Gospel Traditions: The Spreadsheet

After the discussion in the last post about the authorship of the Gospels, I've created a spreadsheet model, in Open Office format, to illustrate the probability calculus for whether the Gospels were written by Matthew, Mark, Luke, and John.  This post probably won't make much sense if you haven't read the comments on the previous one; if you're not a mathy person just skip it.  The spreadsheet file is at the bottom of this post.

It's just a model: I don't claim to have incorporated every possible effect that could be important.  I'm just trying to illustrate how the short chains of oral transmission to Papias and Irenaeus can provide significant information.  This is a dynamic spreadsheet, so if you change one box the rest of the numbers will change accordingly.  The idea is that if you don't accept my numbers, you can change it yourself to see what happens, rather than just griping at me that it should be different.

I've used the following hypothetical likelihoods for each Gospel/early writer to be pseudonymous instead of genuine, from the perspective of a person who isn't sure whether or not to believe Christianity:

Matthew: .1
Mark: .01
Luke: .001
John: .03
Papias: .05
Irenaeus: .001
Eusebius: negligible

This is based on the names attached to the documents, as well as internal evidence and the fact that they were received as genuine by the Church, but not yet taking into account the testimony of Papias through John (concerning Matthew and Mark), as quoted by Eusebius, and Irenaeus disciple of Polycarp disciple of John (for all 4 gospels).

The odds for Matthew are higher than the others because that is the only one where there were significant arguments for pseudonymity, instead of just arguments that it could have happened.  For me, the actual names written on the documents, and their acceptance by the church, are better evidence than any of the evidence against, hence the numbers above.  Papias is more likely to be pseudonymous since other than the few fragments preserved by Eusebius, we have to rely on the judgement of the early church about this.

I haven't taken into account possible lack of independence between the pseudonymity of the 4 gospels.  Although the Gospels do incorporate text from each other, they were almost certainly written by different individuals, so they aren't strongly dependent.  Nevertheless, there's some probability dependence here in their later cultural acceptance by the Church.  If you want to consider different odds for their dependence, you could put that in by hand in the "scenarios" section at the bottom (where e.g. "13" would mean the probability of the  1st and 3rd Gospels are pseudonymous, relative to the probability of all of them being genuine.)

In accordance with the arguments here, I've assigned odds of .001 per century until the first branch point for each document.  I've assumed Eusebius had only one copy of Papias, and I just guessed 4 centuries for Irenaeus since I couldn't figure out the first branch point for him from what I could find online.  I'm pessimistically assuming that if an textual corruption has occured, it makes the entire document unreliable.  I'm also assuming that any given nongospel writer has a .01 chance of being totally unreliable due to e.g. deliberate deception.

Finally, I'm assuming that for any two people related by an oral testimonial link, there's a .05 chance that a given fact about authorship will be garbled.  I'm assuming optimistically there was only one John and that Papias interviewed him directly, but this is balanced by not including any of the numerous other chains back to the apostles which Papias cites.  I'm giving the garbling more odds than any other form of error, but unlike the other errors I'm assuming that because this is inadvertent, if one piece of data in the document is garbled, the rest are all unaffected.

(In order to make the math easier when considering multiple gospels, I had to break out the total errors, the garbling errors and the nongarbling errors into three separate rows).

I haven't included the possibility that we might be wrong about the chains of testimony themselves, but you're free to play around with inserting extra people or changing their dependence or such.

I got the following probability odds for the Gospels being pseudonymous:

Matthew: .007
Mark: .0007
Luke: .00017
John: .0036

And for different numbers of gospels being genuine, the probability price you pay is about 10^{-2} for one Gospel being pseudonymous, 5 \times 10^{-5} for two, 3 \times 10^{-7} for three, and 3 \times 10^{-10} for all four.  This is before considering prior probabilities.

Interdependence between the four Gospels will make these last figures smaller, but I think any reasonable model will have some significant suppression of probability there.

All right then, here it is.  Enjoy!


UPDATE: Replaced incorrect "genuine" with "pseudonymous" above.

UPDATE 2: Fixed a bug in the spreadsheet.  See the comments section below.  This doesn't change things much for the numbers I provided, but might affect things if you change the input assumptions.  See oldspreadsheet to look at the old version.

Posted in Theological Method | 6 Comments

The Gospels Aren't Anonymous

A common criticism of the Gospel accounts is that they are of low historical value because they are anonymous.  This is based on the observation that the authorship is never explicitly mentioned in the main body of the texts of the Gospels attributed to Sts. Matthew, Mark, Luke, or John.

If we use this criterion, then as far as I can tell from briefly sampling my own library and prior reading, the works attributed to these authors also seem to be anonymous:

  • Aristotle's Poetics
  • Plato's Republic
  • Aristophanes' Birds
  • Livy's The Early History of Rome
  • Tacitus' The Annals of Imperial Rome
  • Shakespeare's Hamlet
  • Jane Austen's Pride and Prejudice
  • Kant's Prolegomena to Any Future Metaphysics
  • C.S. Lewis' Mere Christianity

Clearly this criterion is a silly one.  It is customary, both in ancient and modern times, for the authorship of a work to be stated, not in the main body of the text, but in the byline written in the title header, or else on the outside of the book.  The author may choose to begin his book by introducing himself by name to the reader (especially if there is a preface), but this is not the only way that an authorial name can be attached to a book.  It's certainly not the primary method we use today.  We're more likely to find the author's name on the spine or title page than in Chapter One.

According to this apologetics website, author names in the first century were typically written on a tag which was attached to the outside of the scroll, and possibly also in the title at the beginning or end of the document.  I have checked the earliest complete manuscript of the New Testament with my own eyes (via an online scan) and I find that the authors names are clearly stated on the manuscript itself, in a header (e.g. "The Gospel of Mark") which directly proceeds the text itself.  There is no evidence that an earlier form of the Gospels ever circulated which did not contain the author name.  In other words, it is reasonable to assume that the texts were probably written by their stated authors, for roughly the same reason that we assume that the modern-day books on your shelf were probably written by their stated authors.

A better set of definitions is as follows:

  1. A document is anonymous if it was first published (i.e. distributed to a broad audience) in a way that did not have the name attached to the book.  Later readers may speculate about who wrote it, and the text may thereby acquire a traditional authorship, but it is still anonymous as originally written.  Otherwise, if there is a stated author:
  2. Call it genuine if it was really written by that author;
  3. Call it pseudonymous if it was really written by someone else.

Pseudonymous books are not anonymous.  They have a name clearly attached to them, it's just that the name is either fraudulent or fictitious (depending on whether the author intended the reader to be deceived).

A book of the New Testament which actually is anonymous is the Epistle to the Hebrews.  The author was clearly known to the original readers, but in its published form it says nothing about authorship.  Although it is traditionally attributed to St. Paul, the text itself nowhere says this, and there are significant differences in writing style (as I know from personal experience trying to translate it one time—the Greek is difficult and uses classical words; I only got four verses in).  The later history of Hebrews was exactly as you would expect for a truly anonymous document: discussion and disagreement about whether it was really written by Paul or several other proposed candidate authors, and uncertainty about whether or not it should be included in the canon.  From the 4th century on there was  a near consensus that the letter was written by Paul, but this consensus fell apart after the Reformation.  By contrast, the other 13 Pauline epistles have Paul's name clearly stated in the salutation, and are therefore each either genuine or pseudonymous.

Now, it's certainly logically possible that the Gospels were all originally anonymous.  However, achieving this would require a complicated historical process.  It would require believing that for each gospel 1) the gospel was actually written by some person of sufficiently little importance that their name was not recorded, 2) so that the document was published and first circulated without any names attached, 3) then, at some later time, someone speculated that it was really written by the proposed author, 4) belief in this authorship became widespread without any recorded dissent, and 5) all copies of the text which have descendants were modified to include the title.  This has to have happened, not once, but four times.

