# On Bayes’ Theorem and Its Proper Application

By R.N. Carmona

There are times in where one’s assumptions fall into question. This could happen in a number of ways, but one of the more common ways is when one’s assumptions are confronted by facts of the matter. I had assumed that Bayes’ Theorem was well understood by my fellow atheists. Unfortunately, that assumption was wrong. It isn’t well understood. Like some of our academic tools (e.g. Bayes’ Theorem; statistics in general) and even entire disciplines (e.g. philosophy), our perception of things can be tarnished by misuse–specifically, the misuse or more accurately, misapplication of religious apologists and theologians. I see that that’s the case with Bayes’ Theorem.

Bayes’ Theorem is just another way to properly confront one’s assumptions. Nat Napoletano stated that “Bayes’ Theorem is a plan for changing our beliefs in the face of evidence.”1 Before we cross that bridge, however, it will be useful to present the theorem in its entirety. I will then explain its variables. Bayes’ Theorem appears as follows:

P( A | B) = [P (B | A) * P (A)] / P(B)]

Prior to proceeding, some definitions are required. As Mario Triola stated:

One key to understanding the essence of Bayes’ theorem is to recognize that we are dealing with sequential events, whereby new additional information is obtained for a subsequent event, and that new information is used to revise the probability of the initial event. In this context, the terms prior probability and posterior probability are commonly used.2

Given this, we learn that the prior probability “is an initial probability value originally obtained before any additional information is obtained” and the posterior probability “is a probability value that has been revised by using additional information that is later obtained.”3 Another way of putting this is that the theorem shows the relation between a conditional probability with its inverse. In other words, this is the probability of a hypothesis given some observation as related to the probability of an observation given some hypothesis. Let’s work with an example. I’ll first state it as a word problem and then I’ll flesh out the variables.

Let’s first assume that there’s an equal chance that Jack or Zack get the job. Then, let us find the probability that either Jack or Zack get promoted some time after being hired. However, we find that since Zack is less suited for the position, the probability of him being promoted is lower. We therefore have the following probabilities.

P(J): the probability that Jack gets the job. (.5)

P(Z): the probability that Zack gets the job. (.5)

P(pr): the probability of promotion. (.55)

P(t): the probability of termination. (.45)

P(pr|J): the probability that Jack is promoted after getting hired. (.8)

P(pr|Z): the probability that Zack is promoted after getting hired. (.25)

P(t|J): the probability that Jack is terminated. (.2)

P(t|Z): the probability that Zack is terminated. (.75)

As you can see, I already assigned probabilities to each case. I want the numbers to be easy to work with, so that this demonstration is clear. So, let us calculate the probability that one of them is promoted regardless of who’s hired.

P(pr) = P(pr|J) * P(J) + P(pr|Z) * P(Z)

P(pr) = (.8 * .5) + (.25 * .5)

P(pr) = .4 + .125 = .525 or 52.5%

The probability that either of them is terminated can be shown in the same way. It can, however, be simplified as follows:

P(t) = 1 – (.525) = .475 or 47.5%

So Bayes’ Theorem comes in when you want to find out either of the following: the probability that termination occurred given that Zack was hired or the probability that termination occurred given that Jack was hired. Alternatively, you can find out the probability of promotion given that either Zack or Jack was hired. Note that this is *after* the hiring. The equation for the probability that termination occurred given that Zack was hired looks as follows:

P(Z|t) = [P(pr|Z) * P(Z)] / P(pr)

So, P(Z|t) = (.25 * .5)/(.45). P(Z|t) is .278 or 27.8%. Of course, Bayes’ Theorem isn’t used for these kind of cases. This example is simply for ease. As stated earlier, the misapplication of Bayes’ Theorem leads some of us to doubt its utility. Napoletano touched on this exactly. He stated that it has been used by theologians (e.g. Richard Swinburne) to prove the existence of god or to demonstrate that the resurrection actually happened. He does, however, state that “it is difficult to separate out the variables and make them independent”4 in such cases. He also stated the theorem can expose when someone is being irrational. Aside from that, it can expose when someone is being biased–when someone is favoring one hypothesis over the other. In such cases, the theorem is often misapplied.

In medicine, for instance, a correct application will keep one from undergoing a series of medical tests. Like in our example above, doctors will be able to know the probability that you have either x disease or y disease given the symptoms you have. This goes back to showing a relation between the conditional probability and its inverse. This is, in other words, showing the relationship between what we currently know and what we want to know. This is a correct application.

