Bayes’ Theorem and the Virgin Birth of Jesus

17 Feb

(The Annunciation, by Fra Angelico; courtesy Wikipedia)


To continue learning Bayes’ Theorem, I’m attempting to apply it to common arguments I come across. This time, I’m going to attempt to apply it to the virgin birth of Jesus, which is one of the first hurdles that one has to jump over when debating Christians and/or getting into more scholarly discussions of Jesus. While most Biblical scholars believe in the resurrection of Jesus in some fashion, a much smaller percentage of them believe in the virgin birth of Jesus. So dispensing with the idea that Jesus was born from a virgin is probably one of the first signs that one is becoming an educated Christian, to be slightly inflammatory 🙂
To start, here is Bayes’ Theorem again:

P(H | E) = P(E | H) * P(H) / [P(E | H) * P(H)] + [P(E | H) * P(H)]

There are other ways of writing it, but I like this one so far since this is the one I learned.

In order to solve this, we need to know only three terms: The prior probability P(H) and the two conditional probabilities P(E | H) and P(E | ~H). P(H) in normal wording would be the prior probability of someone being born from a virgin. P(E | H) would be the probability of our current evidence given that someone is born from a virgin. And P(E | ~H) would be the probability of our current evidence given some other explanation.

I like to think of P(E | H) being the success rate, and P(E | ~H) being the false positive rate. This language, however, I think only applies to binary tests. If P(E | H) and P(E | ~H) encompass more than two possible outcomes, then success rate / false positive rate language doesn’t apply. Anyway, the success rate divided by the false positive rate gives you how strongly the evidence favors your hypothesis. Or, if that ratio is less than 1, how much it disfavors your hypothesis.

It’s also good to keep in mind the compliments of the success rate and false positive rate. So for example, if some mammogram has a success rate of 80% (meaning that it successfully detects breast cancer in 80 out of 100 women who actually have breast cancer), by necessity this means that the mammogram does not detect breast cancer 20% of the time that a person actually has breast cancer; a false negative rate of 20 out of 100.

Similarly, if the false positive rate of some mammogram is 1% (meaning that it says someone has breast cancer 1 out of 100 times someone does not have breast cancer) this implies that the “true negative” rate is 99% (meaning that it says someone does not have cancer 99 out of 100 times a person actually does not have cancer). This means that we have two ways of determining each conditional probability and not just one (there are other ways using other variables, but I don’t want to get into that just yet!).

Lastly, we have to determine what we mean by “evidence” to get a good grasp on what the conditional probabilities (i. e. P(E | H) and P(E | ~H) ) and the posterior probability P(H | E) mean in normal English. What evidence do we have that Jesus was born from a virgin? Matthew 1.23 (and implied in Luke 1.34; 3.23), which itself is an interpretation of Isaiah 7.14. We can safely take Isaiah 7.14 out of evidence since in context this entire chapter has no inclination that it is attempting to describe events that would take place 700 years after it was written. For the Christians who still think so, I can only assume that they have never actually sat down and read (and understood) the early chapters of Isaiah and only read Isaiah 7.14 without the context of the surrounding paragraphs and chapters.

So the only evidence that we have that Jesus was born from a virgin is Matthew 1.23 (and Luke 1.34; 3.23, but it is my working hypothesis that Luke is following Matt, and Q didn’t exist). Which falls into the larger group of “stories about people born from the union of a woman and a god”.

The prior probability would the number of people in human history that have ever lived who were sired by a god and a mortal woman. Remember, this is prior probability. This is the probability we start with before determining how the evidence affects the hypothesis; the probability of Jesus being born from a virgin before we look at the specific evidence, namely Matt 1.23. Imagine if the entirety of humanity were represented by 1,000,000,000 people standing in a room. How many of those 1,000,000,000 people were born from women and a god? It’s zero, but zero isn’t a probability, so for the sake of this example I’ll just say that ten people in that room of 1,000,000,000 people were born from a virgin (i.e. woman egg + god “sperm”).

