Two Bayesian Fallacies

At INR3 in Kamloops I spoke on applying Bayesian logic to the study of Jesus along with the same principles we apply to dead religions (so as to avoid the “don’t offend the Christians” reaction to controversial claims…claims that would not be controversial if Jesus was not the object of worship of billions of loud, influential people). In Q&A philosopher Louise Antony challenged my application of Bayes’ Theorem to historical reasoning with a series of technical complaints, especially two fallacies commonly voiced by opponents of Bayesianism. I was running out of time (and there was one more questioner to get to) so I explained that I answered all her stated objections in my book Proving History (and I do, at considerable length).

But I thought it might be worth talking about those two fallacies specifically here, in case others run into the same arguments and need to know what’s fishy about them. [Read more...]

Bayesian Atheism Even Lowder

Yesterday’s post inspired someone to point me to another gem in the same category: the ongoing work of Jeffery Jay Lowder at The Secular Outpost on Bayesian Arguments for Atheism and theism. He has a long archive on that topic there and continues to post on debates in religion analyzing them in Bayesian terms. Though his posts are generally at a moderate and not beginner’s level of difficulty, nevertheless a lot of valuable insight is there, and many examples of how to test and frame arguments in religious debates using Bayesian reasoning. Even when he’s wrong, you can learn a lot by thinking about how to articulate what you think his mistake is using the same Bayesian concepts.

Lowder has even assembled a getting-started bibliography of his best posts on how to frame and improve evidential arguments for naturalism using Bayes’ Theorem in his Index of Evidential Arguments for Atheism. This and the ongoing entries he adds on Bayesian reasoning in atheism are definite must-haves on any bookmark list for Bayesian atheism. Enjoy!

Bayesian Atheism

James Lindsay has been doing some great blogging on how to apply Bayesian reasoning to model John Loftus’ Outsider Test for Faith (or OTF). A while ago I asked for recommendations of bloggers that often write about Bayes’ Theorem for a general audience (see Bayesian Blogging), and a few came up there. This is another.

Cover of The Christian DelusionFormulating and extensively defending the OTF is Loftus’ greatest contribution to the philosophy of religion and atheism. His best and most thorough treatment appears as chapter four in The Christian Delusion (a book I always recommend anyway as it contains lots of great chapters by great authors; and two by me). He is writing a whole book on it now. It should be out this year (I’ve seen advanced drafts and it’s good; I’ll blog it when you can buy it). The OTF is featured at Iron Chariots (which provides examples of looser expressions of the concept throughout history) and Loftus discusses it often at Debunking Christianity.

The basic idea is that you can only have a rational faith if you test it by the same standards you apply to all other competing faiths; yet when you do that, your religion tests as false as the others, and the same reasons you use to reject those become equally valid reasons to reject yours. Though this idea has been voiced before, Loftus is the first to name it, rigorize it, and give it an extensive philosophical defense; moreover, by doing so, he is the first to cause a concerted apologetic to arise attempting to dodge it, to which he could then respond. The end result is one of the most effective and powerful arguments for atheism there is. It is, in effect, a covering argument that subsumes all other arguments for atheism into a common framework.

Cover of God Doesn't; We DoLindsay, meanwhile, is an expert mathematician and author of God Doesn’t; We Do: Only Humans Can Solve Human Challenges (2012). His blog of the same title treats a number of issues in support of that book and its argument. I don’t always agree with him. But his blogging on Bayes’ Theorem is great. He started by talking about how Loftus’ OTF can be formulated using Bayes’ Theorem, to show why it can’t be dodged the way Christian apologists want. This led to further blogging on the subject, including a Bayesian analysis of “faith” in general. It’s worth checking out.

The first of these (on which the others build) is:

Here much of his argument is backed formally by my Bayesian models in The End of Christianity (edited by Loftus) for Christianity as a religion (chapter two) and for the design argument generally (chapter twelve); where most of the math is in the endnotes but the Bayesian logic is made explicit in each. These chapters especially explain why the evidence has a much higher consequent probability (a higher “likelihood” in sci-speak) on naturalism than on any kind of theism (much less Christian theism).

Combine those with Lindsay’s post and you should get a clear understanding why atheism is true and Bayesian reasoning proves it. Lindsay’s treatment will be especially helpful in understanding how atheists think like Bayesians all the time even when they don’t know it (and how Christians, in contrast, are really awful Bayesians). I give other examples of Bayesian atheism near the end of my talk Bayes’ Theorem: Lust for Glory (which is still my best intro to BT for beginners), which can supplement all this.

