Proving History: Bayes's Theorem and the Quest for the Historical Jesus Hardcover – Apr 24 2012
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"[C]arefully exposes what happens when sound methodology meets biblical studies. . . . Proving History is a brilliant lesson in the proper proportioning of belief to evidence. Even minimal attention to Bayesian probability theory reveals just how much of Jesus scholarship confuses ‘possibly true’ with ‘probably true.’ The only miracle Richard Carrier has left to explain is why so few appreciate that extraordinary claims require extraordinary support."
-Dr. Malcolm Murray, Author of The Atheist’s Primer
"Carrier applies his philosophical and historical training to maximum effect in outlining a case for the use of Bayes’s Theorem in evaluating biblical claims. Even biblical scholars, who usually are not mathematically inclined, may never look at the ‘historical Jesus’ the same way again."
-Dr. Hector Avalos, Professor of religious studies, Iowa State University, and author of , The End of Biblical Studies
About the Author
Richard C. Carrier, an independent scholar with a doctorate in ancient history from Columbia University, is the author of Why I Am Not a Christian: Four Conclusive Reasons to Reject the Faith; Not the Impossible Faith: Why Christianity Didn’t Need a Miracle to Succeed; and Sense and Goodness without God: A Defense of Metaphysical Naturalism. He has also contributed chapters to The End of Christianity, edited by John W. Loftus; Sources of the Jesus Tradition: Separating History from Myth, edited by R. Joseph Hoffmann; The Christian Delusion: Why Faith Fails, edited by John W. Loftus; and The Empty Tomb: Jesus beyond the Grave, edited by Robert Price and Jeffery Lowder.
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The present volume argues, and argues quite persuasively that historians should employ Bayes's Theorem in their work and of course that includes work on the historical Jesus. Regardless of what you think about that subject, if you are a thinking person, I think you should read this book. If you care about how we know what we know and how likely your beliefs are to be correct you should read this book. In that regard it is excellent. It does have a fairly narrow focus but that focus is on something that has incredibly wide application.
I'm just a lay person interested in science, history, and philosophy among other things. I'm not a professor or specialist in any relevant fields. I found this book an incredibly helpful guide to rigorous thought. This book is definitely not for everyone. Sometimes the author talks too much, but the points are valid and you just need to work through them. This is not light reading, although it is written in a way to be accessible to intelligent readers. You must be willing to put in some work if you are not already well versed in the theory.
I can't wait for the follow up volume where Dr. Carrier actually applies all this to the subject of a historical Jesus. I've now read several of Carrier's books and seen him on some video clips. He's a very articulate man and always seems to have something brilliant to say. I admit that I'm a fan.
Highly recommended to serious readers.
Carrier does not set forth a view of the historical Jesus in this volume. Rather, his goal is to "present a new method that solves the problem... so progress can finally be made in the field of Jesus studies (p.15)". His new method is Bayes's Theorem (BT). One need not be an expert in mathematics or even statistics to follow along in the book: a basic understanding of multiplication, division, and fractions will suffice. However, even if your eyes tend to glaze over once Carrier begins to plug in some numbers in the formula, he still adequately conveys conceptually the arguments he is defending.
The main arguments that an amateur reader like myself can take away from Carrier's work are the following:
1) Contrary to what some (most recently, Bart Ehrman) say, history IS a science. "The fact that historical theories rest on far weaker evidence relative to scientific theories, and as a result achieve a far lower degree of certainty, is a difference only in degree, not in kind (p 48)." Thus, when evaluating historical claims and evaluating when the evidence should cause us to believe the claim, history, like science, is Bayesian.
2) Precision is not necessary to apply Bayes's Theorem. "Rules of thumb" will work just fine and be accurate enough for all historical inquiry.
3) Rather than increasing the amount of disagreement among historians due to quibbling over probabilities, BT will actually expose historians' biases and force them to argue for their premises.
4) All current historical methods in Jesus studies (arguments from evidence, arguments to the best explanation, etc..) reduce to Bayes's Theorem. In other words, whether or not something really is the "best explanation" can only be determined by running the probabilities through BT.
