Tag Archives: Bayes Theorem

What’s the Difference Between Frequentism and Bayesianism? (Part 2)

Witch of Endor by Nikolay Ge

Witch of Endor by Nikolay Ge

In the previous post we began to discuss the fundamental difference between the Bayesian and frequentist approaches to probability. A Bayesian defines probability as a subjective belief about the world, often expressed as a wagering proposition. “How much am I willing to bet that the next card will give me a flush?”

To a frequentist, however, probability exists in the physical world. It doesn’t change, and it isn’t subjective. Probability is the hard reality that over the long haul, if you flip a fair coin it will land heads up half the time and tails up the other half. We call them “frequentists,” because they maintain they can prove that the unchanging parameter is fixed and objectively true by measuring the frequency of repeated runs of the same event over and over.

Fairies and witches

But does objective probability really exist? After reading several books focused on subjective probability published in the past few decades, I couldn’t help noticing that Bruno de Finetti‘s Theory of Probability stands as a kind of watershed. In the preface he says that objective probability, the very foundation of frequentism, is a superstition. If he’s correct, that means it isn’t just bad science; it’s anti-science. He writes: read more »

What’s the Difference Between Frequentism and Bayesianism? (Part 1)

English: Picturing 50 realisations of a 95%-co...

English: Picturing 50 realisations of a 95%-confidence interval (Photo credit: Wikipedia)

As my thesis partner and I gathered up the evidence we had collected, it began to dawn on us — as well as on our thesis advisers — that we didn’t have enough for ordinary, “normal” statistics. Our chief adviser, an Air Force colonel, and his captain assistant were on the faculty at the Air Force Institute of Technology (AFIT), where my partner and I were both seeking a master’s degree in logistics management.

We had traveled to the Warner Robins Air Logistics Center in Georgia to talk with a group of supply-chain managers and to administer a survey. We were trying to find out if they adapted their behavior based on what the Air Force expected of them. Our problem, we later came to understand, was a paucity of data. Not a problem, said our advisers. We could instead use non-parametric statistics; we just had to take care in how we framed our conclusions and to state clearly our level of confidence in the results.

Shopping for Stats

In the end, I think our thesis held up pretty well. Most of the conclusions we reached rang true and matched both common sense and the emerging consensus in logistics management based on Goldratt’s Theory of Constraints. But the work we did to prove our claims mathematically, with page after page of computer output, sometimes felt like voodoo. To be sure, we were careful not to put too much faith in them, not to “put too much weight on the saw,” but in some ways it seemed as though we were shopping for equations that proved our point.

I bring up this story from the previous century only to let you know that I am in no way a mathematician or a statistician. However, I still use statistics in my work. Oddly enough, when I left AFIT I simultaneously left the military (because of the “draw-down” of the early ’90s) and never worked in the logistics field again. I spent the next 24 years working in information technology. Still, my statistical background from AFIT has come in handy in things like data correlation, troubleshooting, reporting, data mining, etc.

We spent little, if any, time at AFIT learning about Bayes’ Theorem (BT). I think looking back on it, we might have done better in our thesis, chucking our esoteric non-parametric voodoo and replacing it with Bayesian statistics. I first had exposure to BT back around the turn of the century when I was spending a great deal of time both managing a mail server and maintaining an email interface program written in the most hideous dialect of C the world has ever produced. read more »

Proving This! — Hoffmann on Bayes’ Theorem

Alan Mathison Turing

Alan Mathison Turing: Genius, Computational Pioneer, and BT Fan (Photo credit: Garrettc)

Misunderstanding a theorem

Over on New Oxonian, Hoffmann is at it again. In “Proving What?” Joe is amused by the recent Bayes’ Theorem (BT) “fad,” championed by Richard Carrier. I’ll leave it to Richard to answer Joe more fully (and I have no doubt he will), but until he does we should address the most egregious errors in Hoffmann’s essay. He writes:

So far, you are thinking, this is the kind of thing you would use for weather, rocket launches, roulette tables and divorces since we tend to think of conditional probability as an event that has not happened but can be predicted to happen, or not happen, based on existing, verifiable occurrences.  How can it be useful in determining whether events  ”actually” transpired in the past, that is, when the sample field itself consists of what has already occurred (or not occurred) and when B is the probability of it having happened? Or how it can be useful in dealing with events claimed to be sui generis since the real world conditions would lack both precedence and context?

I must assume that Joe has reached his conclusion concerning what he deems to be the proper application of Bayes’ Theorem based on the narrow set of real-world cases with which he is familiar. He scoffs at Carrier’s “compensation” that would allow us to use BT in an historical setting:

Carrier thinks he is justified in this by making historical uncertainty (i.e., whether an event of the past actually happened) the same species of uncertainty as a condition that applies to the future.  To put it crudely: Not knowing whether something will happen can be treated in the same way as not knowing whether something has happened by jiggering the formula.

