On the prior probability of Jesus’ resurrection

The resurrection of Jesus of Nazareth after his unjust death stands at the very heart of the Christian faith.

Jesus_resurrection

If materialism is true, it goes without saying that the prior plausibility of a corpse coming back to life through random physical processes is extremely small.

However, some atheist apologists go farther than that and argue that even if God existed, the probability of His raising Jesus from the dead would be incredibly low.

 

Atheistic philosopher Jeffery Jay Lowder (who is a nice, respectful, well-articulated, intelligent and decent man) put it like this:

B3: Approximately 107,702,707,791 humans have ever lived. Approximately half of them have been male.
B4: God, if He exists, has resurrected from the dead at most only one person (Jesus).

B3 and B4 are significant because they summarize the relevant evidence about God’s tendency to resurrect people from the dead (assuming God exists). They show why the resurrection has a low prior probability even for theists. Once we take B3 and B4 into account, the prior probability of the resurrection is less than or equal to 5.0 x 10-12. In symbols, Pr(R | B1 & B3 & B4) <= 5.0 x 10-12.

 

I shall reformulate his argument in a simpler way while emphasising a most problematic hidden assumption.

  1. From the 100 000 000 humans who have ever lived under the sun, none has been resurrected by God’s mighty hands.
  2. Consequently, the probability that a human being chosen at random gets raised from the dead is less than 10-11.

3. God would be as interested in resurrecting Jesus as he would be in resurrecting a random human being.

4. Hence the prior probability of Jesus’ resurrection is less than 10-11.

Although premise 1) might be begging the question against claims of miracles, I shall accept it as true.

Premise 2) is totally uncontroversial. So what truly stands in the way of the conclusion is premise 3).

Why on earth should we assume that Jesus was only a random human being to God? This probability seems unknown to me unless one makes assumptions about the divine Being, i.e. one engages in theology.

(The are good articles written by professional philosopher of science John Norton explaining why epistemic ignorance cannot be represented by a probability distribution [1], [2], [3])

Lowder seems to be aware of this. A (godless) commenter wrote:

“Your estimate of 5.0 x 10-12. assumes that Jesus is a typical human. But if not, if B1A: Jesus is the second person of the Trinity is true, P(B2) becomes much higher, possibly of order 1. In that case the relevant unknown is P(B1A | B1). While that may be small, I doubt if it’s anywhere near as small as 5.0 x 10-12.”

His response was:

“There are not any reliable statistics for the reference class of men who are the second person of the Trinity. Thus, the reference class that must be used is the broadest one for which we have reliable statistics, viz., men.”

But this is clearly begging the question.

  • Why should we  assume that Jesus was a random human being to God?
  • Because this is the only way we can approximately calculate the prior probability of his resurrection.
  • And why should we assume that this value approximates anything if we don’t know whether or not he was just an ordinary man to God?

So I think that unbelievers cannot argue from ignorance here. They should instead give us positive grounds for thinking that Jesus wasn’t special to God.

jesus

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Knowledge-dependent frequentist probabilities

 

This is going to be a (relatively) geeky post which I tried to make understandable for lay people.

Given the important role than epistemological assumptions play in debate between theists and atheists, I deemed it necessary to first write a groundwork upon which more interesting discussions (about the existence of God, the historicity of Jesus, miracles, the paranormal…) will lie.

Bayesianism, Degrees of belief

In other posts I explained why I am skeptical about the Bayesian interpretation of probabilities as degrees of belief. I see no need to adjust the intensity of our belief in string theory (which is a subjective feeling) in order to do good science or to avoid irrationality.

Many Bayesians complain that if we don’t consider subjective probabilities, a great number of fields  such as economy, biology, geography or even history would collapse.
This is a strong pragmatic ground for being a Bayesian I hear over and over again.

Central limit theorem and frequencies

I don’t think this is warranted for I believe that the incredible successes brought about by probabilistic calculations concern events which are (in principle) repeatable and therefore open to a frequentist interpretation of the related likelihoods.

According to a knowledge-dependent interpretation of frequentism I rely on the probability of an event is its frequency if the known circumstances were to be repeated an infinite number of times.

Let us consider an ideal dice which is thrown in a perfectly random way. Obviously we can only find approximations of this situation in the real world, but a computer can reasonably do the job.

In the following graphics, I plotted the results for five series of trials.

FrequencyNormal

FrequencyLog

The frequentist probability of the event is defined as

formel1

,

that is the limit of the frequency of “3” when the number of trials becomes close to infinity.