The task, then, is to show the probability that Jesus is part of that population of people who are born from virgins, given our evidence. Our evidence is stories of people born from virgins. To find that out, we need the success rate and false positive rates for stories of people being born from virgins. And to find that out, we need the number of people out of 1,000,000,000 who have stories of them being born from the union of a god and a woman. There are actually quite a few of these (Achilles, Julius Caesar, Alexander the Great, Helen of Troy, Romulus, etc.). I’ll take a rough estimate and say that 100 people out of 1,000,000,000 have stories of them being born from a god and a woman. Who knows how accurate that is, but I think it’s good enough for pedagogical reasons. The only thing that needs to be accurate, I think, is that stories of virgin births are more frequent than actual virgin births.

Now we look at the success rate and false positive rate.

Out of the 100 people estimated to have stories of them being born from a virgin all throughout both recorded and non-recorded history, how many of those people actually were born from virgins? Remember, this is prior to looking at the evidence for Jesus, so this also seems to be zero. But if that were zero, then the numerator for Bayes’ itself would be zero and that defeats the purpose of this exercise. So for the sake of argument, let’s say that P(E | H) gets it right once out of the 100 times it asserts that someone is born from a virgin. This, by the way, also affects the compliment of P(E | H) which is P(~E | H). That one is the number of people born from a virgin who don’t have stories about them being born from a virgin. Meaning that 90% of 10 people (out of the 1 billion in the room) born from a virgin don’t have stories about it.

On the other hand, out of this group of 100 people, how many people were not born from a virgin? This in reality seems to be 100 out of 100. Again, this is prior to analyzing Jesus so he’s not included. But we also have to take into account the compliment of P(E | ~H) which is P(~E | ~H). That is, the probability of not having a story about you being born from a virgin given that you in fact were not born from a virgin. P(~E | ~H) is the “true negatives” rate which is 1 billion minus the 100 false positives divided by 1 billion. That is 99.99999%. This, in turn, means that P(E | ~H) is 100% – 99.99999% and that’s 0.00001%

We now have our three variables. The prior probability is 10 out of 1,000,000,000. The success rate is 1 out of 100. The false positive rate is 100 out of 999,999,900. In normal English these would be:

1. P(H): What is the prior probability of being born from a virgin? 0.000001%

2. P(~H): What is the prior probability of not being born from a virgin? 99.999999%

3. P(E | H): What is the probability of having a story about being born from a virgin given that you actually were born from a virgin? 10%

4. P(~E | H): What is the probability of not having a story about being born from a virgin given that you actually were born from a virgin? 90%

5. P(E | ~H): What is the probability of having a story about being born from a virgin given that you in fact were not born from a virgin? 0.00001%

6. P(~E | ~H): What is the probability of not having a story about being born from a virgin given that you in fact were not born from a virgin? 99.99999%

The vast majority of humanity falls into the 6th category. There’s also a 7th variable, which is the Total Probability Theorem, or P(E). This is the probability of having a story about a virgin birth period. This number is actually the denominator of Bayes’: [P(E | H) * P(H)] + [P(E | ~H) * P(~H)].

7. P(E): What is the probability of having a story about being born from a virgin? 0.0000101%

This makes sense, because stories of virgin births in and of themselves are pretty rare. If we multiply P(E) by the total number of people in this hypothetical room – 1 billion – we get 101. Which is the 100 false virgin birth stories and the one success.

