Bayes Theorem Greatest Hits (Posts)
(The oft-worn image for posts about Bayes)
My blog was mentioned on Richard Carrier’s blog where he requests blogs that do Bayes. Alright! I have all of my posts about Bayes Theorem tagged as Bayes but a lot of the earlier posts were still me working through learning how to use the theorem properly, so they have some errors. I would correct them, but that would take too much work and I mainly blog through my mobile phone (which can be a pain).
So I thought I would do a roundup of my posts with the most hits that are about Bayes Theorem:
The most popular: Bayes Theorem and the Virgin Birth of Jesus
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… 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 […] 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 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% […]
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, 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 [Matt 1.23] 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.
Absence of Evidence is Evidence of Absence (redux)
[T]here’s the common refrain that “absence of evidence isn’t evidence of absence” but if that were true, then your probabilities won’t be coherent and you’ll Dutch Book yourself above; your terms won’t equal 1.00…
In formulaic terms, your friend [who says absence of evidence isn’t evidence of absence] is saying P(H) = .5. P(H | E) > .5, yet P(H | ~E) = .5. We can model this by throwing in some values in Bayes’ Theorem. P(H) is already .5, so let’s say that P(E) is also .5 and P(E | H) is .6. This becomes:
Evidence is found:
P(H | E) = .6 * .5 / .5
P(H | E) = .6
Let’s try it for the absence of evidence. This would be P(H | ~E) = P(~E | H) * P(H) / P(~E). Again, P(H) is already .5, and to keep it simple let’s again make P(~E) equal .5. In order to make P(H) = P(H | ~E), the conditional probability P(~E | H) should also be .5:
Absence of evidence isn’t evidence of absence: P(H | ~E) = .5 * .5 / .5
P(H | ~E) = .5
But wait, remember that P(E) + P(~E) = 1.00. This means that P(~E) is also necessarily .5. What’s the other axiom? P(E | H) + P(~E | H) should also equal 1.00. But in order to make sure that absence of evidence isn’t evidence of absence, we made P(E | H) = .6, and P(~E | H) = .5. That totals to 1.10: DUTCH BOOKED!!11!!
In reality, if P(E | H) is .6 then P(~E | H) is .4.
For the apostles, in the memoirs composed by them, which are called good news, have thus delivered unto us what was enjoined upon them; that Jesus took bread, and when he had given thanks, said, “This do in remembrance of me, this is my body”. And that, after the same manner, having taken the cup and given thanks, he said, “This is my blood”; and gave it to them alone. Which the wicked devils have imitated in the mysteries of Mithras, commanding the same thing to be done. For, that bread and a cup of water are placed with certain incantations in the mystic rites of one who is being initiated, you either know or can learn. – First Apology ch 66
Here Justin intimates that Mithraists had a similar meal with similar incantations to the Eucharist while he was alive…
Mithraism and Christianity both seem to have started around the same time, in the 1st century… Mithraism seems to have been a mystery cult that Roman soldiers followed. Similarly, Christianity also seems to have started out as a sort of mystery cult. So we do not have any idea whether the Christian Eucharist preceded the Mithraist “Eucharist”, or vice versa.
A basic syllogistic argument might look like this:
P1: Christians borrowed many ideas from their wider pagan matrix
P2: The Eucharist ceremony has a parallel in their wider pagan matrix
C: Therefore Christians borrowed the Eucharist ceremony from their wider pagan matrix
Justin Martyr’s argument (cleaned up to look more respectable than how he presents it [i.e. getting rid of an appeal to demons]) might look like this:
P1: Christians practice the Eucharist ceremony
P2: Mithraists practice a similar ceremony
C: Therefore Mithraists borrowed their ceremony from Christians
Which is more likely? I think I might try Bayes theorem to find out… Concerning the Eucharist we have two options that have the highest probability: Christians borrowed the Eucharist from pagans, or pagans borrowed the Eucharist from Christians. Or more generally, Christians borrowed ideas from pagans, or pagans borrowed ideas from Christians. Here is a list of themes and ideas that Christians borrowed from pagans for our prior probability (that is, prior to the Eucharist):
2. the Logos (from the Stoics)
3. Virgin births
4. Idea of a Heavenly Man (i.e. Platonism and Forms, cf 1 Cor 15.44-49)
5. Gods descending in the form of an avian creature (cf Mark 1.10)
6. Healing the blind with spit (Mark 8.22-26; John 9.1-7)
Here is a list of themes and ideas that pagans borrowed from Christians prior to the Eucharist:
Zero or unknown.
Since followers of Mithras were generally Roman soldiers, we might expect other military themes or ideas to find its way into Christian culture if Christians had contact with and syncretized with Mithraists. It just so happens that words like “Gospel” (from euaggelion [evangelion] > good news > god spell > gospel) and “Parousia” (para ousia, literally a presence next to; usually reserved for the arrival of royalty or a military ambassador) were both originally used in military applications in a Greco-Roman context. On the flip side, if Mithraists syncreticized ideas from Christians, I would expect followers of Mithras to adopt some other Christian culture besides the Eucharist if they indeed did adopt it from Christians. Maybe something like refering to Mithras as a christ. We have no evidence that they did so. Granted, there is very little evidence for the inner workings and language of Mithraism in general so it’s not saying much.
Running through Bayes Theorem, this puts the probability that Christians took the Eucharists ceremony in its current incarnation from Mithraists at 73%. Which means, I’m surmising, that there’s a 73% chance that the Last Supper is not historical, at least in the symbolically eating the flesh and blood of Jesus way. There probably was some sort of original ceremonial meal, as in the Didache and the probable original version of 1 Cor 11, but was overlapped by the current Mithras-like thanksgiving.