The Gospel of John is a special case, because the author is mentioned in the text specifically, although not by name, by the title "the disciple whom Jesus loved".  There's been a lot of ink spilled about the identity of this person, but I think much of it is an effort to deny the obvious.  The scenes with this person strongly indicate that he was one of the Twelve Apostles.  Moreover, the statement of authorship in John 21:24, "This [disciple whom Jesus loved] is the disciple who testifies to these things and who wrote them down. We know that his testimony is true," doesn't make much sense if the identity of the apostle was intended to be mysterious.  But it all makes sense if the Gospel was originally circulated with the name "John" attached to it (as it is in every copy we have).  This isn't a watertight proof by any means, but I think it is significant when combined with the other evidence.

So I think the skeptic would do better to say that the Gospels are pseudonymous.  However, they have to pay a probability price for this too.  In addition to bucking the standard presumption of genuine authorship, we also have testimony from other early documents.  This includes the writings of St. Papias (as preserved in Eusebius) and St. Irenaeus explicitly stating the authorship of the Gospels, and even providing details about how they were written (e.g. that Mark was based on St. Peter's preaching).  These two individuals are connected by very short chains to the original generation of apostles.  In fact, both Irenaeus and Papias assert that Papias was a disciple of John, although some scholars try to argue (in my view unconvincingly) that this was a different John than the apostle.  In any case, he seems to have been related by multiple two-step chains to the original apostles, so it would be hard for a pseudonymous work to fool him.  Irenaeus, in turn, was a disciple of St. Polycarp who was a disciple of John, again a very short chain of transmission.

This is excellent evidence for the traditional authorship of the Gospels, although it may be a more efficient investment of "improbability" for the skeptic to dismiss Papias and Iranaeus rather than accept the fact that two of the four Gospels were written by direct eyewitnesses to the events in question.  (The question of whether apostolic authorship is consistent with textual evidence of date and style will have to wait for another post.)

I'm reminded that at one time it was popular for scholars to believe that Homer never really existed.  Somehow, contrary to all modern experiences of authorship, they thought that the Illiad and Odyssey must have precipitated out of the folk-consciousness, without any definite author.  That's pretty crazy.  But let's suppose that there was some definite author, and we simply don't know what his name was.  Seem plausible?  Well, this would require believing that a class of professional singers, specially trained to memorize over 27,000 lines of poetry, were completely unable to remember a single word accurately.

UPDATE: I should clarify that Papias only refers to the circumstances of the authorship of Matthew and Mark; he does not (at least in the fragment preserved by Eusebius) discuss the authorship of the Third and Fourth Gospels.  Irenaeus, on the other hand, bears witness to all four canonical Gospels.

Posted in Theological Method | 22 Comments

Textual Criticism

A lot of people think that the New Testament documents must be unreliable because (like all published texts from the ancient world) they are based on translations of copies of copies.  So how can we know whether the original version of the text is the same as what's printed in our Bibles?  Isn't it like a game of telephone, where eventually the message becomes distorted beyond recognition?

This issue came up in the comments section of the previous article, but I think it is important enough to justify its own post, especially since there are interesting connections to evolutionary biology.  (Yes, really!)

First let's consider translations.  All modern translations of the Bible are based entirely on direct translation from the original Greek (for the New Testament) or Hebrew (for the Old Testament).  They are not translations of translations (except insofar as the original texts may already be translations of e.g. Jesus' words from Aramaic into Greek).  Now I won't deny that the translation into English may obscure or distort features of the original language.  As the Italian proverb has it, traduttore, traditore.  However, this problem is mitigated by the fact that there exist lots of different scholarly translations of the Bible with different agendas, and we can check them against each other.  Or, if we are really dedicated, learn ancient Greek or Hebrew ourselves.  So let's set aside the translation issue, and just consider the question of whether the Greek text used by the translators is the same as the original Greek text.

In fact, we can be extremely confident that—with a couple important exceptions that I'll discuss below—the Greek New Testament used to make your Bible is in almost all important details the same as in the orignal text penned by the author (or more likely, dictated to a scribe).  And, in cases where the text isn't certain, we almost always know it isn't certain, and a scholarly edition of the Bible will point out the possible variants in a footnote.

So how do we know this?  The answer is using a field of study called "textual criticism", a procedure which is sufficiently methodical and precise that it almost qualifies as Science rather than History.  The idea is that manuscript copies don't just form a straight line, they form a tree with branches.  That's because a single manuscript may be copied more than one time, and hence have multiple descendents.  The root of the tree is the original manuscript (the autograph), and the branches are the various families of manuscripts which descend from it.

It's a lot like the "Tree of Life" in evolutionary biology.  The different species we see can be classified into a tree structure based on how recently they diverged from their most recent common-ancestor species.  By comparing the DNA of different species, you can identify at what points in history various specific mutations must have arisen (e.g. if the mutation affects chimps and humans but not bonobos, it must have occured after the proto-bonobos split off, but before proto-chimps and proto-humans went their separate ways).

Well, it's the same with manuscripts of the New Testament.  From time to time, copying errors were introduced, either accidentally or on purpose.  These errors propagated into the copies of that manuscript, but unrelated manuscripts were unaffected.  What that means, is that the only way an error could go undetected, is if it occured before the first branch point such that we have surviving manuscripts for both branches.  (Or if there was a giant political conspiracy to change every copy of the New Testament in existence, but there's no evidence of any such thing, and once the Chrisitan church had enough centralized political power, the manuscripts were circulated far too widely for this to be feasible.)

If the error occurs after the first branch point, we'll have different manuscripts which say different things in some place.  It's a separate and harder question to figure out which of these variants was the original.  One could just count them (the "majority text") but this might not work since the branch with the error could have proliferated more widely than the others.  Instead, most scholars try to reconstruct the family tree, and weight more heavily manuscripts which don't seem to be closely related to each other (the "critical text").  However, since these variants usually don't affect major points of Theology, mostly I only care that we can identify the passages where there might be a discrepency—in these cases, pick whichever variant is your favorite, and I'll argue based on that edition of the New Testament.

In the case of the New Testament documents, the first branch point has to have been incredibly early.  That's because the New Testament was extremely popular: we have thousands of different manuscripts before the printing press, some of which are quite early, starting with fragments in the early 2nd century, and going onwards from there.  The runner-up is Homer, who was also well-liked, and therefore we have a few hundred copies.  By comparison, for most other ancient Greek and Roman texts (e.g. Aristotle) we're happy if we can find a few dozen copies from the high medieval ages.  The results of textual criticism are usually regarded as reliable in these cases; how much more so when the available "fossil record" is so much more detailed!

Let's try to quantify the probabilities a bit more closely.  I'm not a professional textual critic, so I'm only going to attempt a rough back-of-the-envolope estimate here.

The only way that an error could sneak in unnoticed is if it occured before the first branch point.  Consider as an example St. Paul's letter to the Galatians (which nearly all scholars think was written by Paul himself).  Let us define the "Publisher" of Galatians as the first person who arranged the text to be copied many times, where "many" means enough times to ensure that at least two copies have surviving descendents.  In the case of a famous person such as Paul, whose writings would be in high demand in many churches, it seems likely sufficient to copy the manuscript a mere 5 or 6 times, and then send them to geographically diverse regions in the Roman empire.  Now, the most likely person to be the Publisher is Paul himself.  However, we cannot rule out that the autograph was copied once or twice, maybe even three times, before the Publication.  (More than this seems unlikely to me; why would people be interested enough in a text to copy it once, several times, but not interested enough to distribute multiple copies?)

