Garbage in/Garbage out. Bayes theorem is handy when you are trying to determine the probably of an unlikely event when the alternatives are also unlikely, but you still need valid data.
From what I could tell, the terms of that equation are just full of supposition and numbers plucked out of the air. You can prove anything that way.
I actually agree with most of the arguments Stephen T. Davis made, even though I would disagree with his own conclusions. I didn’t read the first link (requires registration), but if the depiction of the arguments in Davis’s rebuttal are true, then it looks like it was a pretty sloppy piece of work.
But the real problem with using Bayes’ Theorem for something like this is that the use of it assumes that you have reasonable numbers to provide.
These are some of the terms he uses: (where R = the Resurrection hypothesis; K = the background evidence; EH = specific historical evidence; AT = various alternative hypotheses (myth, conspiracy, etc.)
Out of all that is supposed to come a number which represents a probability? Please. You could argue a lifetime over every single one of those terms. And even then, you can’t know, because you don’t know the intent of God. God is the big hidden unknown term here. A belief in a biblical God assumes a directed existence (i.e. he’s controlling the shots to some degree). Bayes’ Theorem assumes events happen within defined limits of probability. The two just don’t work together.
Consider this analogy: If I make a marble statue and throw it in a quarry full of raw marble, and you find it, you could come up with a very intricate equation using Bayes’ theorem to show that it’s vanishingly unlikely that a chunk of marble should take on the perfect shape of, say, a horse. But that equation would be completely wrong in its initial assumptions (that random forces acted on all), and so its conclusion is ridiculous.
This flaw crops up every time overzealous scientists try to use science to either prove or disprove miraculous events.
Out of all that is supposed to come a number which represents a probability? Please.
Well lets see…
Is it in the set [0,1]? Yes. Okay, so far so good.
The problem you seem to be having is with the subjective nature of the probability assessments. To that I say, better get used to it. Subjective Bayesian methods are becoming more and more popular, IMO. With improvements in computer hardware I think Bayesian (Subjective and Objective) will become more and more common.
*You could argue a lifetime over every single one of those terms. *
Sure you could. This is true of even more mainstream uses of Bayes theorem. The advantage with the more mainstream uses is the availability of better data.
A belief in a biblical God assumes a directed existence (i.e. he’s controlling the shots to some degree).
In your opinion. I don’t see why the existence of God (and I don’t believe he (or it) does exist myself) should imply that he is controlling everything. Does the scientist control every aspect of an experiment? No. If God controlled mankind how come he let Adam and Eve partake of the fruit from the Tree of Forbidden Knowledge?
The way to attack this argument is to do a sensitivity analysis. Point out that the probability assessments of the author don’t have to be true.
That is select different prior probabilities and see what happens to the result. What prior probability is necessary for the event to have a zero probability?
I didn’t say he’s directing everything, and I specifically said a ‘biblical’ god, to avoid the inevitable hand-wringing assumptions about the nature of deities.
In other words, we are talking about an event that was initiated and controlled by an omniscient being. You just can’t apply probability theory to that. What are the odds that God would send Jesus down at that exact time? The correct answer is, “Ask God”. We are lacking significant amounts of information (i.e., “What was God thinking?”)
Bayesian analysis doesn’t have to be about subjective vs objective. At its root, it’s simply a technique to establish the probability of an unlikely event, when the alternatives are also unlikely events.
For example, take the dinosaur extinction. If the odds are 1 in a million that an asteroid would hit the earth, it sounds pretty unlikely that it could have caused a dinosaur extinction.
However, we can use Bayesian analysis to narrow that down, because we have the knowledge that an extinction DID occur. Let’s say that we narrowed down the possibilities to three different events, each of which only had a probability of one in a million. By applying Bayes’ theorem, we suddenly discover that the probability of an Asteroid strike, given just those three options, is now .33 instead of 1 in a million.
But if even one of those other probabilities is way off, then the conclusion from the Bayesian analyis will be off by just as much.
The formula for the probability of the resurrection of Jesus contains a whole bunch of terms which are essentially guesses. I have a hard time seeing how any number that comes out of that equation will be any better than the original guesses that went in.
Sure you could. You are not that omniscient being therefore the true state of the world is uncertain to you.
Yes, I know and that is one of the strengths, IMO, of Bayesian analysis. Bayesian analysis allows you to make probability assessments with imprefect information and provides a very neat mechanism for updating your assessments as new information arises. Since we can’t ask God we have to make do with what we have.