5) Current historical Jesus criteria have failed to solve what Carrier calls the "Threshold Problem". In other words, do any of the historical Jesus criteria (dissimilarity, embarrassment, multiple attestation, etc...) in and of themselves tell the historical whether the claim is to be believed? Only by applying BT, argues Carrier.
I highly recommend this book. You need not agree everything Richard Carrier has ever written to recognize that this work is a great contribution to the field of Jesus studies. If it does nothing more than force current scholars in the field on all sides of the debate to abandon the unwarranted certitude many employ to their conclusions and put all arguments through the same, objective test then this book will have served its purpose.
Carrier wastes no time before describing the moribund state of current historical Jesus studies. He cites various analyses which conclude that the recent `method of criteria' fail to produce a consensus. "The entire field of Jesus studies has been left without any valid method". The reason being either invalid criteria, invalid application or a `Threshold Problem' involving the number & weight of criteria and their significance.
THE CONSEQUENCE of this FAILURE is the current multiplicity of plausible Jesus types which abound in the literature. Carrier cites Jesus the Jewish Cynic Sage, Rabbinical Holy Man (or Devoted Pharisee, or Heretical Essene, etc.), Political Revolutionary, Zealot Activist, Apocalyptic Prophet. Messianic Pretender, as well as many other more exotic contenders.
"When everyone picks up the same method, applies it to the same facts, and gets a different result, we can be certain that that method is invalid and should be abandoned."
THE SOLUTION is the application of Bayes's Theorem (BT).
CHAPTER 2: THE BASICS
In WHY HISTORY REQUIRES EXPERTISE, Carrier describes four stages of historic analysis. Textual, literary, source and only last, is historical analysis proper. He then sets down a set of 12 core epistemological assumptions. THE AXIOMS OF HISTORICAL METHOD and discusses them in turn with some illustrative examples mostly derived from ancient times. These are then followed by 12 RULES OF HISTORICAL METHOD which are simply stated without individual comment.
CHAPTER 3: INTRODUCING BAYES'S THEOREM
WHEN DID THE SUN GO OUT? is an interesting example from the Gospels that Carrier analyses both historically and scientifically and then contrasts with a similar hypothetical event from 1983, for the purpose of extolling the different evidentiary probabilities involved. He also introduces the question of lack of evidence or silence from expected sources. Finally concluding that this is "a slam-dunk Argument from Silence" with respect to the nonhistoricity of the Gospel account.
FROM SCIENCE TO HISTORY begins the discussion of BT: "all valid historical reasoning is described by Bayes's Theorem". A gentle nonmathematical exposition canvassing a variety of historical scientific disciplines to purely historical. WHAT IS BAYES'S THEOREM applies more lubricant until at last pg.50 exposes the reader to "this rather daunting equation:", which I shall spare you. There follows immediately a translation "into English" and several pages of explanation where prior probability and what Carrier refers to as consequent probability are discussed.
A BAYESIAN ANALYSIS OF THE DISAPPEARING SUN re-examines the Gospel and 1983 (now assumed fully observed) examples by employing the `daunting equation' in thorough detail with the unsurprising result that the simple arithmetic yields Gospel event 0.01%, 1983 99.9%. As an introduction to BT methodology this is a painless, interesting and instructive exercise and should cause no problem for anyone with a genuine interest in the subject.
WHY BAYES'S THEOREM? further discusses the advantages of employing this methodology and then answers some initial reservations which Carrier has clearly been exposed to over the years.
But what has math to do with history? But math is hard. But history isn't that precise.
Carrier's reply to these legitimate concerns are fulsome and reasonable as he patiently explains the whys & wherefors.
MECHANICS OF BAYES'S THEOREM is "the most math-challenging section of the book". In truth there is very little more in the way of equations, and even then merely a mild extension of the forgoing. Rather there follows an extensive exposition of usage. That is mechanics of prior probability, mechanics of consequent probability, a Venn diagram, consequent probability and historical contingency, the role of conditional probability, the problem of subjective priors, arguing a fortiori, mediating disagreement and a canon of probabilities.