Different values yield different answers!

I’m not sure what’s more breathtaking: the lack of understanding Hoffmann demonstrates — a marvel of studied ignorance — or the sycophantic applause we find in the comments. Perhaps he’s getting dubious advice from his former student who’s studying “pure mathematics” (bright, shiny, and clean, no doubt) at Cambridge who told him:

Its application to any real world situation depends upon how precisely the parameters and values of our theoretical reconstruction of a real world approximate reality. At this stage, however, I find it difficult to see how the heavily feared ‘subjectivity’ can be avoided. Simply put, plug in different values into the theorem and you’ll get a different answer. How does one decide which value to plug in?

You don’t have to do very much research to discover that Bayes’ Theorem does not fear subjectivity; it welcomes it. Subjective probability is built into the process. And you say you’re not sure about what value to plug in for prior probability? Then guess! No, really, it’s OK. What’s that? You don’t even have a good guess? Then plug in 50% and proceed.

It’s Bayes’ casual embrace of uncertainty and subjectivity — its treatment of subjective prior probability (degree of belief) — that drives the frequentists crazy. However, the results speak for themselves.

And as far as getting different answers when you plug in different numbers, that’s a common feature in equations. Stick in a different mass value in F = ma, and — boom! — you get a different value for force. It’s like magic! Good grief. What do they teach at Cambridge these days?

The proper application of BT forces us to estimate the prior probabilities. It encourages us to quantify elements that we might not have even considered in the past. It takes into account our degree of belief about a subject. And it makes us apply mathematical rigor to topics we used to think could be understood only through intuition. Hence BT’s imposed discipline is extraordinarily useful, since we can now haggle over the inputs (that’s why they’re called variables) rather than argue over intuitive conclusions about plausibility — because truthfully, when a scholar writes something like “Nobody would ever make that up,” it’s nothing but an untested assertion.

Bayes’ Theorem ascendant

If you can possibly spare the time, please watch the video after the page break. In it, Sharon Bertsch McGrayne, author of The Theory That Would Not Die: How Bayes’ Rule Cracked the Enigma Code, Hunted Down Russian Submarines, and Emerged Triumphant from Two Centuries of Controversy recounts the story of how Bayes’ Theorem won the day. She tells us how BT is well suited for situations with extremely limited historical data or even no historical data — e.g., predicting the probability of the occurrence of an event that has never happened before. read more »

Putting James the Brother of the Lord to a Bayesian Test

spelt out in blue neon at the offices of Auton...

spelt out in blue neon at the offices of Autonomy in Cambridge. (Photo credit: Wikipedia)

I saw none of the other apostles, except James the brother of the Lord. — Galatians 1:19

On this verse some hang their strongest assurance that Jesus himself was an historical figure. Paul says he met James, the brother of the Lord (assumed to be Jesus), so that is absolute proof that Jesus existed. That sounds like a perfectly reasonable conclusion. So reasonable, in fact, that some quickly denounce as perverse cranks any who deny this “obvious meaning”.

But should historians be content with this? Is it being “hyper-sceptical” to question this explanation?

We need to keep in mind some fundamental principles of historical research and explanations from the professional historians themselves. Renowned conservative historian, Sir Geoffrey Elton, warns against deploying such simplistic methods as citing a single piece of evidence to make a case. In this instance, the case is about evidence for the historicity of Jesus.

Historical research does not consist, as beginners in particular often suppose, in the pursuit of some particular evidence that will answer a particular question (G.R. Elton, The Practice of History, p.88)

If that’s what historical research is not, Elton goes on to explain what it is:

it consists of an exhaustive, and exhausting, review of everything that may conceivably be germane to a given investigation. Properly observed, this principle provides a manifest and efficient safeguard against the dangers of personal selection of evidence. (p.88)

Since I am currently reading and reviewing Richard Carrier’s Proving History: Bayes’s Theorem and the Quest for the Historical Jesus I am taking time out in this post to see what happens if I test this “obvious” interpretation of Galatians 1:19 by means of Bayesian principles. Carrier argues that Bayes’ Theorem is nothing more than a mathematical presentation or demonstration of what goes on inside our heads when we are reasoning about any hypothesis correctly. So let’s try it out on the conclusions we draw from Galatians 1:19.

The way it works is like this. (But keep in mind I am a complete novice with Bayes’ theorem. I am trying to learn it by trying to explain what I think I understand so far.) I see a verse in Paul’s letters that appears to have a simple explanation. I think of myself as a geologist looking at strata in a rock face and I think about all I know about strata and the evidence in front of me and with all that in mind I try to work out how that strata came to look the way it does. This verse is like that strata. My task is to test a hypothesis or explanation for how it came to be there and to appear as it does.