Wow, is that a reduction to absurdity or what? Bayesian analysis is more of a methodological viewpoint than it is just a simple technique. I have a belief about some event. I formulate that belief probabilistically. I get new data and use Bayes Theorem to update my beliefs. To say it is just a technique is to ignore the huge philosophical gap between the Bayesian veiwpoint and other statistical methodologies.
Which contains very large amounts of subjectivity in it. Shame on you for not pointing them out. Why did you assign zero probabilities to the fourth, fifth, etc. events that could cause extinction?
Ahhh I think I see it. You have a fundamental misunderstanding. The result is not literally True. In fact, a person from the Bayesian viewpoint wouldn’t say it is True, but that given the prior this is probability. You can then do a sensitivity analysis (i.d. different priors) and look at the result. Further, as you gather new information (data) you can update your results using Bayes Theorem.
Yes, subjective probability assessments. This does not bother me, apparently it does you. However, as I noted the same thing takes place with the use of Bayesian methods in more mainstream applications.
Well the Bayesian would say, does it predict well? For any model that is the ultimate criterion. Does it predict well. If not then the model, no matter how spiffy, is no good.
And while we are at it, why did you pick a uniform probability distribution for your prior?
That’s hardly a sufficient critirion for whether it’s a probability. What, exactly, does .97 represent? Does it mean that .97 of the time when someone says “The resurrection happened”, they’ll be right? I think “confidence assessement” would be a better term for this number than “probability”, seeing as how the resurrection is not a statistic.
Yes, but you aren’t predicting anything when talking about the resurrection. In this case, you’re trying to assign a probability to a past event, by making assumptions about the probabilities of the events surrounding it or supporting it. There is no way to test this model. And since the numbers going in are completely arbitrary, I fail to see the value of the analysis at all.
Just look at the terms used in the equation. Submitted as evidence are things like the frequency of miracles happening in the bible, alternate explanations for the events surrounding the resurrection, etc. But this misses a larger, meta-analysis of all this, which is that the entire edifice could be concocted out of whole cloth. We don’t need the resurrection to explain subsequent events in the bible, because we don’t even know if those events happened at all. We can’t count the frequency of miracles because A) if there is a god, we don’t know if the bible is an accurate account of ALL miracles, or just the ones necessary to support the theology, and b) if there is no god, those stories are fabrications or exaggerations.
I fail to see how any of this stuff is reasonable enough to serve as terms in an equation like this, other than for purely speculative reasons (i.e., "Let’s say that the probability of Jesus rising was X. What does that say about other possibilities?). That may be a valid approach and an epistemelogical tool, but is not ‘proof’ of anything at all, even if you can jigger the numbers to make the resultant probability come out to .999999
But I’ll admit this is a novel application of Bayes’ Theorem, and perhaps I’m missing something here. I’ve used it before for concrete analysis of very specific controlled situations (for example, games of chance). I also didn’t read your first link, because I don’t want to register for the New York Times.
Sam: Re: The first link. You didn’t miss much, IMHO. One could easily read the article without understanding what a Bayesian approach is, aside from a nifty-sounding piece of terminology.
Sam *Garbage in/Garbage out. Bayes theorem is handy when you are trying to determine the probably of an unlikely event when the alternatives are also unlikely, but you still need valid data. *
AFAIK, the usefulness of Bayes theorem is not limited to low probability events.
Bayes’ theorem lays out the method for combining disparate probabilistic information. If the quality of that information is suspect, the results will be as well.
More generally, a Bayesian approach can be useful when one assessing probabilities in order to make a decision. The tricky part is setting the probability distributions in the first place.
(I’ll set aside the issue of whether Bayesian computer software is sufficiently advanced to cover the whole range of plausible probability distributions. Mainly because I don’t know the answer.)
To evaluate the likelihood of the resurrection, I would advocate an old fashion weighing of the evidence, something like that undertaken .here or here.
kesagiri, I highly recommend that you take a look at The Foundations of Statistics, by Savage. This book describes the philosphy underlying Bayesian statistics. Most of it is non-technical. It’s a classic.
Bayesian statistics describes a personal probability. That is, one person’s beliefs. One begins with certain prior assumptions and modifies them appropriately in the light of additional information. The modification results in an a posteriori probability. However, note that this modified probability is still a personal probability. Bayes Rule tells you how to modify your beliefs in a coherant fashion, but it doesn’t say that anyone else must agree with your beliefs.