To estimate the rate at which copying produces errors, let's consider the Textus Receptus (the line of descent which gave us the King James Version), circa 1500.  Let's suppose this is about 50 copies removed from the first branch point (that's about one copy per 30 years)  There are about 8,000 verses in the New Testament.  Of these, two large sections of the Gospels, the longer ending of Mark (12 verses) and the Woman caught in Adultery in John (12 verses), were probably not in the original manuscripts (although they are early enough that they may well date back to the first century).  In addition to these, Wikipedia lists 100 verses (I counted, that's the exact number) with major discrepencies between the Textus Receptus and the accepted critical text.  There would be a lot more if we counted minor variants, but since most of the "major differences" on this list don't significantly affect interpretation, I'll draw the line here.

Putting all this together, we arrive at a "major error rate" of about .0003 per verse per copy.  If we allow for 3 copies before the first branch point, you still have to pay about 10^{-3} in probability ratios for each (noncontiguous) verse of the New Testament you'd like to dynamite.  Since the most important claims in the New Testament are typically found in several places, that's a very inefficient form of skepticism.

So the evidence suggests that a modern critical New Testament is at least about .999 accurate to the original text.  To be clear, this doesn't tell us anything about whether the books were really written by their traditional authors, or whether the original claims are true; that's a subject for other posts.  It does tell us, though, that the original claims were in fact made, by some first century person (whoever wrote the book in question), who was able to pass himself off to some audience, as a person with apostolic credentials.

Posted in Theological Method | 8 Comments

Christianity is True

Last time I wrote a long discussion of what I think is the best evidence for and against Christianity.

When it comes to the historical testimony to the Resurrection, I analyzed the data in terms of a set of minimal facts which I think even a "liberal" biblical critic ought to accept.   By the "liberal" view I mean someone who thinks that the Gospels, Acts, half the letters of Paul, and the general epistles were pretty much all written in the closing decades of the 1st century, by people other than their traditional authors—but that nevertheless the texts are based on identifiable earlier sources which go back to some real, and much earlier, written or oral traditions about Jesus.  (Because some parts of the Gospels are obviously rooted in actual historical events.)

This was a concession, not my actual beliefs: in reality I think this "liberal" view, despite its acceptance by the majority of secular biblical critics, is supported mostly by circular reasoning, and is more than a little reminiscent of the claim that Bacon wrote Shakespeare.  Nevertheless, if a good case can be made for Christianity even under such hostile circumstances, then obviously things would work even better on a more "conservative" view.  For a Bayesian approach based on a somewhat more conservative view of biblical criticism, see this paper by Timothy and Lydia McGrew (they argue for the conservative biblical view, but don't incorporate it into their probability analysis).

I also made no reference to the "inspiration" of the Scriptures by the Holy Spirit; rather I am treating them as ordinary historical documents containing what is claimed to be testimony.  Just to be clear, I do believe that the entire Bible is inspired by the Holy Spirit, and that therefore all of it is an important communication from God to us.  I disagree with fundamentalists in that I don't think the point of the Bible is to communicate scientific facts, or even matters of historical trivia.  For this reason it doesn't bother me that the Gospel accounts contain apparent contradictions: although most of these contradictions would probably be resolvable if we knew the full circumstances, it's not important to me whether if the documents contain mild inaccuracies regarding trivial matters, as all genuine eyewitness testimony does.

So now I have to analyze how I think the probability analysis actually goes.  To summarize the previous post, the main factors weighing against Christianity are:

(A1) prior probability due to specificity of Christian doctrine,
(A2) prior probability due to the weirdness of Christian doctrine,
(A3) prior probability due to Jesus being only one of billions of people.
(B) the Argument from Evil

while some of the factors weighing in favor are:

(C) the Fine-Tuning version of the Argument for Design.
(D) circumstantial facts of Jesus' life prior to his death, making him more likely to be the Messiah
(E) multiple testimonies to the Resurrection,
(F) modern-day miracles.

This is by no means a complete list, but I think it contains most of the highlights.  I've excluded some other arguments for Theism, and various Christian experiences, as less likely to be persuasive to someone coming from a skeptical point of view, even though I rate some of these things as significant.

Each of these factors leads to a shifting of the relative odds of Christianity : Naturalism.  (I'm assuming here that Naturalism is the main rival hypothesis to Christianity for skeptics of a "scientific" sort.)  Since these factors are approximately independent, each of them multiplies the relative odds by some number.  Since we aren't going to agree on precise ratios, I'm going to just estimate how persuasive these are to me, in terms of orders of magnitude (i.e. powers of 10).

I think that (B) and (C) are both good for about 2-4 orders of magnitude and therefore they approximately cancel each other out.  Each of them, despite being grounded in certain observable facts about the universe, is a controversial philosophical argument about metaphysics, and I think it's quite rare for any such argument to establish a conclusion with an error rate less than 10^{-3} or 10^{-4}.  You could say that I'm skeptical about Philosophy, but that's not really fair since shifting the odds by a factor of 1000 should count as a "success" even if this can be easily overwhelmed by more definitive evidence.  For similar reasons I don't think (A2) should be much less than 10^{-4}.

(A3) and (D) cancel each other out nearly exactly, because if Jesus is in a very small group of people more likely to be the Messiah, then it doesn't matter how many other people there are in the control group.   See the discussion of the prosecutor's fallacy.

(E) contains a factor which cancels out (A3) exactly.  That's because the Christian testimony is to Christianity specifically, and not to any other religion.  Whereas a made up religion is going to be selected from some large set of possible religions.  See the discussion of the Farmer Jones case in the comments of this thread.

The remaining part of (E) has to do with how plausible the testimony is, leaving aside its specificness.  Now the testimony of a single individual is has to be worth at least 2 orders of magnitude, because most of the time people don't lie about things.

Somewhat counterintuitively, the value of testimony towards implausible claims like miracle reports is actually somewhat greater than this, because people are less likely to make up claims that are implausible.  There are many people who claim to have crossed the street, but only a few people claim to be abducted by UFO's.  The number of people who claim to have seen people physically resurrected is quite low; certainly much less than 10^{-2} of the population.  To the extent that this effect exists, (E) also partially cancels (A2).  But this cancellation only works up to a certain point, since nothing says that the rate of false claims has to mirror the true a priori probabilities.  For the small fraction of the population who is willing to make absurd claims, there is no limit to how unlikely their assertions may be.  So some residue of (A2) may remain.

What remains to be considered in (E) is the gross improbability of seeing testimony as strong as this about anything false.  The multiplicity of eyewitnesses, the priority of the women, the apparent unanimity in the pre-existing group of Twelve (minus Judas), and the fact that they were willing to die for their testimony all weigh in its favor.  All of these facts are strongly supported by the historical data.  Their initial skepticism and doubt, mentioned explicitly in almost all of the Gospel accounts, weighs heavily in its favor, so the skeptic is probably better off trying to argue (somewhat less implausibly) that this was made up later to prove a point.  The lack of any reasonable ulterior motive also makes the testimony less likely given Naturalism.

St. Paul presents a special case: since we know he was previously hostile, his testimony is more or less independent of the others.  That in conjunction with the unusual circumstances surrounding his conversion is a factor of 10^{6} all by itself; the group testimony of the remaining apostles is at least 10^{8}, and that's rather generous, since on that hypothesis we'd expect to find about 100 similarly strong examples of false dramatic claims by large groups, in the most recent generation of 10^10 people alone.

Taking everything together I think that the implausibility of a set of testimony looking like (E) given Naturalism is at least 10^{-14}, and much lower if one takes a more conservative approach towards biblical criticism.  If this is right, then (E) pretty much overwhelms every other factor in play, and (F) is just icing on the cake.  Therefore, with high probability, the Resurrection seems to have really happened.  In other words, (some unspecified version of) Christianity is far more probable than Naturalism is.