This is a mathematical abstraction which never exists in the real world, but from the 6000-th trial onward the frequency is a very good approximation of the probability which will converge to the probability according to the central limit theorem.

Actually my knowledge-dependent frequentist interpretation allows me to consider the probability of unique events which have not yet occurred.

For example, a Bayesian wrote that “the advantage of this view over the frequency interpretation is that it can deal with cases where there is no relative frequency to draw on: for example, Gigerenzer mentions the first ever heart transplant patient who was given a 70% chance of survival by the surgeon. Under the frequency interpretation that statement made no sense, because there had never actually been any similar operations by then.“

HeartTransplant

I think there are many confusions going on here.
Let us call K the total knowledge of the physician which might include the different bodily features of the patient, the state of his organs and the hazard of the novel procedure.

The frequentist probability would be defined as the ratio of surviving patients divided by the total number of patients undergoing the operation if the known circumstances underlying K were to be repeated a very great (actually infinite) number of times.formel2Granted, for many people this does not seem as intuitive as the previous example with the dice.
And it is obvious there existed for the physician no frequency he could have used to directly approximate the probability.
Nevertheless, this frequentist interpretation is by no means absurd.

The physician could very well have used Bayes’s theorem to approximate the probability while having only used other frequentist probabilities, such as the probability that the body reacting in a certain way would be followed by death or the probability that introducing a device in some organs could have lethal consequences.

Another example is the estimation of the probability it is going to rain tomorrow morning as you will wake up.

Wetter
While the situation you are confronted with might very well be unique in the whole history of mankind, the probability is well defined by the frequency of rain if all the circumstances you know of were to be repeated an extremely high number of times.

Given this extended, knowledge-dependent variant of frequentism, the probabilities of single events are meaningful and many fields considered as Bayesian (such as economical simulations, history or evolutionary biology) could be as well interpreted according to this version of frequentism.

It has a great advantage: it allows us to bypass completely subjective degrees of belief and to focus on an objective concept of probability.

Now, some Bayesians could come up and tell me that it is possible that the frequentist probabilities of the survival of the first heart transplant patient or of the weather does not exist: in other words, if the known circumstances were to be repeated an infinite number of times, the frequency would keep oscillating instead of converging to a fixed value (such as 1/6 for the dice).

Fluctuations

This is a fair objection, but such a situation would not only show that the frequentist probability does not exist but that the Bayesian interpretation is meaningless as well.

It seems utterly nonsensical to my mind to say that every rational agent ought to have a degree of belief of (say) 0.45 or 0.87 if the frequency of the event (given all known circumstances) would keep fluctuating between 0.01 and 0.99.
For in this case the event is completely unpredictable and it seems entirely misguided to associate a probability to it.

Another related problem is that in such a situation a degree of belief could be no nothing more than a pure mind state with no relation to the objective world whatsoever.

As professor Jon Williamson wrote:
Since Bayesian methods for estimating physical probabilities depend on a given prior probability function, and it is precisely the prior that is in question here, this leaves classical (frequentist) estimation methods—in particular confidence interval estimation methods—as the natural candidate for determining physical probabilities. Hence the Bayesian needs the frequentist for calibration.”

But if this frequentist probability does not exist, the Bayesian has absolutely no way to relate his degree of  belief to reality since no prior can be defined and evaluated.

Fortunately, the incredible success of the mathematical treatment of uncertain phenomenons (in biology, evolution, geology, history, economics and politics to name only a few) show that we are justified in believing in the meaningfulness of the probability of the underlying events, even if they might be quite unique.

In this way, I believe that many examples Bayesians use to argue for the indispensability of their subjectivist probabilistic concept ultimately fail because the same cases could have been handled using the frequentist concept I have outlined here.

However this still leaves out an important aspect: what are we to do about theories such as the universal gravitation, string theory or the existence of a multiverse?
It is obvious no frequentist interpretation of their truth can be given.
Does that mean that without Bayesianism we would have no way to evaluate the relative merits of such competing models in these situations?
Fortunately no, but this will be the topic of a future post.
At the moment I would hate to kill the suspense 🙂

A mathematical proof of Bayesianism?

This is going to be another boring post (at least for most people who are not nerds).

However before approaching interesting questions such as the existence of God, morality and history a sound epistemology (theory of knowledge) must already be present. During most (heated) debates between theists and atheists, people tend to take for granted many epistemological principles which are very questionable.