Lindsay continued blogging under the tag “Math” and what’s there so far is all Bayes’ Theorem stuff. Maybe that won’t always be the case, but keeping tabs on that tagged subject going forward might lead you to more gems about Bayesian reasoning. So far there are three other posts:

  • A Bit More Clarity on Bayes’s Theorem and Loftus’s Outsider Test for Faith (which shows how a BT-formulated OTF forces believers to confront facts that plain descriptions of the OTF might not; in short, it’s the probability of the evidence, and not just the prior probability, that’s the problem, although the OTF shows both are a problem for any honest believer)
  • Continuing My Bayesian Argument–The Role of Evidence (where he defends the OTF against accusations that it would lead to weird conclusions in other domains, which a BT analysis shows is actually not true; although he incorrectly applies the term a priori here: the prior probability in the OTF is not a priori, but based on background evidence regarding the number of observed religious faiths; a priori knowledge is by definition not based on any such evidence, and in particular neither are a priori probabilities; for an actual example of the latter, see my note 8, pp. 406-07, in TEC)
  • Defining Faith via Bayesian Reasoning (which builds a Bayesian definition of faith, when faith is used in any sense other than as a synonym of belief; this also provides an example of how many of Loftus’ rebuttals of critics of the OTF can be framed in Bayesian terms to show why he is right and they are not)

Good stuff so far. So I’m adding this to my list of Bayesian bloggers worth keeping an eye on. Another to add is Jeff Lowder.

Miracles & Historical Method

Fan photo of Dr. Carrier in shadow before stage screen showing slide that says 'Conclusion: Christians Were Big Ass Liars'Video of my talk for this year’s Skepticon is now available on YouTube. See Miracles and Historical Method. Description:

Carrier talks about how to think critically about history generally, using miracles as an entertaining example. Builds on his talk last year on Bayes’ Theorem, but this time it’s more about method than math, and surveys a lot of real-world examples of miracles from the ancient world (pagan, Jewish and Christian). Summarizes some of what is covered in much more detail in his book.

Understanding Bayesian History

So far I know of only two critiques of my argument in Proving History that actually exhibit signs of having read the book (all other critiques can be rebutted with three words: read the book; although in all honesty, even the two critiques that engage the book can be refuted with five words: read the book more carefully).

As to the first of those two, I have already shown why the criticisms of James McGrath are off the mark (in McGrath on Proving History), but they at least engage with some of the content of my book and are thus helpful to address. I was then directed to a series of posts at Irreducible Complexity, a blog written by an atheist and evolutionary scientist named Ian who specializes in applying mathematical analyses to evolution, but who also has a background and avid interest in New Testament studies.

Ian’s critiques have been summarized and critiqued in turn by MalcolmS in comments on my reply to McGrath, an effort I appreciate greatly. I have added my own observations to those in that same thread. All of that is a bit clunky and out of order, however, so I will here replicate it all in a more linear way. (If anyone knows of any other critiques of Proving History besides these two, which actually engage the content of the book, please post links in comments here. But only articles and blog posts. I haven’t time to wade through remarks buried in comment threads; although you are welcome to pose questions here, which may be inspired by comments elsewhere.)

Ian’s posts (there are now two, A Mathematical Review of “Proving History” by Richard Carrier and An Introduction to Probability Theory and Why Bayes’s Theorem is Unhelpful in History; he has promised a third) are useful at least in covering a lot of the underlying basics of probability theory, although in terms that might lose a humanities major. But when he gets to discussing the argument of my book, he ignores key sections of Proving History where I actually already refute his arguments (since they aren’t original; I was already well aware of these kinds of arguments and addressed them in the book).

[Read more...]

McGrath on Proving History

James McGrath has reviewed my book Proving History. We’ve argued before (e.g. over claims Bart Ehrman made), so there is backstory. But his review is unexpectedly kind and praising at points, and he likes the overall project of explaining the underlying logic of history as fundamentally Bayesian and making productive use of that fact. He does conclude with some select criticism, though, and that is what I will respond to here. [Read more...]

Bayesian Blogging

This is a request to all fans of Bayes’ Theorem out there: I’m looking for the best blogs and websites substantially devoted to discussing all things Bayesian.

Of course I know about Less Wrong, the brainchild of Eliezer Yudkowsky, which often discusses Bayesian reasoning and is a fabulous website for learning about human reason, and cognitive biases and how to overcome them, and other related subjects (it should be regular reading for most people keen on those subjects). But I also just discovered the awesome blog Maximum Entropy by Tom Campbell-Ricketts (since he asked me about the famous anecdote of Laplace, “Sir, I have no need of that hypothesis,” which might be apocryphal, but I directed him to what evidence there is for it). This blog is a Bayesian paradise of great posts, often quite advanced (so not for beginners or mathphobes)–but for people getting into the groove of these kinds of things, a fun resource.