So the explanation, or hypothesis, that I decide to test is: That James, whom Paul meets according to this letter, was a sibling of Jesus. That’s my initial explanation for this verse, or in particular this phrase, “James the brother of the Lord”, being there.

It seems pretty straightforward, surely. This should be easy enough to confirm.

So let’s set it out in the theorem structure. read more »

Carrier’s “Proving History”, Chapter 3(a) — Review

provinghistoryI have been studying the first half of Richard Carrier’s chapter 3, “Introducing Bayes’s Theorem”, in his recent book Proving History: Bayes’s Theorem and the Quest for the Historical Jesus. I mean studying. I want to be sure I fully understand the argument before tackling the second half of the chapter, headed Mechanics of Bayes’s Theorem, which promises to be “the most math-challenging section of the book” (p. 67). Maths is not my most outstanding strength so I want to be sure I get the basics clear before moving into those waters. I have come to a point where I can enjoy playing little mind-games with Bayes’ Theorem for the sake of reinforcing my understanding. Last night on the TV news was dramatic story of an unexpected resignation of a leading Australian political figure. So I found myself piecing all I heard, how I heard it and what I knew etc. into a Bayes’ equation and calculating the probability that the story was true. Kind of fun. At least for the moment before the novelty factor wears off.

Result: While I believe I can see Richard’s point some of my niggling questions have not yet gone away.

When did the sun go out?

Carrier begins by setting out our reasoning when we read in the Gospels that darkness covered the whole earth for three hours at the time of the crucifixion of Jesus. What he is seeking to do is to take readers through the processes they would undergo in order to conclude that such an event almost certainly never really happened.

To make the scenario work he posits at least a barely conceivable natural cause for the event: “a vast dense cloud of space-dust swiftly drifting through the plane of the solar system . . .” — Wouldn’t the Sun’s gravity prevent that? But I’m happy to go along with the exercise for sake of argument nonetheless.

The critical point for Carrier is that what would convince us that such an event really had happened in the past is if we could find records testifying of the event across all world cultures thousands of miles apart from Britain to China.

There could not fail to have been mention or discussion of such a remarkable and terrifying event across many of these cultures among their surviving textual traditions and materials. (p. 43).

The key point is that we know in advance that this is the evidence we would expect to find IF such an event had happened.

And if indeed that were the case, we would surely have adequate warrant to believe the sun was blotted out for three hours on the corroborated day . . .

What Carrier is preparing his readers for is to accept that reasoning about historical events is fundamentally similar to reasoning in the sciences. If such and such a hypothesis (or explanation) is true then we would predict (or expect) certain events (or evidence) to be manifest.

Then there is the converse. If such a hypothesis (explanation) were true, we would NOT expect to find a universal silence in the surviving records:

[A] single claim in a single religion repeated only in its own documents (and documents relying on those), is extraordinarily improbable — unless the event was entirely made up. . . . This is a slam-dunk Argument from Silence, establishing beyond any reasonable doubt the nonhistoricity of this solar event . . . (p. 44)

My niggling question:

I follow the reasoning. But in my mind, rather than taking me into the realm of mathematics, it all leads back to my own argument about how historians know anything at all about the persons and events of the past. read more »

Richard Carrier’s “Proving History: Bayes’s Theorem and the Quest for the Historical Jesus” Chapter 1 (A Review)

Till now I’ve always been more curious than persuaded about Carrier’s application of Bayes’s Theorem to what he calls historical questions, so curiosity led me to purchase his book in which he discusses it all in depth, Proving History: Bayes’s Theorem and the Quest for the Historical Jesus.

Before I discuss here his preface and opening chapter I should be up front with my reasons for having some reservations about Carrier’s promotion of Bayes’ theorem. (Allow me my preference for Bayes’ over Bayes’s.) I should also say that I’d like to think I am quite prepared to be persuaded that my resistance is a symptom of being too narrow-minded.

My first problem with Carrier’s use of the theorem arises the moment he speaks of it being used to “prove history” or resolve “historical problems”. For me, history is not something to be “proved”. History is a quest for explanations of what we know has happened in the past. Historical problems, to my thinking, are problems having to do with how to interpret and understand what we know has happened in the past. The milestone philosophers of the nature of history — von Ranke, Collingwood, Carr, Elton, White — have certainly spoken about history this way.

I have always understood that where there is insufficient data available then history cannot be done at all. Ancient history, therefore, does not allow for the same sorts of in-depth historical studies as are available to the historian of more recent times. Historical questions are necessarily shaped (or stymied altogether) by the nature and limitations of the available sources.