Posted in Theological Method | 103 Comments

Let Us Calculate

In this post I am going to outline what I consider the most relevant evidence for and against Christianity, in the form of a Bayesian probability analysis.  Now, just to be clear in advance, I'm not even going to take seriously the Jesus-myth crank theory that Jesus never existed.  No reputable scholar, no matter how hostile to Christianity, believes this, and one can only believe it by completely discounting almost all of the available evidence, including from nonbiblical sources.  And even the most skeptical or "liberal" biblical critics usually assume that at least some parts of the Gospels go back to real information about Jesus.  So I'm going to assume that Jesus existed, and that we know at least some minimal facts about him and his immediate followers.

Consider the proposition Christianity*, defined as something roughly like the following: "There exists one benevolent God who came to Earth in the form of a particular unique human being, died, came back to life again in resurrected form, and then departed again".  Note that: 1) Christianity* differs from Christianity in that it does not indicate which particular human being this is, whereas of course Christianity says that Jesus is that person;  2) We can, if we choose, add additional Christian doctrines to the proposition Christianity*, but I am going to argue that this doesn't matter very much for the end probability result, so long as the additional doctrines are supported by empirical testimony of the kind that I described last post, and we are willing to accept some rate of error in the correctness of our theological interpretations.  3) Although the conditional

P(\mathrm{Christianity}\,|\mathrm{{Christianity\!\ast} + facts\,about\,history})

is high, because Christianity is as far as I know the only historically credible example of Christianity*, nevertheless the prior probabilities of Christianity and Christianity* are quite different since there's lots of people, and Jesus is only one of them.

Let us now consider the prior probability of Christianity.  The prior probability (A) consists of a product of 3 terms which come from the following kinds of improbabilities in Christianity:

_____A1) Implausibility due to the fact that Christianity* makes a series of several specific claims, whereas prior probability of various hypotheses needs to be distributed amongst all possible kinds of religious claims.  (The size of this factor depends on how many doctrines we include in Christianity*, but I've been arguing in the comments to this thread that the large factor in A1 will be exactly cancelled in cases involving testimony.)
_____A2) Whatever implausibility is inherent in Christianity* due to the fact that it postulates new entities, which behave in weird ways.
_____A3) Implausibility due to the fact that if Christianity* is true, there are billions of people who might have been the special person, and Jesus is only one such person.

Next we have to consider whatever posterior arguments there may be for, or against, generic Theism (with a benevolent deity).  In my opinion there's only one really good philosophical argument against, namely the Argument from Evil:

_____B) One would expect a universe with a benevolent God to have little or no evil in it.  (If we define Christianity as including the existence of evil, this factor would be incorporated into the prior probabilities instead, but that won't make any difference.)

There's several possible arguments for Theism, some good and some bad, but I think the best one is the Argument form Design, which can be made surprisingly precise in the form of the Fine-Tuning Argument.  Eventually I'd like to talk about this in much more detail, but a short summary is here:

_____C) Since, in our current understanding of physics, the overwhelming majority of possible configurations of the constants of nature do not permit life, life is much more probable if those laws were selected by a deity interested in producing life.  The most plausible naturalistic explanations have to propose additional, ad hoc entities, e.g. a gazillion extra universes with different constants, or unknown physical mechanisms with surprising properties.

Next we turn to the arguments for Christianity specifically.  One could consider arguments from religious experiences, but to avoid wallowing in subjectivity I'll focus on miracle claims, particularly the historical testimony to the Resurrection.

Now remember from the Proscutor's Fallacy how even rare events can be expected to occur so long as one draws from a large enough pool of candidates.  In the story I told, the prosecutor needs both the DNA evidence (something strongly correlated with guilt when taken on an individual basis) and also independent circumstantial evidence that the person who matched is among a small group of people more likely to have committed the crime.  We discussed how this circumstantial evidence may be quite weak, in the sense that it only suggests the possibility of guilt without in any way proving it.  Nevertheless, it can make the critical difference when combined with stronger evidence.

This is important because factor (A3) is large, and we need to be able to cancel it out by showing that Jesus is more likely to be special than other people.  This evidence has to be independent of the Resurrection claim, so it's best to look at features of Jesus' earthly life prior to that time.  Exactly how one goes about this will depend on which fraction of the New Testament one accepts, but even from a "liberal" perspective I think it's extremely probable that at least some of the following facts are true:

  • Jesus was Jewish (Note here that P(\mathrm{Judaism} |\mathrm{{Christianity\!\ast} + facts\,about\,B.\!C.\,history}) is moderate to high since Judaism is one of the few successful monotheistic religions, and if Judaism is true the Messiah should be Jewish—indeed, descended from King David, but that is more difficult to prove.)
  • During his lifetime Jesus claimed to be, either the Son of God in a unique sense, or the Messiah, or at the very least he made some highly unusual claims about himself which were understood by some as implying such things.
  • Jesus was one of the most provocative and insightful moral teachers who ever lived.  (Of course, if Jesus is the Incarnation of a benevolent God, we'd expect him to be the most insightful moral teacher.  But this is difficult to prove because there are disagreements about morality.  Nevertheless, judging based on the fact that his teaching is widely regarded as great even by saintly people from different religious traditions, it's fair to say he at least makes the top 100 list.)  Note also that even the "liberal" interpretation of New Testament scholarship accepts many of the sayings in the Gospels as really going back to Jesus, and even if some of the insightful sayings were made up by his disciples, those disciples would have had to become unusually morally insightful somehow in order to do that.
  • At least some people prior to Jesus' death believed he had performed miracles involving healings, exorcisms, and/or power over Nature.  Otherwise it's difficult to explain the enormous number of realistic-seeming scenes in the Gospels where the crowds are pushing and shoving each other in order to try to get healed, and Jesus has to continually keep escaping from them and hiding in various places to get any rest at all.  Note also that the event where Jesus miraculously feeds 5,000 people is contained in all 4 gospels.

The following data point would also suffice for circumstantial evidence, for those willing to accept more details from the Gospels:

  • Some facts about Jesus' life could reasonably be interpreted as fulfilling some of the prophecies found in the Hebrew Scriptures, to a greater degree than expected by chance.  (Note that this counts as circumstantial evidence even if the fulfillment of the prophecies is intentional, because even if this is easy to do, very few people bother to do it.  So it still picks out Jesus as special.  But it doesn't count if the fulfillments of prophecy are fictional reading-back into Jesus' life as a result of the Resurrection experience.)

Let's call this set of facts, and any similar ones, as (D).  Regardless of the precise details, the conjunction of these is sufficient to pick out Jesus as one of a very small number of people who were especially likely, prior to their deaths, to be the special person of Christianity*.  Consequently, the factor (D) cancels out the factor (C).

Since there are multiple circumstantial facts in (D), one could conceivably use them all by themselves to mount a cumulative case that Jesus has the features expected of the Messiah.  However, this would be tricky to do, since the 5 factors listed above aren't very independent of each other.

Next we turn to the testimonial evidence for the Resurrection.  Now, no one claims to have seen the actual instant when Jesus came back to life again, but the New Testament does claim that the tomb was found empty (except for angelic messengers) and that hundreds of people witnessed him physically alive afterwards, on several different occasions before the Ascension.

That's going by the New Testament, but what are the core facts that even someone fairly skeptical of the Gospels should accept?  There has to be some plausible origin story for how Christianity came into being: there must be some core set of claims which led to the later formation of the New Testament.