This is why I spend a certain amount of my time exploring such questions, as a groundwork for more applied discussions.

I highly recommand all my reader to first read my two other posts on the concept of probability before reading what follows.

Bayesianism is a theory of knowledge according to which our degrees of belief in theories are well defined probabilities taking on values between 0 and 1.

According to this view, saying that string theory has a probability of 0.2 to be true is as meaningful as saying that a normal dice randomly thrown has a probability of 1/6 to produce a “3”.

Bayesians like asserting over and over again that it is mathematically proven to say we ought to compute the likelihood of all beliefs according to the laws of probability and first and foremost Bayes formula:

BayesGeneral

Here I want to debunk this popular assertion. Bayes theorem can be mathematically proven for frequential probabilities but there is no such proof that ALL our degrees of belief behave that way.

Let us consider (as an example) the American population (360 millions people) and two features a person might have.

CE (Conservative Evangelical): the individual believes that the Bible contains no error.

God-said-it-believe-it-1

FH (Fag Hating): the individual passionately hates gay people.

homobashing

Let us suppose that 30% of Americans are CE and that 5.8% of Americans hate homosexuals.

The frequencies are f(CE) = 0.30 and f(FH) = 0.058

Let us now consider a random event: you meet an American by chance.
What is the probability that you meet a CE person and what is the probability that you meet a FH individual?
According to a frequentist interpretation, the probability equals the frequency of meeting such kinds of persons given a very great (actually infinite) number of encounters.
From this it naturally follows that p(CE) = f(CE) = 0.30 and p(FH) = f(FH) = 0.058

Let us now introduce the concept of conditional probability: if you meet a Conservative Evangelical, what is the probability that he hates faggots p(FH|CE)? (the | stands for „given“).

If you meet a fag-hating person, what is the probability that he believes in Biblical inerrancy p(CE|FH)?

To answer these questions (thereby proving Bayes theorem) it is necessary to get back to our consideration of frequencies.

Let us consider that 10% of all Conservative Evangelicals and 4% of people who are not CE hate faggots: f(FH/CE) = 0.1 and f(FH/CE) = 0.04. The symbol ⌐ stands for the negation (denial) of a proposition.

The proportion of Americans who are both conservative Evangelicals and fag-haters is f(FHCE) = f(FH/CE)*f(CE) = 0.1*0.3 = 0.03.

The proportion of Americans who are NOT conservative Evangelicals but fag-haters is f(FH∩⌐CE) = f(FH/⌐CE)*f(⌐CE) = 0.04*0.7 = 0.028.

Logically the frequency of fag-haters in the whole American population is equal to the sum of the two proportions:

f(FH) = f(FHCE) + f(FH∩⌐CE) = 0.03 + 0.028 = 0.058

But what if we are interested to know the probability that a person is a conservative Evangelical IF that person hates queers p(CE|FH)?

This corresponds to the frequency(proportion) of Conservative Evangelicals among Fag-Haters: f(CE|FH).

We know that f(FHCE) = f(CE∩FH) = f(CE|FH)*f(FH)

Thus f(CE|FH) = f(FH∩CE) / f(FH)

FagBayesFrequencies

Given a frequentist interpretation of probability, this entails that

FagBayes

which is of course Bayes theorem. We have mathematically proven it in this particular case but the rigorous mathematical demonstration would be pretty much the same given events expressable as frequencies.

If you meet an American who hates gays, the probability that he is a Conservative Evangalical is 51.72% (given the validity of my starting values above).

But let us now consider the Bayesian interpretation of probability (our degree of confidence in a theory) in a context having nothing to do with frequencies.

Let S be “String theory is true“ and UEP “an Undead Elementary Particle has been detected during an experience in the LHC“.

StringTheory

In that context, the probabilities correspond to our confidence in the truth of theories and hypotheses.

We have no compelling grounds for thinking that

StringBayes

, that is to say that is the way our brains actually work or ought to work that way in order to strive for truth.

The mathematical demonstration used to prove Bayes theorem relies on related frequencies and cannot be employed in a context where propositions (such as S and UEP) cannot be understood as frequencies.
Considering ALL our degrees of beliefs like probabilities is a philosophical decision and not an inevitable result of mathematics.

I hope that I have been not too boring for lay people.

Now I have a homework for you: what is the probability that Homeschooling Parents would like to employ my post as an introduction to probability interpretation, given that they live in the Bible Belt  p(HP|BB)?

Image Of Thomas Bayes