So, we start off with our prior probability of 0.000001% (10 out of 1 billion). How much does our evidence — Matt 1.23 — increase or decrease our prior probability of 0.000001%? Bayes:

= P(H) * P(E | H) / [P(H) * P(E | H)] + [P(~H) * P(E | ~H)]

= 10 out of 1 billion * 1 out of 10 / [10 out of 1 billion * 1 out of 10] + [999,999,990 out of 1 billion * 100 out of 999,999,990]

= 0.000001% * 10.0% / [0.000001% * 10.0%] + [99.999999% * 0.00001%]

= 0.0000001% / [0.0000001%] + [0.00001%]

= 0.0000001% / 0.000010100%

= 0.990099019703951%

So, due to the evidence at hand, we went from 0.000001% probability of being born from a virgin (i.e. 10 out of 1 billion) to 0.990099019703951% probability of being born from a virgin. This is still not very good evidence for Jesus’ virgin birth; it’s less than 1%. Especially since this still means that P(~H) is 100% – 0.990099019703951%, which is 99.009900990001%. Meaning that there is a 99.009900990001% chance that Jesus was not born from a virgin. We would need more evidence to continually corroborate and update that probability.

But this was all done assuming that virgin births have actually occurd in real life and that stories of virgin births actually have at least one positive hit with a real virgin birth. Also, a prior probability of 10 out of 1 billion is absurd. This would mean that there are around 70 people alive today who were born without male sperm. I only made such an assumption to privilege the virgin birth hypothesis a bit.

With Jesus’ virgin birth, we actually have zero known instances of people being born from virgins yet multiple false positives of people being born from virgins. So the false positive rate (which would stay the same as above) is actually much higher than the success rate which would make the likelihood ratio less than 1. We would need some other evidence that better attests to virgin births, other than just stories. This much is obvious, since extraordinary claims (like a virgin birth) require extraordinary evidence (stories about virgin births are not extraordinary).

A exacerbating factor is the low prior probability. Even if stories of virgin births had a much higher success rate, it would move the prior probability negligibly. Let’s see what happens when we have a 100% success rate:

P(E | H), the success rate, is 10 out of 10. P(E | ~H), the false positive rate, is 100 out of 999,999,990.

= P(H) * P(E | H) / [P(H) * P(E | H)] + [P(~H) * P(E | ~H)]

= 0.000001% * 100.0% / [0.000001% * 100.0%] + [0.00001% * 100.0%]

= 0.000001% / [0.000001%] + [0.00001%]

= 0.000001% / 0.000011%

= 9.09090917355372%

The prior probability moved up to a little over 9%. Again, this is with unrealistic numbers, such as all stories of virgin births besides Jesus’ story being true.

In closing, even if our success rate for stories about people being born without male seed were 100% true, due to the low prior probability and the high false positive rate this is not enough to make it a good argument. Ignoring the prior probability and the false positive rate while only concentrating on the success rate (in this case, the Bible) is nothing less than the base rate fallacy (see also Prosecutor’s Fallacy and also the False Positive Paradox).

This low prior probability also applies to the resurrection of Jesus. As I like to say, the virgin birth of Jesus is no less believable than the resurrection of Jesus, so people really have no warrant for choosing one over the other. The above run-through of Bayes’ with the virgin birth equally applies to the resurrection of Jesus, simply substitute “virgin birth” with “resurrection”. Both suffer from low prior probabilities, low success rates, and high false positive rates. And for both, even if the success rate were insanely high, this high success rate isn’t high enough to make the prior probability of virgin birth/resurrection from the dead a rational belief since the false positive rate will always be higher than the prior probability (this situation creates the False Positive Paradox). For both, we would need multiple lines of corroborating high success rate evidence to move the prior probability to a reasonable level.


Posted by on February 17, 2012 in Bayes


8 responses to “Bayes’ Theorem and the Virgin Birth of Jesus

  1. Jon Hanson

    February 18, 2012 at 8:45 am

    This is great stuff. It's amazing how Bayes' Theorem brings these issues into such clear focus, I really do think it can revolutionize the religious debate and I love seeing people like you at the forefront pushing this stuff.

    I really can't imagine where a Christian would criticize this, it seems like they'd have to argue that the evidence for the Bible as a top flight source of information is just off the charts.