It is a theorem that under certain assumptions, two different people, each starting from their own prior, will come close together, after adjusting their priors sufficiently many times. However, the supposed Bayesian proof of God doesn’t do all that.
Flowbark: Sorry if I said or implied that it was only useful for low probability events. The example I used showed how Bayesian analysis could be used to determine a likely outcome given several unlikely options, but certainly it’s not limited to that.
December: I don’t know that I buy that explanation either.
Let me explain a situation where I’ve used a Bayesian analysis regularly. I used to be a professional poker player. I was often faced with a situation where a player would raise me, and I’d have to try to figure out what he might have. A Bayesian analysis can help here. For instance, let’s say the guy is an ultra-tight, unimaginative player. He raises me on the river. representing a straight flush (say, the board is paired, and I know that he thinks I have a full house). Now, I know the odds of a straight flush in that situation may be 100,000 to 1, but I also know that the odds of him raising me in that situation with a lesser hand is almost as unlikely. Given the various comparisons, you can know within a high degree of likelihood that the guy actually has a very unlikely hand.
Another, more valuable application would be in things like disease detection. Let’s say you have a test for a disease that is 98% accurate. But the disease you are testing for is only found in one person in a million. Using a Bayesian analysis, you can show that even if a test for that disease comes up positive, the odds that the person has it are still very, very low. The smart money in that case would be that the test returned a false positive.
And this differs from what I have written exactly how? I haven’t anywhere claimed that this calculation is correct (i.e. as in a Truth has been discovered) except maybe in a methodological sense (i.e. did the author correctly apply Bayes Theorem). I quite agree that the subjective nature of the priors means that it is open to alot of interpretation.
Exactly, or as Priorer put it, two researches agree to disagree (i.e. different priors) and let the data settle the dispute.
Sadly this is true. Unfortunately with the reporting of a probability of .97 it is likely many people will come away with the incorrect conclusion. I think until this has been put through a sensitivity analysis it is just another example of Creationism run amok. Dembsky, Behe, all the rest, and now this fellow.
It means that given the priors the probability that Jesus rose from the dead is .97.
Eeeee! How dare you use that word with regards to Bayes theorem. Down Vile one! Bayes theorem does indeed return correct probabilities, but the result is open to a great deal of interpretation.
Lets look at Sam’s example of the dinosaur extinction event. He has settled on 3 possible cuases. Then he also seemed to decide that each cause would result in extinction with a probabiltiy of 1. Then he decided that the prior for each of the three events is the uniform distribution (i.e. each event has a probability of 1/3 occurring).
Now does that sound reasonable? Not to me. How come each event had to have a probabiltiy of 1 attached to causing extinction? Because is happened? I think not.
A few years ago, my reinsurance company was offered the opportunity to share a certain portion of another insurance company’s business. I received a consulting actuary’s report showing that their business was profitable. I was surprised, because I know most companies were losing money in that line of insurance.
When I checked the other actuary’s work, I discovered that he had used a Bayseian method. His predicted result depended very heavily on his initial assumption. It turned out that his starting point was so rosy, that his prediction was still pretty good, even though actual results had been bad.
I asked the actuary where he obtained his starting values. He replied, “Management estimate.” I suppose that’s a good way to retain one’s clients.
Had I been doing the study, my prior would have been to assume bad results, because that what other companies were reporting.
“How We Know What isn’t So”
Tells us all we need to know.
Tells as all the about the ways
Human brains can misuse Bayes.
Let belief get in our mind
Then we’ll promptly strive to find
Evidence that we think might
Serve to show that we are right
While ignoring all day long
Evidence that proves us wrong.
Bayesian proofs that God is nigh us
Suffer from selection bias.
Suppose you have a coin, which you know is biased one way or the other. You know for certain that it will either {land heads 2/3 of the time} or {land tails 2/3 of the time}. Let’s say that you believe these two possibilities are equally likely. Then, you assign 50% probability to each case. That’s your prior distribution: 50-50.
Now you flip the coin once and it comes up heads. IF the coin were biased toward heads, this would have probability 2/3. IF it’s the coin biased toward tails, the probability would have been 1/3. Your a posteriori probabilies on the coin are odds of 2 to 1, by Bayes Rule. I.e., probabilities of 2/3 that it’s biased toward heads, 1/3 biased toward tails. In other words, the probability of the coin being biased toward heads is 66.7%.