It would be extremely unparsimonious to postulate the existence of a completely different form of Christianity, of which we amazingly have no records, which transmuted into a completely different thing in just a few decades.  And besides, even the more skeptical biblical critics say that we have genuine letters of St. Paul dating from the late 40's or 50's (recall that the Crucifixion took place around 30), which make reference to doctrines and events taking place many years earlier.  Using just Paul's letters which are agreed to be authentic, together with some not-particularly-controversial chronological details from the Acts of the Apostles, one can date Paul's conversion to just a few years after the Crucifixion, sometime in the 30's.

Furthermore, while Paul might have been able to foist new doctrines off onto Gentile converts to Christianity, it's not very likely he would be able to seize control over the leaders of a previously existing religious group, which he had formerly persecuted, and convince them not only to accept the Resurrection, but also to claim to be witnesses of it.  As Paul says here:

For what I received I passed on to you as of first importance: that Christ died for our sins according to the Scriptures, that he was buried, that he was raised on the third day according to the Scriptures, and that he appeared to Peter, and then to the Twelve.  After that, he appeared to more than five hundred of the brothers at the same time, most of whom are still living, though some have fallen asleep.  Then he appeared to James, then to all the apostles, and last of all he appeared to me also, as to one abnormally born.

For I am the least of the apostles and do not even deserve to be called an apostle, because I persecuted the church of God.  But by the grace of God I am what I am, and his grace to me was not without effect. No, I worked harder than all of them—yet not I, but the grace of God that was with me.  Whether, then, it was I or they, this is what we preach, and this is what you believed.  (1 Cor 15:3-11)

Unless one thinks that Paul made up all of these people, it's clear from this list that the Christian belief in the Resurrection predated Paul's conversion.  So, using Paul's letters and the extremely broad details common to all four gospels (allowing for distortion as time passes), I think one should accept the following Resurrection facts (E) as nearly certain, even from a fairly "liberal" biblical criticism perspective:

  • Jesus had an inner circle of twelve disciples, who (except for Judas) later were important leaders in the early church, especially St, Peter, St. James, and St. John.
  • Jesus really died on the Cross.  (The theory that he swooned and then recovered is monstrously improbable.  Not only does it require Jesus to be sufficiently "dead-seeming" that the Romans and his friends didn't realize it, he then has to somehow regain enough health to persuade people he's not only alive but triumphant over death.  It doesn't explain Paul's conversion, and it's grossly inconsistent with any of the Gospel narratives.  Since the skeptic is still going to have to explain these things with some independent hypotheses, it's rather a stretch to have a medical improbability on top of all that.)

Furthermore, within a few months or years, the following were all true:

  • At least some Christians believed that the tomb was found empty after the Resurrection.
  • At least some female disciples believed (or else claimed to believe) that they were the first people to see Jesus, after his Resurrection.  (Since women were not considered reliable witnesses in the 1st century, male disciples would be unlikely to invent this detail out of whole cloth.  It must go back to some original historical fact.)
  • Some group of male disciples, including at the very least the Twelve Apostles, believed (or else claimed) that Jesus had appeared to Peter, and to the whole group on more that one occasion, after his death.  (This is only going with the groups mentioned in both the Gospels and Paul.  Paul's list includes St. James, the brother of Jesus, and more than 500 other witnesses, while the Gospels include two additional disciples on the road to Emmeus.)
  • Furthermore, either (a) many of these male disciples were originally inclined to be skeptical, or (b) the early Christians liked to invent stories portraying the faith of their most cherished leaders as weak.  Furthermore, they knew that persisting in this testimony was likely to lead to their deaths, which would have been obvious from (i) the fate of Jesus, (ii) the fact that their testimony was inherently uncomplimentary to the Jews and the Romans, and (iii) Paul himself (see below).
  • We have Paul's testimony against himself that before his conversion he "persecuted the church of God and tried to destroy it" (Gal. 1:13).  Then, as a result of an experience which he interpreted as a Resurrection Appearance of Jesus speaking to him, he become a Christian and immediately began to preach the new faith.  (If one additionally accepts the accounts in Acts 9, this experience was accompanied by phenomena which affected his travelling companions, and resulted in his temporary blindness until 3 days later, when someone named Ananias came to see him as a result of a vision of his own, after which "something like scales" fell from his eyes.)

This last fact is staggeringly implausible from a naturalistic point of view, even ignoring the additional details from Acts.  This can be seen by imagining yourself in the situation beforehand and then asking how surprised you'd be if it happened.  It's a lot like Hitler converting to Judaism after being struck by lightening, and then later being accepted into the Jewish community and becoming a highly respected rabbi.  That's a 1-in-a-million event right there, folks.  If Christianity isn't right, it's still true that they were fantastically lucky in the case of Paul, at a very critical moment in their history.

If we were to accept even one of the additional corroborating circumstances in Acts, the event becomes even harder to explain naturalistically.  Yet in general the book of Acts (which records 3 different versions of the conversion of Paul), even if it was not written by St. Luke, is filled with so much realistic detail and mundane trivia that it's impossible for me to believe it doesn't incorporate at least some memoirs which go back to the time of Paul himself (especially in the chapters which use the "we" pronoun).

Finally, some of the Apostles were not only willing to die for their testimony, they did die for it.  Christian tradition says that all of the Twelve except for John were martyred, but the later lives of many of the apostles are known only through fanciful legends.  However, at least 4 cases are fairly certain:

  • Peter, Paul, the Apostle James (brother of John), and the other James (brother of Christ) were all martyred.  The Apostle James was killed by Herod in the 40's, as described in the book of Acts (hard to imagine why they'd make this up).  That Peter and Paul were martyred in Rome is described in several early post-New Testament Sources, and the martyrdom of the other James in Jerusalem is described by the Jewish historian Josephus.

From Paul's uncontested letters we can also know this:

  • Paul believed (or at least claimed) that he and certain other Christians sometimes had the power to do miracles, and that these miracles had been witnessed by the churches he writes to.  (Additionally, the book of Acts records several miracles by Peter and Paul).

Finally, if Christianity is true, one would expect that God would continue to sometimes perform miracles even up to the present day.  And there is indeed evidence for this.  But it would take a whole 'nother post to go into any details, so I'll just reserve a parking space for this by calling it (F).

Posted in Theological Method | 3 Comments

Christianity and Observations

My first pillar of Science is that it is based on repeatable observations.  In order to see how Christianity measures up, we need to examine whether it is based on observations, and whether it is repeatable.

Observations ultimately boil down to sense-data experienced by individual human beings.  You can't go directly from observations to theories without a certain amount of interpretation, but that's true in Science too.  In order to be regarded as accurate, theories need to be based on empirical evidence, which has to be specific enough to prove that that theory is accurate, rather than just some similar theory in the class of theories.  Not all aspects of the theory need to be directly confirmed to be accepted (otherwise we could never use a theory to make predictions in new circumstances).  So long as the untested aspects of the theory are closely bound to aspects of the theory which are testable, the theory has a whole can be considered empirical.

Now Christian theology is empirical in this sense: that most of the core doctrines are the most reasonable explanations of certain ordinary sense-data reported by actual human beings.  By "ordinary" sense data, I mean things which appear to be interactions with the normal shared waking world, rather than sense-data seen in dreams or hullucinations, experienced in an altered state of consciousness.  (If Christianity is true, God does sometimes communicates using dreams and such, but the most important information was not given primarily in this way.)  The experiences might be "visions" in the sense of revealing truths about things in Heaven, but they are not "visions" in the sense of being subjective and intangible features of a single individual's private experiences.

