Isn't the use of Bayes Theorem predicated on building a cumulative case? Odds are calculated for each individual element, that contributes to an overall score.
Carrier writes in OHJ, page 15:
With all that background work done, then I must determine the relative prior probabilities of each hypothesis, by ascertaining the clearest and most applicable reference class for which we have sufficient data (Chapter 6). Then I must determine the consequent probability of all the evidence on each hypothesis, which I will do by breaking the evidence down into categories and treating each separately but cumulatively, after first attending to questions of dating and authorship (Chapter 7). The four categories I develop will then each receive a chapter, in the order of latest to earliest: the extrabiblical evidence, both secular and non-canon ical (Chapter 8); the canonical Acts (Chapter 9); the canonical Gospels (Chapter 10); and finally the canonical Epistles (Chapter II). I will then tie everything together and argue for a conclusion (Chapter 12).
Also I found this webpage called "How to Construct a Cumulative Case Argument" on a Philosophy/Theology website which states:
The cumulative case uses the prime principle of confirmation: Whenever we are considering two competing hypotheses, an observation counts as evidence in favor of the hypothesis under which the observation has the highest probability. This principle is sound under all interpretations of probability...
The Likelihood Principle can be derived from the so-called odds form of Bayes’ Theorem, which also allows one to give a precise statement to the degree to which evidence counts in favor of one hypothesis over another. The odds form of Bayes’ Theorem is P(h1|e)/P(h2|e) = [P(h1)/P(h2)] x [P(e|h1)/P(e|h2)]. The Likelihood Principle, however, does not require the applicability or truth of Bayes’ Theorem and can be given independent justification by appeal to our normal epistemic practices.
See also Jeffrey Jay Lowder's webpage "Intelligent Design Arguments and Cumulative Cases" here:
How Correct Cumulative Cases Work
To make things simple, let’s pretend there are two facts, F1 and F2, and we want to argue that F1 & F2 combine into a cumulative case for one theory (H1) over another (H2). How to do? Let’s do this in plain English, taking it one step at a time. The first step should be obvious.
(1) Fact F1 favors theory H1 over H2.
The next step is where I’ve seen a lot of people make mistakes. They will argue:
(2) Fact F2 favors H1 over H2.
The problem is that this approach doesn’t connect or link F1 and F2 in the needed way for a cumulative case. Instead, they should argue:
(2′) Given that fact F1 is true, fact F2 favors H1 over H2.
If both (1) and (2′) are true, then F1 combines with F2 to make a cumulative case for H1 and against H2.
But Neil is correct in that most usages of cumulative cases (as described as "cumulative cases") deal with apologetic claims (either for theism or atheism), which was a surprise to me.
ETA Looking at more examples, it seems the commonality in the use of cumulative cases is around questions of philosophy. So are arguments using Bayes Theorem "cumulative cases" or not? I'm not sure. BT seems to fit the definition though.
It is really important, in life, to concentrate our minds on our enthusiasms, not on our dislikes. -- Roger Pearse
Here's an question: Why did the Roman Empire decline and fall?
Like the historical Jesus question, this is a very, very politicized question that each generation tends to answer in line with its own ideological interests. It is also, I think, a question that will only yield to a cumulative case. It is not that "smoking guns" or "sexy explanations of everything" haven't been tried, and some of them even gain widespread acceptance for a time, but I get a sense from the recent scholarship on the subject (what admittedly modest amount I have studied it) that scholars these days are piecing together archeological evidence (say, things that show evidence of (de)population, (less) wealth, military strength, or plague), combining that with the scanty literary record and considering co-incidental factors (say, "barbarians and Persians got stronger" at the same time as "plague and provincial depopulation happened") that combined are enough to take an empire down.
All that is considerably less sexy than an romantic, dashing, (and evidence-lite) sweeping 19th Century style view of history that finds grand trends of everything. But it is also more likely to yield answers when the evidence is problematic.
TO BE DELETED (what I wrote in the quote)!!!
I found the mathematical equations (which are only applicable to a spreadsheet), and the results are:
Two supporting pieces of evidence at 50% chance to be valid will give me 75% chance my conclusion is TRUE.
135 supporting pieces of evidence at 1% chance to be valid will give me 75% chance my conclusion is TRUE.
I could not get up to 200 because my spreadsheet cannot handle monstrously long numbers, but studying the pattern in the 120 to 150 range, for 200 pieces of evidence at 1%, that would put the conclusion at close to 90%.
I would only need 3 pieces of evidence at 50% to get close to this 90%.
Just a thought about cumulative case:
Is
a) having 200 independent pieces of evidence, estimated for each to have a chance to be 1% correct (regarding the overall desired conclusion)
the same than
b) having 4 independent pieces of evidence, estimated for each to have a chance to be 50% correct (regarding the overall desired conclusion)?
My feeling is b) would give strong support to that overall desired conclusion, but a) would not, not even any significant degree of support.
That can be visualized as follows:
For a) have 200 dices with each 100 equal facets, but only one facet showing "true". Roll the 200 dices. Then, add up the ones showing "true". Repeat many times.
For b) have 4 dices with each 6 equal facets (the minimum for a cubic & rollable dice), but only 3 facets (out of 6) showing "true". Roll the 4 dices. Then add up the ones showing "true". Repeat many times.
I would think an average of about 2% for a) and about 100% for b) is a reasonable assumption for the two results.
I have no doubt that concept can be put in a mathematical equation. But, of course, rating the pieces of evidence & demonstrating their independence between each other is totally another matter.
Cordially, Bernard
Last edited by Bernard Muller on Thu Feb 12, 2015 9:47 am, edited 1 time in total.
I believe freedom of expression should not be curtailed
cienfuegos wrote:I don't see where you have proposed different probabilities. How do these points change the numbers in the Bayesian calculations? A lot of people have criticized the Rank-Raglan prior, including me at first, but I think ultimately Carrier made his case on that(and a lot of critics just don't understand it).
So show us the numbers. How did each particular argument affect Carrier's numbers and how do your proposed changes alter it.
It's a principal problem with Carrier's approach. The objectivity of his Bayesian calculations is preceded by the subjectivity of assigning numbers. This result leads to his calculations, though objective themselves, being subjective because of the input. These calculations only tell you whether you made a right or wrong conclusion from the probabilities you guess a certain data point has, completely ignoring that these probabilities are drawn from not much more than thin air.
You may as well assign a probability of 0.99 to a historical or mythic Jesus from the get-go. Which is only marginally worse regarding the result and is also what people usually do. The positive aspect of the Bayesian approach is just that you keep your argumentation straight and that people immediately see if you neglect some obvious point.
