Part 5: The Problem of Setting Prior Probabilities

How do we fairly determine the plausibility or prior probability of something? With unusual historical questions like the resurrection of Jesus, this issue gets even more pointed because we don’t have direct sense access to the phenomena at hand, sense experience that could overcome extremely low plausibility assessments.

For things like the resurrection, we can weigh them as so absurdly implausible or unlikely as to make it impossible that any obtainable level of evidence can change our minds. Such a belief would then be unfalsifiable, which isn’t particularly reasonable.

One way of trying to figure out the evidential strength of a position is to use Bayes’ Theorem.

On the left side is what we are trying to figure out, the probability of the hypothesis (H) after we look at some particular evidence (E). In order to do this we need to know first what the probability of hypothesis H is regardless of this particular evidence. This can make an enormous difference. So if something is really probable already, like there being papers on my desk, I’m not going to need much evidence to confirm that. However, if someone said the Hope Diamond was sitting on my desk, I’d need a lot more evidence to make that seem reasonable.

Here the question becomes, “What is the proper level of probability to assign to the resurrection before we look at the historical data?”

One common way of assigning a probability to the resurrection is to look at the number of resurrections that have happened previously—not a lot. We can’t say that number is 0, because 0 multiplied by anything is zero and our bias becomes unfalsifiable. So we pick something small-say 1 out of the number of humans that have ever existed, say 100 billion (yes, I picked the round number to make the math easier ;).

Now suppose that someone you knew had died, appeared to you later, and the experience was incredibly convincing. You were 99.9% sure that person had appeared to you in the flesh (this is the Pr(E|H) part). Not only that, it seems quite unlikely for you to have a conversation and meal with dead person, say only a 1 in a 1000 chance (.1%) that would happen. Well, if we put our numbers into Bayes theorem and calculate, we discover that the odds that person rose from the dead are still much less than 1%.[1] Why? Because the odds of it happening are so ridiculously low to begin with, that even if you’re 99.9% sure you saw that person alive, it barely makes a dent in the overall probability.

Now this seems a bit weird to many of us and it shows us a few issues with the objectivity of analysis. The first is this—how do we calculate the appearance to us as data? Is the whole experience just one datum? Or, does every single moment of being 99.9% convinced count independently? If each new second (split second? microsecond? picosecond?) counts as its own piece of data, then sure, we can overcome a 1 in 100 billion deficit in fairly short order.

This does highlight the power that our own sense experience has. We can overcome ridiculously low priors if we see things with our eyes over time. An example of this comes from the gospel stories themselves. The disciples were not expecting Jesus to come back. They thought it was ridiculously implausible and improbable that Jesus would be alive again. However, because of seeing Jesus or hearing others whose reliability they’d personally verified over a long time, they overcame their negative bias and came to believe in Jesus’ resurrection. However, we are in a different place and this causes problems for us, something we’ll discuss in a bit.

Another issue is this, why should we pick 1 out of 100 billion as our starting place? Have we verified all those deaths and non-resurrections? Is there other relevant information that would change that number? This is the reference class problem, trying to choose the best starting place.[2] Choose the wrong reference class, the wrong starting point, and the whole analysis will be skewed. The math itself doesn’t tell you what the starting point should be, just how to calculate the numbers. But our choice of numbers can be quite subjective and radically skew the analysis.

For example, lets calculate the odds of Trump sleeping in the White House tonight (this article was written when Trump was president) given that he is spending the night in Washington DC. Well, since there are 8 billion people on the planet, and only say, 10 people sleep at the White House, we could say our starting probability, our prior probability, is 10 in 8 billion or .000000125%. Now suppose he’s in the DC about 80% of the time and, obviously, when he’s sleeping in the White House he’s in DC . If we plug all those numbers into Bayes Theorem we get a .000000156% chance that Donald Trump is sleeping at the White House. Now this is considerably higher than before, but seems very wrong. There’s a less than 1% chance that DJT is sleeping in the White House when we know he’s spending the night in DC? Really?

The problem here comes from the choice of prior probability. 10 in 8 billion isn’t the most relevant stat for DJT. Why choose that starting point? When we look for the prior probability, the Pr(H), it’s actually supposed to include every piece of background knowledge (B). So you are really looking for Pr(H,B) the probability of the hypothesis taking into account everything that is relevant to that hypothesis except for the new pieces of evidence you’re trying to evaluate. Now, the problem here is that, strictly speaking, it’s impossible. There is way too much stuff in the world with way too many complicated connections to take all of it into account. However, it’s obvious that we left out some pretty important stuff about Donald Trump in coming to our prior of 10 in 8 billion.

A different way of looking at the same problem is this. The way that we calculated Trump’s sleeping is equivocal; there are actually two different hypotheses but we’ve covered up the difference and used H to refer to both of them. In the first case H [in the Pr(H)] is referring to the probability that some random person is in the White House. However, in the other cases it is referring to the probability that Donald Trump is sleeping in the White House—two very different things. The prior probability of Donald Trump sleeping in the White House is much much greater than 10 in 8 billion.[3]

A similar issue arises in the Jesus resurrection debates. We are not looking at the probability that some random person rose from the dead but that Jesus Christ rose from the dead. So what then should be the starting probability that Jesus Christ rose from the dead? Well, that is a much less obvious question to answer—and it will almost certainly be much much more probable than some random person rising from the dead.

We get into why this is true more in the next post.

[1] Don’t believe me? Do the math yourself!

[2] See, for instance, Alan Hajek, “The Reference Class Problem Is Your Problem Too,” Synthese 156, no. 3 (May 2007): 563–85.  

[3] Considering that he lives there, it is actually quite likely.