So maybe a Roman soldier joined Christianity after having been a Mithraist and ported the Mithraist “Eucharist” into Christianity. It might even be that, the first gospel Mark was written by a Roman soldier. It’s certainly possible and explains some other oddities/Latinisms in Mark (like the term Syro-Phoenitian), but is it probable? Who knows. That would take some more involved Bayesian judo.
Say your friend has two die. One has six sides numbering 1 – 6 and the other is a trick die that has a 1 on all faces. She rolls one of the die at random and it ends up with a 1. What is the probability that the die that she rolled was the normal 6 sided one or the trick die?
For the normal 6 sided die, our probability distribution is P(One | Normal) + P(Two | Normal) + P(Three | Normal) + P(Four | Normal) + P(Five | Normal) + P(Six | Normal) = 1.00. If it is a fair die, then the probability for P(One | Normal) = 1/6 or .1667.
For the trick die, our probability distribution is P(One | Trick) = 1.00.
We can then go through Bayes’ to see what the probability is for her rolling each:
P(Normal | One) = P(One | Normal) * P(Normal) / [P(One | Normal) * P(Normal)] + [P(One | Trick) * P(Trick)]
= .1667 * .5 / [.1667 * .5] + [1.00 * .5]
= .0834 / [.0834] + [.5]
= .0834 / .5834
P(Trick | One) = P(One | Trick) * P(Trick) / [P(One | Trick) * P(Trick)] + [P(One | Normal) * P(Normal)]
= 1.00 * .5 / [1.00 * .5] + [.1667 * .5]
= .5 / [.5] + [.0834]
= .5 / .5834
So upon rolling a 1, the probability that she rolled the normal sided die is .1429 and the probability that she rolled the trick die is .8571.
This is the problem with positing hypotheses that can equally explain multiple exclusive outcomes, even if there is a high initial probability of that hypothesis being true. If we had a 100 sided die, and a 90% chance of picking that die, upon rolling a 1 there would only be a .1337 probability that the 100 sided die was picked, in contrast to a .7426 probability that the trick die was picked. A 200 sided die would do worse. 300, even worse. Etc.
1. Prior beliefs influence whether or not the argument is accepted.
A) I’ve often drunk alcohol, and never gotten drunk. Therefore alcohol doesn’t cause intoxication.
B) I’ve often taken Acme Flu Medicine, and never gotten any side effects. Therefore Acme Flu Medicine doesn’t cause any side effects.
Both of these are examples of the argument from ignorance, and both seem fallacious. But B seems much more compelling than A, since we know that alcohol causes intoxication, while we also know that not all kinds of medicine have side effects.
2. The more evidence found that is compatible with the conclusions of these arguments, the more acceptable they seem to be.
C) Acme Flu Medicine is not toxic because no toxic effects were observed in 50 tests.
D) Acme Flu Medicine is not toxic because no toxic effects were observed in 1 test.
C seems more compelling than D.
3. Negative arguments are acceptable, but they are generally less acceptable than positive arguments.
E) Acme Flu Medicine is toxic because a toxic effect was observed (positive argument)
F) Acme Flu Medicine is not toxic because no toxic effect was observed (negative argument, the argument from ignorance)
Argument E seems more convincing than argument F, but F is somewhat convincing as well.
For example, in order to flip a coin and get three flips of heads in a row, I would have to first flip two heads in a row. In order to flip two heads in a row I have to flip heads on the first flip. Three heads depends on two heads which depends on one heads. Since the probability of flipping heads once is .5, and each additional heads depends on the previous heads, they all multiply together: .5 * .5 * .5 = .125.
What is the probability that I will flip three heads in a row given that I have flipped heads once?
P(Flipping Three Heads In A Row | Flipping Heads Once) = P(E | H) * P(H) / P(E)
= 1.00 * .125 / .5
= .125 / .5
Given that I have flipped heads once, my prior has moved from .125 to .25.
The Fine Tuning Argument is an Argument for Atheism (Summerized)
According to this apologetics website the probability of the current arrangement of our universe’s constants is the equivalent of picking one red dime out of a pile of 1037 dimes. Or, P(Current Universal Constants) = 0.0000000000000000000000000000000000001.
Back to our Total Probability formula:
0.0000000000000000000000000000000000001 = P(Current Universal Constants | Christian God) * .7412 + P(Current Universal Constants | Non Christian God, Atheism) * .2588.
0.0000000000000000000000000000000000001 = ???? * .7412 + P(Current Universal Constants | Naturalism, Atheism) * . 2588.
It looks like the equation has to be P(Current Universal Constants | Non Christian God, Atheism) > P(Current Universal Constants | Christian God) in such a manner that makes the Total Probability equal to 0.0000000000000000000000000000000000001. Since P(Current Universal Constants | Christian God) is basically zero — the majority of the probability capital goes into P(Other Universal Constants | Christian God) — this means that P(Current Universal Constants | Non Christian God, Atheism) is equal to a miniscule amount more than P(Current Universal Constants). At this point, it might as well be equal to P(Current Universal Constants).
Since P(Current Universal Constants | Christian God) is basically, zero, this means that the probability of the Christian god’s existence given the current universal constants is also basically zero. It’s not actually zero because zero isn’t a probability. I’d like to say that I’m the first one to make that argument, but it already looks like other people have come to a similar conclusion about the fine-tuning argument [being an argument for atheism].
I’ll also keep this post in my Pages on the sidebar for quick reference.