Here's an example.  One of the most esoteric doctrines in all Christianity—the one that may at first sight seem to have the least to do with the actual physical world—is the doctrine of the Trinity: that God consists of a loving communion between the Father, Son, and Holy Spirit.  But the Gospel of St. Luke tells us that

When all the people were being baptized, Jesus was baptized too. And as he was praying, heaven was opened and the Holy Spirit descended on him in bodily form like a dove. And a voice came from heaven: “You are my Son, whom I love; with you I am well pleased.” (Luke 3:21-22)

There is a specific series of sense-data which communicates something about the relationship between these three entities, two of which are normally invisible.  (The Son would also have been invisible, but for his Incarnation as a human being.)  Combining this with other similar events, such as the Transfiguration and Pentecost, and the fact that Jesus claimed to be divine but also prayed to his Father, the Church came to the conclusion that the Trinity was the best explanation of the observed facts.  If it had been based on philosophical speculation, they would have come up with some more logical way to divine divinity into three aspects (like say, Creator-Preserver-Destroyer).

Note that here I am addressing the question of whether the claimed empirical facts support the theological claim.  Although some might argue that the claimed facts support some other claim (e.g. space aliens), most skeptics would probably think that's pretty silly.  It's a completely different question whether the purported testimony is actually true, whether it really happened or else was made up by somebody.  The accuracy of the accounts is a very important question, but before I go into it, I wanted to make sure to get clear what kind of factual support Christianity claims to have.

Here's a parable to illustrate why this distinction is important.  Water normally boils at 100 ^\circC.  Suppose I claim that if I drop an orange into water, it boils at 150 ^\circC instead.  I'm pretty sure that's false, but suppose you claim it is true.  And you say that you know it's true because you went into the laboratory and tested it carefully.  Now that doesn't mean I have to believe your claim.  I could accuse you of lying, or making stupid mistakes in the lab.  But if I make fun of you for "believing without any evidence at all" and make derogatory comments about the tooth fairy and Santa Claus, then I haven't understood the nature of your claim.

Admittedly, the degree to which Christian doctrine is founded on sense-data depends on the specific doctrine.  For example, the Second Coming will involve a lot of sense-data when it happens, but because it concerns the future the sense-data hasn't been observed yet.  It is based primarily on his promise that he will do so.  It is related to his Ascension into Heaven, which is itself partially based on sense-data (so far as his leaving Earth is concerned).  The fact that he has the power to rise from the dead and ascend into Heaven is certainly relevant to assessing how likely he is to keep this promise to return to Earth.

The appeal to sense-data becomes particularly clear in the case of the Resurrection.  Here are all the accounts of Resurrection Accounts contained in the four Gospels, Acts, and 1 Corinthians.  (The appendix of the Mark account is a latter addition, which was almost certainly not written by St. Mark.  It might well go back to the first century, but for obvious reasons I won't be putting any weight on it.  I've included it solely for the sake of completeness.)

Note how the accounts refer specifically to the fact that Jesus had a tangible body which could be viewed by multiple people with multiple senses simultaneously, and that he eats fish in order to reassure the disciples that he is not a ghost.  This despite the fact that he was also capable of instantaneously appearing and disappearing.  Note also the role that doubt and skepticism plays in several of these stories, as well as the fact that Jesus could eventually be recognized, but was frequently not recognized at first.

This is most obvious, of course, in the "Doubting Thomas" incident I quoted yesterday.  St. Thomas had an opportunity to believe in the Resurrection on the testimony of others first, but he rebuffs this by saying that he needs to experience the sense-data himself in order to believe it.  He demands that the experiment be repeated.  Jesus appears again a week later, and offers the evidence, but at the same time rebukes him gently by pronouncing a blessing on those who believe without seeing.

At this point the story is likely to raise all sorts of questions to a skeptical mind.  Why should faith play a role at all?  Why not just empiricism?

There's some deep issues here, but let's start with this: Jesus isn't requiring belief without empirical sense-data.  He's asking for belief based on the empirical sense data of other people besides yourself.  That may involve faith in the sense of trusting others, but not in the sense of "belief without evidence".

Leaving aside for the moment questions of why God would choose to arrange things that way, the situation itself is not that unusual.  It is how things always work when you're studying History prior to the 20th century, which is not repeatable.  It's how things usually work in Science itself, since only a very few people have replicated the fundamental experiments in physics personally.  Most of us believe in scientific experiments on the testimony of other people.  As a theorist I certainly haven't done all those experiments myself; in fact, I haven't even interviewed most of the people who did them!

This year, the Higgs boson was discovered at the LHC.  This is Science, so the results have to be "repeatable".  That means, if you have billions of dollars, your own team of hundreds of scientists, and a decade of your life to devote to it, you too can discover the Higgs boson.  But are you going to wait for that to happen before you believe it?  No, you believe it now, on faith!  And what I mean by faith here is not a vaporous sentimentality, but trust based on good evidence that the community of particle physicists is reliable in this particular respect.

(I apologize to anyone who was expecting a Bayesian analysis in this post.  As you can see, I'm still working up to it.)

Posted in Theological Method | 5 Comments

The Gospel

It's a little bit strange liturgically to have Easter on Christmas Day, but that's how things worked out in this series.  I'm going to quote a passage from the Gospel of St. John, which illustrates the Resurrection claim and connects to many of the issues I'm going to discuss in this series.  (I know that many biblical critics believe that the Fourth Gospel isn't a historically reliable source, but for reasons that will be explained later, I don't agree with them.)  Note that this narrative occurs after the Crucifixion, so according to the Gospels, Jesus has already performed a bunch of miracles in public, and then been killed.

I won't add any more commentary here—that's coming later.  So then, folks, hear the word of the Lord:

Early on the first day of the week, while it was still dark, Mary Magdalene went to the tomb and saw that the stone had been removed from the entrance.  So she came running to Simon Peter and the other disciple, the one Jesus loved, and said, “They have taken the Lord out of the tomb, and we don’t know where they have put him!”

So Peter and the other disciple started for the tomb. Both were running, but the other disciple outran Peter and reached the tomb first. He bent over and looked in at the strips of linen lying there but did not go in.  Then Simon Peter, who was behind him, arrived and went into the tomb. He saw the strips of linen lying there, as well as the burial cloth that had been around Jesus’ head. The cloth was folded up by itself, separate from the linen.  Finally the other disciple, who had reached the tomb first, also went inside. He saw and believed.  (They still did not understand from Scripture that Jesus had to rise from the dead.)

Then the disciples went back to their homes, but Mary stood outside the tomb crying. As she wept, she bent over to look into the tomb and saw two angels in white, seated where Jesus’ body had been, one at the head and the other at the foot.

They asked her, “Woman, why are you crying?”

“They have taken my Lord away,” she said, “and I don’t know where they have put him.”At this, she turned around and saw Jesus standing there, but she did not realize that it was Jesus.

Woman,” he said, “why are you crying? Who is it you are looking for?”

Thinking he was the gardener, she said, “Sir, if you have carried him away, tell me where you have put him, and I will get him.”

Jesus said to her, “Mary.”  She turned toward him and cried out in Aramaic, “Rabboni!” (which means Teacher).

Jesus said, “Do not hold on to me, for I have not yet returned to the Father. Go instead to my brothers and tell them, ‘I am returning to my Father and your Father, to my God and your God.’ ”

Mary Magdalene went to the disciples with the news: “I have seen the Lord!” And she told them that he had said these things to her.

On the evening of that first day of the week, when the disciples were together, with the doors locked for fear of the Jews, Jesus came and stood among them and said, “Peace be with you!” After he said this, he showed them his hands and side. The disciples were overjoyed when they saw the Lord.

Again Jesus said, “Peace be with you! As the Father has sent me, I am sending you.”  And with that he breathed on them and said, “Receive the Holy Spirit.  If you forgive anyone his sins, they are forgiven; if you do not forgive them, they are not forgiven.”

Now Thomas (called Didymus), one of the Twelve, was not with the disciples when Jesus came. So the other disciples told him, “We have seen the Lord!”