Criteriology (I take the term from Scot McKnight‘s discussion of the historical methods of biblical scholars in Jesus and His Death) has always looked to me like a fallacious attempt to get around the problem of having insufficient data to yield any substantive answers to questions we would like to ask. We don’t know what happened? Okay, let’s apply various criteria to our texts to see if we can find out what “very probably really did happen”.

Carrier’s introduction of Bayes’ theorem has always appeared to me to be an attempt to salvage some value from a fundamentally flawed approach to “history” — the striving to find enough facts or data with which to begin to do history.

I should add that I do like Carrier’s offering of hope that Bayes’ theorem can promote more rigorous and valid thinking and applications of criteria. But I can’t help but wonder if in the end the exercise is an attempt to patch holes in the Titanic with admittedly very good quality adhesive tape.

What is really accomplished if we find only a 1% probability for the historicity of Jesus? Improbable things really do happen in the world. Otherwise we would never know chance and always be living with certainty. Or maybe I’m overlooking something about Carrier’s argument here.

Not that I’m a nihilist. I do believe we have lots of useful evidence to assist us with the study of Christian origins. I think scholars are agreed that pretty much all of that evidence speaks about a Christ of faith (a literary figure) and not an historical figure. That’s where our historical enquiry must begin — with the evidence we do have. After we analyse it all and frame such questions as this sort of evidence will allow us to ask then we can begin to seek explanations for Christian origins. This will probably mean that we will find answers that do not address the life and personality of someone who is hidden from view. Our understanding will address religious developments, ideas, culture, literature, social developments. We will probably be forced to conclude — as indeed some historians do — that if there is an historical Jesus in there somewhere he is irrelevant to our enquiry.

So that is where I am coming from.

Let’s see if I am being too narrow-minded. Here is my reading of Carrier’s preface and opening chapter. read more »

Demystifying R. Joseph Hoffmann, and the war over Bayes’ theorem

Some academic jargon is nothing more than the modern equivalent of a sorcerer’s mumbo jumbo — designed to awe while hiding the fact that there is really nothing meaningful on offer at all

Updated 5th July to add link to Richard Carrier's post taking Hoffmann to task.

R. Joseph Hoffmann has let a crotchety side to his nature show as he publicly attempts to humiliate a younger scholar who, in exchanges with the aging don, has exposed a dint of mediocrity in his intellect.

The casus belli is, at least on the surface, the place of Bayes’ theorem in historical Jesus studies.

Now Hoffmann’s writing is surely more renowned for its thick overlays of esoteric intellectual jargon and rhetoric than for its content. The reason is pure mathematics. More people can read his posts than can understand them. (Stanislav Andreski wrote that this sort of intellectual jargon as the modern equivalent of earlier efforts to bamboozle the uninitiated and impress the elite: various uses of medieval Latin, witch-doctor mumbo-jumbo, etc.)

But on the positive side, one does get a sense that he is thoroughly enjoying himself as he shows off his verbal wit, and there’s nothing wrong with that. Everyone has a right to enjoy themselves (or “oneself”, as I am sure the don would prefer me say).

But what the hell is he trying to say when he burgeons like a Baroque artist doing abstract?

I’m sure he won’t like any attempt at simplification, but then why would any biblical scholar be bothered with a blog like mine when the guild does not even consider it to be an honest discussion of the Bible and Christian origins anyway.

All Hoffmann means to say is that he thinks: read more »

Bayes’ theorem and the Jesus mythicism-historicity conflict

Bayes theorem

I showed pictures of adorable or scary animals to counteract the inherent boredom of reading math. Like, for instance, the hippopotamus.

Richard Carrier is well known for his advocacy of the use of the Bayes’ theorem in historical Jesus studies. (Find the link to Bayes’ Theorem for Beginners here or go direct to the pdf article here.) Carrier has enumerated its advantages, and I highlight the ones that are my own personal favourites (all quotations are from the pdf article, Bayes’ Theorem for Beginners):

1. Helps to tell if your theory is probably true rather than merely possibly true

2. Inspires closer examination of your background knowledge and assumptions of likelihood

3. Forces examination of the likelihood of the evidence on competing theories

4. Eliminates the Fallacy of Diminishing Probabilities

5. Bayes’ Theorem has been proven to be formally valid

6. Bayesian reasoning with or without math exposes assumptions to criticism & consequent revision and therefore promotes progress

The reason 2, 3 and 6 stand out for me is because they are at the heart of my past criticisms of historical Jesus studies that typically begin with assumptions of historicity, and avoid (or fail to comprehend or even attack) alternative explanations that do not support those assumptions. One does not really need Bayes theorem to expose your assumptions to criticism, but the formality of this method does potentially encourage stronger awareness of where we may be failing to do this adequately. read more »