You could continue flipping the coin and revising your probabilities after each flip, by the use of Bayes Rule. If you continued to get more heads than tails, you would become more and more certain that your coin was biased toward heads. At some point, that probability might be 97%.
Note two key problems:
If someone else started with a different prior, the two of you wouldn’t ever agree (although your estimates and his would become closer and closer.)
Suppose lots and lots of information about the coin were available, not just a series of flips. Then you’d need to decide which information to use and which to ignore. Someone else might choose to utilize different information than you did; the two of you would never agree.
The book cited above argues that human psychology is such that we tend to focus on information that supports opinions we already hold. The religious person will modify his opinion, by using information tending to indicate that God exists. he will become more and more certain that he’s right.
The atheist will look for evidence confirming his beliefs. He will become more and more certain that he’s right.
I do know a little bit about Bayesian analysis. My question was about the specific statement, “the probability that Jesus rose from the dead is .97”. I think that this has to be interpreted as a subjective probability (after all, the resurrection either happened or it didn’t, so the actual probability can only be zero or one), but I wanted kesagiri to explain exactly what (s)he meant by that. Thanks for the explanation, though; I’m sure somebody will find it useful.
Yes, prior probability elicitation is one of the problem areas of Bayesian analysis. However, in the case above the author should have provided different scenarios with different priors and then let management pick which one they felt was best.
I agree that assuming the “best case scenario” is going to be the default is bad planning.
Prior probabilities can depend on one’s subjective beliefs as well as other information that is at hand. For example, suppose you are trying to forecast market share for an item your company makes, but you find out that a major competitor just decided to close up shop. What will happen to your market share? Probably go up? By how much? Well that is where the subjective intervention comes in. You would ideally look around at a similar situation and use that as a base case.
Traditional forcasting (i.e. the Classical/Frequentist method) does not allow for this kind of infomation to be readily included in the analsysis.
You are having some similar problems that Sam Stone had, IMO.
Suppose I flip a coin. It is sitting here on my desk (i.e the flip is done and no more probability exists…for me). Now, is it heads or tails? You are still uncertain as to what the outcome is. So phrasing your answer probabilistically is still valid, IMO. Look at Sam’s dinosaur example. He implicitly assumed that each of his three extinction hypotheses would result in the extinction of the dinosaurs with a probability of 1. Why? Lets go back to my coin. It landed heads. Does this mean that everytime I flip it (I believe it is a fair coin, it is one I got out of a change box at home) it will land heads with probability 1?! No.
Sloppy Bayesian analysis will give you sloppy results.
Yes, the resurrection probability above is a subjective probability. That is why I think a sensitivity analysis would be illuminating. Suppose we start changing the priors and observe what happens.
LOL. Lets suppose this is indeed true (and it sure sounds good to me). However, this is not just a failing of Bayesian analysis. Anybody who claims otherwise is being misleading at best.
Some background, there are (to over generalize a bit) two large schools of thought regarding statistics; the Bayesian and the Frequentist.
Now, in many cases the results that you obtain using Frequentist methods are also attainable using Bayesian methods given you select the appropriate priors. That is, the Bayesian method subsumes the Frequentists results (although the interpretations are quite different).
Also, the idea that there is no subjectivity in the Frequentist method is patently false. Why did you select the model you did? The data you are using? The tests that you apply? And so on and so forth. One fo the big differences is that the Bayesian analyst is just alot more upfront with his subjective decisions. It is right there in the prior. With the Frequentist analyst it is less obvious providing, IMO, a false patina of objectivity.
So the above criticism is a general one and not something that can be leveled just at the Bayesian point of view. To claim otherwise, IMO, is intellectually dishonest.
In the classic view of Savage, probability for an individual is defined by the odds at which he would be willing to make a bet. Under the example I gave, the a priori probability would have been 50-50 for you if you were willing to bet on either possibility with no odds either way. Bayes law gives a consistent formula for modifying your probability distribution after observing the flip. After the flip, presumably, you would be indifferent to betting either way, provided that the odds were 2 to 1.
kesagiri, I didn’t mean to specifically attack Bayesian analysis. I’m a confirmed Bayesian, as are most casualty actuaries.
But, when proving resurrection,
Watch for bias in selection.
Why did you assign zero probabilities to the fourth, fifth, etc. events that could cause extinction?