The Wikipedia article on Bayes’ Theorem has already become too advanced to recommend to beginners. The Stanford Encyclopedia of Philosophy entry isn’t any better that way, but at least it discusses the application of the theorem to philosophy (epistemology in particular) and has a more extensive bibliography. My own Bayesian Calculator page (which is continually in development) will perhaps be more helpful, with more plain English explanation and some actual calculators you can fiddle with to see what happens. And total beginners should start with my Skepticon video Bayes’ Theorem: Lust for Glory! (that blog article gives the links plus additional resources about the video). Lots of good links are also assembled at Alexander Kruel’s A Guide to Bayes’ Theorem.

But none of these are blogs or websites that regularly produce discussion and articles about Bayesian reasoning. And I’m looking for the best of the latter. I’m looking for more stuff like Less Wrong or Maximum Entropy. If there is any. It can be basic intro level stuff, or advanced, but it should be good reading either way, the kind of place a general Bayesian might want to visit monthly to see what’s going down. So if anyone reading this has recommendations, please plop them in the comments section!

[I should add that I think all Bayesians should also familiarize themselves with the lists of cognitive biases and logical fallacies at Wikipedia, to contemplate how these can model misuses of Bayes' Theorem or be corrected or avoided by using Bayes' Theorem. FallacyFiles also has a useful taxonomy of logical fallacies. But I'm also interested in lists or sites dedicated to common errors or fallacies in reasoning about probability specifically.]

Limited Comments Policy: Because this post is a resource request, only comments that supply relevant hyperlinks (or names of websites) will be posted. Everything else will be deleted. Comments on other subjects should be posted within an appropriate blog thread (see the topic index for my blog down the right side of this page).

New Bayesian Calculator

Thanks to Cam Spiers (who has produced an interesting selection of free javascript Bayesian Calculators), I have updated my own Bayesian Calculator page using the most basic of those. This might be updated again in coming months. Right now it only allows running calculations with two-digit probabilities from .01 to .99 (or 1% and 99%), so you can’t use it for odds outside that range (for example, you can’t see what happens when the prior is 1 in 1,000 or 1 in 1,000,000 or when a consequent is even closer to 100% than 99%). But future versions of the page might have those features.

For people new to the whole idea of Bayes’ Theorem and Bayesian reasoning, you should first check out my talk at Skepticon last year: Bayes’ Theorem: Lust for Glory! For a more thorough treatment (using historical reasoning as a running example), which is also aimed as much as possible at lay readers, there is now of course my book: Proving History: Bayes’s Theorem and the Quest for the Historical Jesus.

The Jesus Tomb and Bayes’ Theorem

Finally, a mathematician actually gets the math right on the Jesus Tomb hypothesis. Conclusion? We have not found the tomb of Jesus. For those who already know the backstory and want to jump right to it, read Bayes’ Theorem and the “Jesus Family Tomb” by physicist Randy Ingermanson. He approached the problem like a physicist dealing with any old problem in data analysis (the problem is not so much different from how particle accelerator data are analyzed). He was assisted by political scientist Jay Cost, another who has good experience running Bayesian models like this. This expands on Ingermanson’s work on this published under peer review as Randall Ingermanson, “Discussion of: Statistical Analysis of an Archaeological Find,” Annals of Applied Statistics 2.1 (2008): 84-90 (responding to Feuerverger).

Backstory: James Tabor and some others have been pushing the claim that a tomb uncovered in the Talpiot district of Jerusalem (hence now called the Talpiot tomb) is the actual burial place of Jesus (and we not only have his “coffin,” but his DNA! As well as evidence he had a child named Judas by Mary Magdalene, also buried therein, also with her DNA!), and they published a book and a documentary arguing their case. (I’m just being colloquial. The tomb’s not full of coffins, of course, but ossuaries, a cultural analog). They had a mathematician backing them (Dr. Andrey Feuerverger), but his math has been consistently bogus from day one. For example, even though we have vastly better odds of randomly getting a name in a group of ten-to-thirty bodies than in a group of five, he kept running the math for five, even though there were ten-to-thirty bodies buried in that tomb. He also adopted a number of dubious (and some outright refuted) factual assumptions (for example, regarding the names of the women in the tomb: see, as one instance, the penultimate paragraph of my previous article on this tomb). By these devices, he found the odds were 600 to 1 in favor of this being the actual tomb of Jesus.

What happened: Ingermanson and Cost apply the correct math (Bayes’ Theorem, valid historical premises, proper treatment of variables, and correct mathematical models, e.g. acknowledging that more than five people were buried there). They find that by standard historical assumptions, the odds are 1 in 19,000 against the Talpiot tomb being the tomb of Jesus, and even by more generous assumptions the odds are 1 in 1,100 against (I put my own assumptions into their model and came up with 1 in 200 against), while even the most fanatical “I desperately want this to be the tomb of Jesus” estimator can only get odds of 1 in 18 that the Talpiot tomb is the tomb of Jesus. Thus, it probably isn’t, even if we are ridiculously generous to the hypothesis that it is.

