the creatures, but the backlit glow makes it hard to see shapes. We count the Ecru colored creatures and find 10, but we cannot discern what kind of creature they are. Later the full moon rises and in its cool light we can easily see all the creatures on the hill and can discern their shapes, but our night vision makes it impossible for us to see the color of these creatures. There are 36 total creatures, 24 of which are hares. This gives us Pr(H) = 24/36 = .66; two thirds of the creatures are rabbits. Similarly, 10/36 creatures are Ecru, Pr(E) = .28. As yet we cannot get to Pr(H|E), how many of the Ecru creatures are Hares. But you suddenly remember, Hares around here tend to be Ecru a quarter of the time Pr(E|H) = 1/4 = .25. This now lets us calculate Pr(H|E) according to Bayes theorem.

We discover that the probability that an Ecru creature is a hare is .6. Looking at the same hilltop with different information still gives us the same Pr(H|E)—just as it should. In the first case it was easier, we could more directly get to Pr(H|E) without Bayes theorem. In the second case we needed Bayes. This fits with my position on Pr(H|E), sometimes you can learn it by means of Bayes theorem, but other times you can’t. Sometimes we can have better access to Pr(H|E) via a different means.

Now, perhaps you might object that this is not dealing with hypotheses and hence is an unfair or irrelevant example. What happens when we are dealing with hypotheses? That is the topic for the next post.