CHAPTER 4: BAYESIAN ANALYSIS OF HISTORICAL METHODS
As specified Carrier proceeds to use BT to analyse;
The Argument From Evidence (AFE)
The Argument to the Best Explanation (ABE)
The Hypothetico-Deductive Method (HDM)
Then is given a Formal Proof of Universal Applicability which is quite brief, except for the caveats and explanatory discussion, but eventually all is well. Next follows,
Bayesian Analysis of the `Smell Test', and the most fun of all,
Bayesian Analysis of the Argument from Silence.
CHAPTER 5: BAYESIAN ANALYSIS OF HISTORICITY CRITERIA
Carrier identifies "at least eighteen distinctive criteria", such as Dissimilarity, Embarrassment, Coherence, etc.
Embarrassment receives the most extensive treatment and falls under the BT axe for a variety of reasons. There follows a SPECIFIC INADEQUACY OF THE CRITERION OF EMBARRASSMENT involving a detailed examination of; Jesus' crucifixion by Romans, Jesus birth in Nazareth, John's baptism of Jesus, Jesus' ignorance of the future, Did Jesus know he was the Son of Man?, Jesus betrayal by Judas Iscariot, And so on ...
The remainder of the criteria fall with increasing rapidity, as do some OTHER CRITERIA.
However, a BAYESIAN ANALYSIS OF EMULATION CRITERIA survive (with modification) and prove most instructive when "Daniel in the lion's den" becomes "Jesus in the empty tomb".
Finally in BAYESIAN DEMONSTRATIONS OF AHISTORICITY "... at first glance it seems surely "Jesus existed" would win out as the most probably hypothesis on BT. In my next volume (On the Historicity of Jesus Christ) I'll reveal that on second glance, that conclusion is not so obvious, and might even be wrong".
CHAPTER 6: THE HARD STUFF
The final chapter addresses "deeper issues regarding the application and applicability of Bayes's Theorem generally". It contains some new maths but is mostly concerned with technical aspects of BT and its use in historical research.
The book is well written with a clear and logical progression of argument. The mathematical development could hardly be more benign and there are many illustrative and entertaining examples to elucidate the details of both methodology and application. A brief Appendix provides a handy summary of the maths. The extensive notes constitute more than 10% of the book and there is a useful index. From a technical and logical perspective it very adequately covers the ground required to underpin Carrier's next volume.
We can begin by looking at Dr. Carrier's "proof" about BT (beginning on p. 106), which starts as follows:
“P[remise] 1. BT is a logically proven theorem.
P2. "No argument is valid that contradicts a logically proven theorem.
C[onclusion] 1. Therefore, no argument is valid that contradicts BT."
Granting the truth of this (it's false, but this isn't the time to get into the difference between a sound and a valid argument), then the only historian I know of who actually gives arguments that contradict BT is Dr. Carrier. To understand why, let's look at Dr. Carrier's own sources.
On p. 50 Dr. Carrier refers the reader via an endnote (no. 9) to “several highly commendable texts” on BT. The one he states gives “a complete proof of the formal validity of BT” is Papoulis, A. (1986). Probability, Random Variables, and Stochastic Processes. (2nd Ed.). I don’t have the 2nd edition, but I do have the 3rd & 4th and as this proof is trivial I really could use any intro probability textbook. Papoulis begins his "complete proof of the formal validity" (as opposed to proof of informal validity? or incomplete proof?) by defining a set and probability function for which the axioms of probability hold. A key axiom is that any set of possible outcomes must sum or integrate to 1 (simplistically, for those who haven't taken any calculus, integration is a kind of summation). For example, imagine an individual named "Anna" is drawing cards from a pack. Lets imagine that
"It wasn't the Jack of Diamonds
Nor the Joker she drew at first
It wasn't the King or the Queen of Hearts
But the Ace of Spades reversed"
The probability of drawing the card she did is 1/53 (it includes the joker). This is true for the other 52 cards as well. The probability that she would draw a card from the deck that was in the deck is 53/53 or 1. For a "regular" deck, it would be 52/52 or 1 (no joker). This is intuitive and obvious, but the important point is that it also follows from the fact that a normal deck has 52 cards and the probability for drawing any one of them is 1/52, hence the probability of drawing a card is given by the sum of the probabilities of drawing each individual card, or 1/52 summed 52 times. Dr. Carrier’s appendix (p. 284), says of something related to probabilities that they “must sum to 1", just like the possible outcomes of drawing cards from a deck do (52/52=1). What he apparently doesn’t understand is why they "must" do so or what this entails. It means that in order to use BT to evaluate how probable some outcome, result, historical event, etc., is, one must consider every single one.