In science, the theorem can be used to arrive at the probability of a hypothesis given observations. In other words, given MOND (Modified Newtonian Dynamics), what is the probability that some phenomena would be the case? Chris Wiggins explains:

One primary scientific value of Bayes’s theorem today is in comparing models to data and selecting the best model given those data. For example, imagine two mathematical models, A and B, from which one can calculate the likelihood of any data given the model (p(D|A) and p(D|B)). For example, model A might be one in which spacetime is 11-dimensional, and model B one in which spacetime is 26-dimensional. Once I have performed a quantitative measurement and obtained some data D, one needs to calculate the relative probability of the two models: p(A|D)/p(B|D). Note that just as in relating p(+|s) to p(s|+), I can equate this relative probability to p(D|A)p(A)/p(D|B)p(B). To some, this relationship is the source of deep joy; to others, maddening frustration.5

Religious apologists have misused the theorem. This is most apparent in their absurd calculations. “Richard Swinburne, for example…estimated the probability of God’s existence to be more than 50 percent in 1979 and, in 2003, calculated the probability of the resurrection [presumably of both Jesus and his followers] to be “something like 97 percent.”6 These numbers are, as Dawkins stated, “not measured quantities but personal judgments, turned into numbers for the sake of the exercise.”7 Put simply, it’s blatant bias.

This abuse of the theorem is likely behind people’s suspicion of the theorem. Whether or not a theist or an atheist can use Bayes’ Theorem to arrive at a probability of god’s (non-)existence or of the resurrection is perhaps an open question. There are, however, accurate historical applications. It must first be pointed out that Bayes’ Theorem cannot be used in isolation. “The only way to argue for a claim is to look at the evidence and compare that claim to alternative claims.”8 More importantly, however, assumptions reduce prior probability. Rather than reduce their prior probability, religious apologists and theologians inflate it. This is perhaps at the core of people’s suspicions. Your suspicion, however, isn’t in Bayes’ Theorem itself but rather in how it is being applied. It must be noted that a reduction in the prior implies a reduction in the posterior. Richard Carrier explains:

[I]f the evidence doesn’t match your claim so well (as theism always doesn’t, e.g. the Problem of Evil, or to continue the last analogy, the fact that when miracle claims are properly investigated they never turn out to be genuine), you can invent “excuses” to make your claim fit that evidence better. This means, adding assumptions to bolster your claim. But what Bayes’ Theorem teaches us is that every such assumption you add reduces the probability that your claim is true, rather than increasing it (as that excuse was supposed to do). Because adding assumptions to any claim C reduces the prior probability of C, and any reduction in the prior probability of C entails a reduction in the posterior probability of C, which is simply the probability of C. This is because the prior probability of several assumptions is always necessarily lower than the prior probability of any one of them alone, a basic fact of cumulative probability, which when overlooked produces the Conjunction Fallacy.9

When it comes to the existence of god or the occurrence of a miracle or a purported past occurrence like the resurrection, a neutral probability of .5 can serve as our prior. An atheist will no doubt assume that the prior probability of Jesus’ resurrection is much lower. A theist will assume it’s much higher. If, however, we set our assumptions aside, our posterior probability also goes unaffected. This will allow the numbers to do the work, so to speak. Though it remains difficult to separate the variables and make them independent, our results aren’t laden with bias. Ultimately, we can do closer analysis of Richard Swinburne’s misapplication or Richard Carrier’s application of Bayes’ Theorem. Such analyses are, however, cumbersome and for our purposes, unnecessary.

Ultimately, if you have any suspicions, I must reiterate, they do not lie with the theorem itself but rather with the way it is being applied. We must always mind the source. Not in the sense that we’re dismissive in an *ad hominem* or perceived guilt by association sort of manner, but in the sense that we’re acquainted with their biases. If, for instance, s/he is a believer in god, if it turns out that the calculations are suspicious or even dubious, we must single out the bias. There is, for instance, no way that the probability of the resurrection of Jesus is “something like 97 percent.” This is *prima facie* ludicrous and tenuous. It, of course, must be questioned. Common sense dictates that if the probability of his resurrection is that high, the probability of all miracles, regardless of which religious adherent makes the claim, is comparably high. This certainly isn’t the case, especially when we account for the dubious epistemic support for such claims (e.g. eyewitness testimony). I repeat, this irresponsible and to be blunt, non-academic misapplication must be questioned. We, however, must learn to do so responsibly and that isn’t possible when we’re directing our suspicions at the theorem rather than the theorist.

**Works Cited**

1 Napoletano, Nat. “Bayes’ Theorem for Everyone 01 – Introduction”. *YouTube*, *YouTube, LLC*. 29 Jan 2012. Web. 1 Mar 2015.

2 Triola, Mario F. “Bayes’ Theorem”. *University of Washington*. ND. Web. 1 Mar 2015.

3 Ibid. [2]

4 Ibid. [1]

5 Wiggins, Chris. “What is Bayes’s theorem, and how can it be used to assign probabilities to questions such as the existence of God? What scientific value does it have?”. *Scientific American*. 4 Dec 2006. Web. 1 Mar 2015.

6 Ibid. [5]

7 Ibid. [5]

8 Carrier, Richard C. “If You Learn Nothing Else about Bayes’ Theorem, Let It Be This”. *Freethought Blogs*. 26 Apr 2014. Web. 1 Mar 2015.

9 Ibid. [8]