But he said to them, “Unless I see the nail marks in his hands and put my finger where the nails were, and put my hand into his side, I will not believe it.”

A week later his disciples were in the house again, and Thomas was with them. Though the doors were locked, Jesus came and stood among them and said, “Peace be with you!” Then he said to Thomas, “Put your finger here; see my hands. Reach out your hand and put it into my side. Stop doubting and believe.”

Thomas said to him, “My Lord and my God!”

Then Jesus told him, “Because you have seen me, you have believed; blessed are those who have not seen and yet have believed.”

Jesus did many other miraculous signs in the presence of his disciples, which are not recorded in this book. But these are written that you may believe that Jesus is the Christ, the Son of God, and that by believing you may have life in his name.

(John 20)

Merry Christmas, everyone!

Posted in Theological Method | Leave a comment

Can Religion be Based on Evidence?

So I'd like to get kicking soon on the project of actually presenting the positive evidence for Christianity.  In my view the best evidence is the historical testimony of the apostles to Jesus' Resurrection (along with other ancient and modern miracle claims).  However, some people have problems with this because it isn't scientific, and they think that only a "scientific" proof of miracles should qualify as evidence.

The idea that Science is the only very reliable way to gather empirical data is called (usually pejoratively) Scientism.  It is closely related to Naturalism, the belief that the world consists entirely of a certain class of physical things, of a sort which can be scientifically analyzed.  However, the two are not the same, since Scientism is a claim about there being only one good methodology for learning about the world.  A Naturalist is free to believe that there are valid nonscientific methods for learning about the world, as long as they also think those methods don't reveal the existence of any entities they'd consider supernatural.  (There's a bit of a definition problem in defining what exactly natural vs. supernatural can mean, but we more or less know what kinds of things this sort of person doesn't believe in: gods, miracles, spirits or ghosts of any kind, psychic powers, destiny, reincarnation etc.)

Well, Scientism in its strongest form is obviously stupid, since as I pointed out here there exist several other kinds of evidence-based inquiry that involve different methodologies.  Here's another rebuttal by atheistic philosopher Richard Chapell, who points out that Scientism isn't even logically consistent with itself.  So, there may or may not be good reasons to believe in religious claims, but "Science" taken by itself is not one of them.

Well, that was easy.  Maybe too easy.  Because, after all, someone could say this:  Even if there are nonscientific methods of inquiry, hasn't Science at least taught us something about the way the world is?  And hasn't it taught us something about what kinds of evidence are reliable?  Maybe there isn't a sharp contradiction between Science and Religion, but maybe there are things that make it more difficult for a scientifically-minded person to accept religious claims.  I think a lot of people have this idea at the back of their heads, and I'm going to try to address it in my future posts.

For further reflections on the relationship between Science, History, Philosophy, and the various arguments for and against Christianity, see here:

Can Religion be Based on Evidence?

It also explains briefly why I think the Historical Argument for Christianity is quite strong, although I plan to go into that in considerably more detail here.

(Erratum: there are a couple things I'd phrase differently if I were re-writing this essay now.  First of all, my parenthetical statement about "overcredulous Catholics, Pentecostals, and missionaries to Third World nations" was intended as a statement of a skeptical point of view rather than my own view, although there certainly are some overcredulous people in the groups named.  And this book has convinced me that modern day miracles are more frequent than I had previously thought.  Also, the phrase "tortured to death" should really be replaced with "tortured or killed"—in fact the whole sentence is too strongly written, and should make clearer who exactly it refers to.  For now read it referring to "several of the key eyewitnesses", I guess.)

Posted in Scientific Method | 10 Comments

Bayes' Theorem

Today I'd like to talk about Bayes' Theorem, especially since it's come up in the comments section several times.  It's named after St. Thomas Bayes (rhymes with "phase").  It can be used as a general framework for evaluating the probability of some hypothesis about the world, given some evidence, and your background assumptions about the world.

Let me illustrate it with a specific and very non-original example.  The police find the body of someone who was murdered!  They find DNA evidence on the murder weapon.  So they analyze the DNA and compare it to their list of suspects.  They have a huge computer database containing 100,000 people who have previously had run-ins with the law.  They find a match!  Let's say that the DNA test only gives a false positive one out of every million (1,000,000) times.

So the prosecutor hauls the suspect into court.  He stands up in front of the jury.  "There's only a one in a million chance that the test is wrong!" he thunders, "so he's guilty beyond a reasonable doubt; you must convict."

The problem here—colloquially known as the prosecutor's fallacy—is a misuse of the concept of conditional probability, that is, the probability that something is true given something else.  We write the conditional probability as P(A\,|\,B), the probability that A is true if it turns out that B is true.  It turns out that P(A\,|\,B) is not the same thing in general as P(B\,|\,A).

When we say that the rate of false positives is 1 in a million, we mean that

P(\mathrm{DNA\,match}\,|\,\mathrm{innocent}) = .000001

(note that I'm writing probabilities as numbers between 0 and 1, rather than as percentages between 0 and 100).  However, the probability of guilt given a match is not the same concept:

P(\mathrm{innocent}\,|\,\mathrm{DNA\,match}) \neq .000001.

The reason for this error is easy to see.  The police database contains 100,000 names, which is 10% of a million.   That means that even if all 100,000 people are innocent, the odds are still nearly equal to .1 that some poor sucker on the list is going to have a false positive (it's slightly less than .1 actually, because sometimes there are multiple false positives, but I'm going to ignore this since it's a small correction.)

Suppose that there's a .5 chance that the guilty person is on the list, and a .5 chance that he isn't.  Then prior to doing the DNA test, the probability of a person on the list being guilty is only 1 : 200,000.  The positive DNA test makes that person's guilt a million times more likely, but this only increases the odds to 1,000,000 : 200,000 or 5 : 1.  So the suspect is only guilty with 5/6 probability.  That's not beyond a reasonable doubt.  (And that's before considering the possibility of identical twins and other close relatives...)

Things would have been quite different if the police had any other specific evidence that the suspect is guilty.  For example, suppose that the suspect was seen near the scene of the crime 45 minutes before it was committed.  Or suppose that the suspect was the murder victim's boyfriend.  Suppose that the prior odds of such a person doing the murder rises to 1 : 100.  That's weak circumstantial evidence.  But in conjunction with the DNA test, the ratio becomes 1,000,000 : 100, which corresponds to a .9999 probability of guilt.  Intuitively, we think that the circumstantial evidence is weak because it could easily be compatible with innocence.  But if it has the effect of putting the person into a much smaller pool of potential suspects, then in fact it raises the probability of guilt by many orders of magnitude.  Then the DNA evidence clinches the case.

So you have to be careful when using conditional probabilities.  Fortunately, there's a general rule for how to do it.  It's called Bayes' Theorem, and I've already used it implicitly in the example above.  It's a basic result of probability theory which goes like this:

P(H\,|\,E) = \frac{P(H)P(E\,|\,H)}{P(E)}.

The way we read this, is that if we want to know the probability of some hypothesis H given some evidence E which we just observed, we start by asking what was the prior probability P(H) of the hypothesis before taking data.  Then we ask what is the likelihood P(E\,|\,H), if the hypothesis H were true, we'd see the evidence E that we did.  We multiply these two numbers together.

Finally, we divide by the probability P(E) of observing that evidence E.  This just ensures that the probabilities all add up to 1.  The rule may seem a little simpler if you think in terms of proability ratios for a complete set of mutually exclusive rival hypotheses (H_1,\,H_2\,H_3...) for explaining the same evidence E.  The prior probabilities P(H_1) + P(H_2) + P(H_3)\ldots all add up to 1.  P(E\,|\,H) is a number between 0 and 1 which lowers the probability of hypotheses depending on how likely they were to predict E.  If H_n says that E is certain, the probability remains the same; if H_n says that E is impossible, it lowers the probability of H_n to 0; otherwise it is somewhere inbetween.  The resulting probabilities add up to less than 1.  P(E) is just the number you have to divide by to make everything add up to 1 again.