Dr. Carrier wishes to use BT to evaluate the probability that particular events occurred ~2,000 years ago. For example, on pp. 240-42 he considers the possibility that Jesus was a “legendary rabbi” in terms of the “class” of legendary rabbis and information we have on such a class. Do we know how many such rabbis existed and who they were (the way I know how many cards there are in a deck as well as the "name" of each, e.g., "ace of spades")? No. Ergo, Bayes' Theorem is unusable.
There is another basic property of BT Dr. Carrier seems to have missed. As Papoulis clearly states, BT is only valid for events/outcomes that are mutually exclusive. A simple example of mutual exclusivity is the coin toss. If I toss a coin, the probability that I will get heads and the probability that I will get tails are "exclusive" because I cannot get BOTH heads AND tails given one toss.
Often, both of these requirements (“must sum to 1” and mutual exclusivity) are given together: the set of outcomes must be collectively exhaustive and mutually exclusive, or BT can only be used if
1) all possible outcomes are known
2) one and only one outcome of this set of all possible outcomes can occur.
This makes BT useless for most purposes, including historiography. However, Dr. Carrier isn’t really using BT. As his references show (as well as his description of BT throughout his book), he is actually using something called Bayesian inference/Bayesian analysis. However, this renders almost completely irrelevant every conclusion in his "proof" about BT, because he isn't using it. Thus, by misusing and inaccurately describing what BT is, he comes about as close to contradicting it as a historian can. Also, because he conflates BT with Bayesian inference/analysis, it doesn’t matter if “BT is a logically proven theorem” as he isn't using BT. Finally, there is no “complete proof of formal validity” for some Bayesian inference/analysis “theorem” Dr. Carrier could use in place of the first premise in his proof.
We may also wonder how much Dr. Carrier actually understands about the nuances of Bayesian methods, as in his presentation of the “frequentist vs. Bayesian” debate. To keep things simple, let’s just say that this is an ongoing debate arguably going back to Thomas Bayes but which is definitely over a century old.
Dr. Carrier describes the dispute as follows: “The debate between the so-called ‘frequentists’ and ‘Bayesians’ can be summarized thus: frequentists describe probabilities as a measure of the frequency of occurrence of particular kinds of event within a given set of events, while Bayesians often describe probabilities as measuring degrees of belief or uncertainthy.” (p. 265). We must grant Dr. Carrier the requisite lenience given his necessary simplifications, because were this really his view it would be laughably wrong:
“Frequentist statistical procedures are mainly distinguished by two related features; (i) they regard the information provided by the data x as the sole quantifiable form of relevant probabilistic information and (ii) they use, as a basis for both the construction and the assessment of statistical procedures, long-run frequency behaviour under hypothetical repetition of similar circumstances.”
Bernardo, J. M. & Smith, A. F. (1994). Bayesian Theory. Wiley.
"Undoubtedly, the most critical and most criticized point of Bayesian analysis deals with the choice of the prior distribution, since, once this prior distribution is known, inference can be led in an almost mechanic way by minimizing posterior losses, computing higher posterior density regions, or integrating out parameters to find the predictive distribution. The prior distribution is the key to Bayesian inference and its determination is therefore the most important step in drawing this inference. To some extent, it is also the most difficult. Indeed, in practice, it seldom occurs that the available prior information is precise enough to lead to an exact determination of the prior distribution, in the sense that many probability distributions are compatible with this information...Most often, it is then necessary to make a (partly) arbitrary choice of the prior distribution, which can drastically alter the subsequent inference."