  2. Julien Rousseau

    February 18, 2012 at 9:23 pm

    There is another line of evidence, though I doubt the christians will want to accept it as corroborating evidence:

    God told mohammed that Jesus was born of a virgin.

    So, how many christians are ready to accept the koran as true to have a second, independent line of evidence? *crickets chirping*

  3. Jake

    March 1, 2012 at 10:51 pm

    Hi J,

    I think you're pulling a bait and switch on “H” in the Bayes analysis. You define the prior probability p(H) as the probability of any old human being born of a virgin. You assume, quite rightly, that based upon how human sexual reproduction works, this is biologically impossible. Though you are charitable and allow for some miracles, so that's good that you're not being dogmatic about it.

    So assuming Jesus was just some human, the analysis may stand. However, if Jesus is God, then that changes H altogether. Now we're not talking about human sexual reproduction. We're talking about the probability that God incarnated into the womb of a virgin. So your prior probability is irrelevant.

    So sorry to burst your bubble.

  4. J. Quinton

    March 2, 2012 at 8:54 pm

    Jesus being god is an extraordinary claim, just like virgin births and resurrections from the dead. We would need extraordinary evidence to support it.

  5. Jake

    March 3, 2012 at 6:06 am

    Could you define “extraordinary claim” and “extraordinary evidence”?

    To me it sounds like you're saying that for any claim you find implausible, you can always demand more evidence. Sounds pretty dogmatic to me.

  6. Jake

    March 3, 2012 at 1:46 pm

    I also think you're being very dishonest about other virgin birth stories. Even just following your Helen of Troy link outs one of your lies:

    “In most sources, including the Iliad and the Odyssey, Helen is the daughter of Zeus and Leda, the wife of the Spartan king Tyndareus.[11] Euripides' play Helen, written in the late 5th century BC, is the earliest source to report the most familiar account of Helen's birth: that, although her putative father was Tyndareus, she was actually Zeus' daughter. In the form of a swan, the king of gods was chased by an eagle, and sought refuge with Leda. The swan gained her affection, and the two mated.'

    The story is that Helen's mother mated with a swan. Not a virgin. Other less common stories indicate that both the mother and Zeus transformed to bird before mating. Still not virgin.

  7. J. Quinton

    March 4, 2012 at 8:03 pm

    “Extraordinary” always means low probability. An extraordinary claim is any P(H) that has a low prior probability. Extraordinary evidence is any event — P(E) — that has a low probability. A third and overlooked factor is the success rate which is the conditional probability P(E | H). This is the part of the equation that has to be high.

    So if P(H) was the hypothesis that Jesus is god, and P(E) was the event of an amputee growing a limb back we would have the equation like so:

    P(E | H) * P(H) / P(E)

    That's what it means when one says that extraordinary claims require extraordinary evidence. If the claim is extraordinary but the evidence is ordinary (i.e. high probability) then it doesn't help.

    I went in depth with it a bit here (which is posted in the above post).

  8. Mike Gage

    March 20, 2012 at 8:57 pm

    Jake, it sounds to me like you want to sneak some question begging into the prior probability. Remember that you determine your priors before considering the specific case. In some contexts that may work if everyone has a shared understanding of God's existence, etc. but it's not likely going to be a part of a skeptics priors. It's not a part of our background knowledge. On another note, using a frequency analysis, like Laplace's Law is a quite acecptable way of coming up with a prior in an unbiased way. That would give an even lower number than the charitable one here.I would suggest putting the idea that God wants to specifically raise Jesus from the dead into your evidence consideration. For example, a discussion of Jesus saying he was the son of God or maybe even was God would be included in the evidence section, I would say, not the the prior probability. If God wants to do something, then you could make the case that pretty much guarantees it and maybe even stack the evidence as 100% for and 0% against. Then, it would actually work out in your favor. Of course, that's a difficult case to make, but if you can't make it, then you give no reason for us to significantly update the low probability outcome.

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