If you're comparing two rival hypotheses, P(E) doesn't matter for calculating their relative odds, since it's the same for both of them.  It's easiest to just compare the probability ratios of the rival hypotheses, because then you don't have to figure out what P(E) is.  You can always figure it out at the end by requiring everything to add up to 1.

For example, let's say that you have a coin, and you know it's either fair (H_1), or a double-header H_2.  Double-headed coins are a lot rarer than regular coins, so maybe you'll start out thinking that the odds are 1000 : 1 that it's fair (i.e. P(H_2) = 1/1,001).  You flip it and get heads.  This is twice as likely if it's a double-header, so the odds ratio drops down to 500 : 1 (i.e. P(H_2) = 1/501).  A second heads will make it 250 : 1, and a third will make it 125 : 1 (i.e. P(H_2) = 1/126).  But then you flip a tails and it becomes 1 : 0.

If that's still too complicated, here's an even easier way to think about Bayes' Theorem.  Suppose we imagine making a list of every way that the universe could possibly be.  (Obviously we could never really do this, but at least in some cases we can list every possibility we actually care about, for some particular purpose.)  Each of us has a prior, which tells us how unlikely each possibility is (essentially, this is a measure of how surprised you'd be if that possibility turned out to be true).  Now we learn the results of some measurement ESince a complete description of the universe should include what E is, the likelihood of measuring E has to be either 0 or 1.  Now we simply eliminate all of the possibilities that we've ruled out, and rescale the probabilities of all the other possibilities so that the odds add to 1.  That's equivalent to Bayes' Theorem.

I would have liked to discuss the philosophical aspects of the Bayesian attitude towards probability theory, but this post is already too long without it!  Some other time, maybe.  In the meantime, try this old discussion here.

Posted in Scientific Method | 3 Comments

All points look the same

I've told you so far that the gravitational field is encoded in a 4 \times 4 matrix known as the metric.  Here it is, displayed as a nice table:

 g_{ab} = \left( \begin{array}{cccc} g_{00} & g_{01} & g_{02} & g_{03}\\ g_{01} & g_{11} & g_{12} & g_{13} \\ g_{02} & g_{12} & g_{22} & g_{23} \\ g_{03} & g_{13} & g_{23} & g_{33} \end{array} \right)

There's 10 components because the matrix is symmetric when reflected diagonally.  The 4 diagonal components (g_{00}, g_{11}, g_{22}, g_{33}) tell you how to measure length-squared along the four coordinate axes.  For example, the length along the 1-axis is given by

\Delta s = \sqrt{g_{11}} \Delta x^1,

where \Delta x^1 is the coordinate difference in the 1-direction.  The remaining 6 off-diagonal terms keep track of the spatial angle between the coordinate axes.  If you know enough Trigonometry, you can figure out that the angle \theta between e.g. the 1-axis and the 2-axis is given by this formula:

\cos(\theta) = \frac{g_{12}} {\sqrt{g_{11} g_{22}}}

However, I've also said that the metric depends on the choice of coordinates, which is arbitrary.  We can use this freedom to choose a set of coordinates where the metric looks particularly simple at any given point.   We can start by choosing our four coordinate axes to be at right-angles to each other.  This gets rid of all those funky off-diagonal components of the metric, which involve two different directions:

 g_{ab} = \left( \begin{array}{cccc} g_{00} & 0 & 0 & 0\\ 0 & g_{11} & 0 & 0 \\ 0 & 0 & g_{22} & 0 \\ 0 &0 & 0 & g_{33} \end{array} \right)

If any of the four remaining numbers happen to be 0, we say that the metric is degenerate.  This would correspond to a weird geometry in which you can move in one of the directions for free without it affecting your total distance travelled.  Since we all know that's not the way the real world works, we'll ignore this possibility.

We can also rescale the tick marks along any coordinate axis.  This allows us to multiply each diagonal component of the metric by a positive real number.  So if say g_{22} is positive, we can choose coordinates where it's +1, and if it's negative, we can choose coordinates where it's -1.  This gives us:

 g_{ab} = \left( \begin{array}{cccc} \pm 1 & 0 & 0 & 0\\ 0 & \pm 1 & 0 & 0 \\ 0 & 0 & \pm 1 & 0 \\ 0 &0 & 0 & \pm 1 \end{array} \right)

Since it also doesn't matter what order we list the four coordinate directions, all that matters is the total number of +'s and -'s.  This choice is called the signature of the spacetime.

Now if you remember my very first post on spacetime geometry, + directions in the metric correspond to spatial dimensions, while the funny - sign is what makes for a time dimension.  But the real world has one time dimension, everywhere.  No matter how far you travel, you'll never find a place (so far as we know) where there isn't any time direction, or where there are extra time dimensions.  So that means that the correct signature for spacetime has (-, +, +, +) along the diagonal, which is called Lorentzian (a.k.a. Minkowskian) signature.  (If we had wanted to describe a timeless four-dimensional space, we would instead select the Riemannian (a.k.a. Euclidean) signature (+, +, +, +).)  We conclude that for any point of spacetime, you can always choose a set of coordinates such that the metric takes a special form that we'll call \eta_{ab}:

 g_{ab} = \left( \begin{array}{cccc}-1 & 0 & 0 & 0\\ 0 & +1 & 0 & 0 \\ 0 & 0 & +1 & 0 \\ 0 &0 & 0 & +1 \end{array} \right) = \eta_{ab}.

In other words, if you zoom in on any point, you recover Special Relativity.  So after all this fidgeting around, we end up with a somewhat profound conclusion: in General Relativity, every point of spacetime looks the same as every other point.

This is related to what Einstein called the Equivalence Principle, which says that at short enough distances, the effects of acceleration are indistinguishable from being in a gravitational field.  We all know from personal experience that riding in an elevator can make us weigh more or less, and from TV that astronomers in the Space Shuttle are weightless when they're in free fall.  In other words, you can always choose a coordinate system in which there is no gravitational force at any given point.

(Lewis Carroll actually described this principle several decades before Einstein in Sylvie and Bruno, which includes a description of a tea party taking place in a freely-falling house.  Then he describes what happens if the house is being pulled down with a rope faster than gravity would accelerate it, and explains how you could have a normal tea party as long as you have it upside-down.  I like this book better than his more famous classics, but don't read it unless you can withstand LD20 of Victorian sentimentality about fairy children.  Also, Carroll didn't go on to discover a revolutionary theory of gravity based on this principle.)

It might seem now like everything has become too simple.  If the metric looks the same at every single point, then why did we even bother with it?  Where's the information in the gravitational field?  Well, it's true that for any one point, there's a coordinate system where the metric looks just like \eta_{ab}.  But there's no coordinate system for which the metric looks like \eta_{ab} everywhere at once.  (Unless there's no gravitational field anywhere, in which case Special Relativity is true).  If you make the metric look simple in one place, it has to look complicated somewhere else.

So in order to describe the gravitational field properly, we have to find a way to compare the metric at different points.  We can do this using something called parallel transport.  I'll give more details later, but basically it tells us how an object moves in a gravitational field when we carry it along a path through spacetime.  When we carry the object around a tiny loop so that it returns to its original position, we might find that it comes back rotated compared to its original orientation.  If so, we say that the spacetime contains curvature.  If the spacetime contains curvature, this is a fact about the gravitational field which is invariant, i.e. objectively true.  You can't eliminate it just by changing your coordinates.

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