So much for that. Done and dusted.

The Lame That Would Not Die!

What is The Lame? Unfortunately no one can be told what The Lame is. You have to see it for yourself. No, just kidding. It’s the claim that “Science Requires a Christian Worldview.” JT just blogged that, responding reasonably enough to a repeat of a standard Christian apologetic shibboleth (and, as he callously and shamelessly threatened therein, did indeed email me the link in question as if to annoy me, like the gangster cad that we all know he is; for shame). I realized I should probably collect a resource list of all I’ve written in refutation of it. This is that list.

First, I pretty much kick the legs out from under it with the extensive historical argument (since non-Christians invented science, and that centuries before Christianity even existed, obviously science does not require a Christian worldview) in “Christianity Was Not Responsible for Modern Science,” The Christian Delusion (2010), pp. 396-420. You really don’t have to read anything else on the subject, frankly.

Second, I refute one component of the philosophical case, the claim that the universe must have been designed to be understood or the human brain designed to understand it, in “Neither Life Nor the Universe Appear Intelligently Designed,” The End of Christianity (2011), pp. 279-304 (key pages: pp. 298-302). That’s a short but compact and effective refutation, with references.

Third, I take on the entire Argument from Reason (which is a kind of umbrella argument that includes the claim that science only makes sense if Christianity is true, by arguing that reason would not exist but for God) in an extensive philosophical critique of Victor Reppert’s Argument from Reason. But the most pertinent sections of that are my refutation of the original version of the “Science Needs Christianity” argument from (Surprise!) C.S. Lewis. Those are the sections on where the “Five Axioms of Science” came from, and preceding that, on why “Our Mind Is Reliable Enough for Inductive Logic to Work.” And following both, I refute the more general claim that “Only Theists Can Invent Science” (although I give an even clearer answer to that in the Christian Delusion chapter, item 1 above).

Fourth, I have refuted the claim that the mathematical nature of the universe entails it was intelligently designed, in my critiques of Steiner and Howell. But of those, my refutation of Steiner (Fundamental Flaws) is less fun to read than my refutation of Howell (Our Mathematical Universe), in which I refute Howell’s attempt to rehabilitate Steiner; and really, if you’ve read the latter, you don’t need so much to read the former (unless you are really geeking out on the ontology of scientific theories, which is totally cool if you are).

Now you can add to all that JT’s response, which covers a lot of the most common sense rebuttals. The only weakness of which is that he doesn’t give the best response to the claim that “the atheist worldview cannot account for the uniformity of nature on which to base the scientific process.” He rightly points out that an argument from ignorance is a fallacy, and that Christians don’t really believe in the uniformity of nature (remember those miracles they keep going on about?), and if anyone is going to suss this, it’s going to be actual cosmological scientists, not hack armchair theologians.

But there is one argument one can make that kind of dodges those otherwise obvious points: the evidence e is “the uniformity of nature,” and the explanation h “God made it that way” makes e highly probable whereas one might suppose ~h “a god did not make it that way” does not make e highly probable, therefore e is an argument for god. Not that this must be a conclusive argument; having evidence for something is not the same as that something being true. For example, you can have evidence for someone committing a crime that in fact they didn’t commit–like fingerprints on a murder weapon, which could have gotten there in other ways besides having used it to kill the vic. But still.

The real problem is that ~h is a stand-in for all other theories of the evidence. Because h and ~h together must include all logically possible explanations of the evidence. And since h is only one of them (“God did it”); then necessarily ~h contains all other explanations. Many of which do make e highly probable. We don’t have to pick one, either. We can say “I can think up ten different explanations, other than God, which all guarantee that e will obtain” (for ten such examples see below). And if those all have a higher prior probability than “God did it,” then God is no longer the better explanation. In fact, it then becomes one of the worst. Note that we don’t have to know or even claim that any of those explanations is true. It’s still the case that more probably one of them is true, than that h is true, regardless.

I outline several of these possible explanations in Sense and Goodness without God (especially in section III.3 on “The Nature and Origin of the Universe,” pp.  71-96, and most especially, pp. 86-88), and all of them are more plausible than “God did it,” which means, all have a higher prior probability, because all of them are based on established precedents or simpler assumptions (on this point in general see my End of Christianity chapter again, item 2 above, pp. 282-84). Accordingly, I’ll count this as my fifth listed resource.

So there you have it. A complete kit for battling The Lame.