Robert, C. P. (2001). The Bayesian Choice: From Decision-Theoretic Foundations to Computational Implementation (Springer Texts in Statistics). (2nd Ed.). Springer.
The “frequency” part of “frequentist” does have to do with kinds of events, but frequencies are the measure of probability, not the reverse. To illustrate, consider the “bell curve” (the graph of the normal distribution). It’s a probability distribution. Now imagine a standardized test like the SATs which is designed such that scores will be normally distributed and have this bell curve graph. The bell curve is the graph of a probability function (technically, of a probability density function or pdf), and it is formed by the frequency of particular scores. We know that it is very improbable for a person’s score to be perfect or near perfect, and the graph shows this because the right-hand end is nearly flat, indicating scores close to perfect are very infrequent outcomes.
What’s key is that the data are obtained and analyzed but the distribution is only used to determine whether the values the analysis yielded are “statistically significant". Bayesian inference reverses this, creating fundamental differences. The process starts with a probability distribution. The prior distributions obtained represent uncertainty and make predictions about the data that will be obtained. Once the new data is obtained, the model is adjusted to better fit it. This is usually done many, many times as more and more information is tested against an increasingly more accurate model. The key differences are
1) the iterative process
2) the use of models which make predictions
3) the use of distributions to represent unknowns and (in part) the way the model will “learn” or adapt given new input.
So why don’t we find any of this in Dr. Carrier’s description of Bayesian methods? Why do we always find ad hoc descriptions of “priors”? Because Dr. Carrier wants to use Bayesian analysis but apparently doesn’t understand what “priors” actually are or how complicated they can be in even simple models:
“In many situations, however, the selection of the prior distribution is quite delicate in the absence of reliable prior information, and generic solutions must be chosen instead. Since the choice of the prior distribution has a considerable influence on the resulting inference, this choice must be conducted with the utmost care.”
Marin, J. M., & Robert, C. (2007). Bayesian Core: A Practical Approach to Computational Bayesian Statistics. (Springer Texts in Statistics). Springer.
“While the axiomatic development of Bayesian inference may appear to provide a solid foundation on which to build a theory of inference, it is not without its problems. Suppose, for example, a stubborn and ill-informed Bayesian puts a prior on a population proportion p that is clearly terrible (to all but the Bayesian himself). The Bayesian will be acting perfectly logically (under squared error loss) by proposing his posterior mean, based on a modest size sample, as the appropriate estimate of p. This is no doubt the greatest worry that the frequentist (as well as the world at large) would have about Bayesian inference — that the use of a “bad prior” will lead to poor posterior inference. This concern is perfectly justifiable and is a fact of life with which Bayesians must contend...We have discussed other issues, such as the occasional inadmissibility of the traditional or favored frequentist method and the fact that frequentist methods don’t have any real, compelling logical foundation. We have noted that the specification of a prior distribution, be it through introspection or elicitation, is a difficult and imprecise process, especially in multiparameter problems, and in any statistical problem, suffers from the potential of yielding poor inferences as a result of poor prior modeling.”
Samaniego, F. J. (2010). A Comparison of the Bayesian and Frequentist Approaches to Estimation. (Springer Texts in Statistics). Springer.
The “stubborn and ill-informed Bayesian” is in a much better position than Dr. Carrier. Dr. Carrier has conflated BT with Bayesian analysis and mischaracterized the distinctions between the Bayesian and frequentist approaches. Instead of prior distributions his “priors” are best guesses. Instead of real belief functions we find “here’s what I believe”. No considerations are given to the nature of the data (categorical, nominal, and in general non-numerical data require specific models and tests, Bayesian or not). Whether all this is due to simplification for his audience or no, it cannot serve as a foundation for any sound methods.