Probability Theory: Bayes’ Theorem & the Inference to the Best Explanation.

Screen Shot 2017-08-10 at 2.39.13 PM

Image Credit: Retail 360

Bayes’ theorem, named after the English mathematician Thomas Bayes (1701-1761), is a way to figure out conditional probability, namely, the probability of an event happening given specific conditions. For instance, the probability of me finding a parking space would surely be linked to the time during the day at which I want to park, where I want park, and what events are going on at that time.

Bayes’ theorem is one useful way that probability theorists are able to calculate the probability of particular statements or events. Using the theorem, probabilities calculated will range between the values 0 and 1. 1 represents the highest possible probability and 0 the lowest possible probability. If one calculates a value higher than 0.5 (>.5) then it indicates a positive probability of a statement or event whereas a calculation of less than 0.5 (<.5) suggests an improbability. 0.5 would suggest an exact balance between the two. As per the formula, the probability of an event or a statement is designated as Pr (A.B) (1). This indicates the probability of an event or statement of A given B, or A on B with both A and B standing in for specific statements or events. It is likewise important to observe that it is impossible to calculate probability with absolute precision and certainty. Thus, it is best to use phrases such as “highly probable” (>>.5) or “highly IMprobable” (<<.5) when arguing for an hypothesis, and that one hypothesis is more probable than competing hypotheses.

Screen Shot 2017-08-06 at 11.34.51 PM.png

Another way to formulate Bayes’ Theorem

Bayesian approaches are often used in theology and the philosophy of religion, especially when it concerns arguments for God’s existence, calculating God’s existence, the problem of evil, and the problem of miracles. According to Chris Wiggins, an associate professor of applied mathematics at Columbia University, “Arguments employing Bayes’s theorem calculate the probability of God given our experiences in the world (the existence of evil, religious experiences, etc.) and assign numbers to the likelihood of these facts given existence or nonexistence of God, as well as to the prior belief of God’s existence–the probability we would assign to the existence of God if we had no data from our experiences” (2).

In his book The Resurrection of God Incarnate, philosopher Richard Swinburne, in his application of of Bayesian calculations, estimated the probability that Jesus’ resurrection stood at a high 97%, hence making it highly probable (3). There are a number of values that Swinburne used within his Bayesian calculation, for example, he weighs heavily on how Jesus lived, a key piece of evidence he argues is very relevant to whether we should believe he rose from the dead. Swinburne also factors in the evidence supporting Jesus’ resurrection, that the evidence we possess (the diversity of the resurrection appearances, empty tomb etc.) is the sort of evidence we’d expect if Jesus really did rise from the dead. Philosopher William Craig argues similarly asking, “What would the probability be of the resurrection not happening in hindsight of these four facts?” To which he answers that“It is highly, highly, highly, improbable that we should have that evidence [four facts] if the resurrection had not occurred” (4). However, not all Christian scholars have found Swinburne’s Bayesian calculations, and the values he assigns to it, necessarily convincing (5). Likewise, naturalists such as John Mackie and Richard Dawkins have attempted to use Bayesian calculations to disprove or argue against God’s existence.

Inference to the Best Explanation.

An alternative to Bayesian reasoning is what is known as the inference to the best explanation. This begins with the need and desire to be able to explain data. In order to explain the data we bring together a pool of options that try to account for it. From the pool of options we select an explanation which, we would argue, tends to best explain the data. This is a very common practice that most of us make use of on a daily basis when we entertain hypotheses that might explain some phenomena or another.

For example, if Jill puts her money on the windowsill, leaves her room, and then comes back to find it gone, she is likely to entertain a number of hypotheses to account for the phenomenon of its disappearance. If the window atop the windowsill is slightly ajar, she might suspect that someone jumped over her small garden fence, put a hand through the window, and made off with her money. Alternatively, someone within the house, perhaps a housemate in one of the adjacent rooms could have snuck in and stolen it. However, that Jill locked he bedroom door with a key, the only key matching the lock, makes that an unlikely explanation. Perhaps it was the domestic worker, but the domestic worker was in Jill’s room only once that day and an hour before she put her money on the windowsill. Jill is presented with a number of options looking to explain her money’s disappearance.

Philosopher Gilbert Harman explains that “In general, there will be several hypotheses which might explain the evidence, so one must be able to reject all such alternative hypotheses before one is warranted in making the inference. Thus one infers, from the premise that a given hypothesis would provide a “better” explanation for the evidence than would any other hypothesis, to the conclusion that the given hypothesis is true” (6).

Thus, with Jill, she has a set of three options to explain the possible reason behind her lost money. She rules out the possibility of a house mate stealing it given the locked door. It is also not possible that the domestic worker is the guilty thief given that the worker was not in the room at the moment within which the money disappeared. The most viable option, with these other two being dismissed, is that someone put a hand through the open window as not only seemingly evidenced by the open window itself but also by the inability of the other options to make sense of the situation.

The criteria used to determine which option in a pool of options is the best one, is debated. For example, as we’ve examined in a separate essay before, many hold that properties such as explanatory power, explanatory scope, plausibility, ad hoc–ness, and so on, are good means at getting to a best explanation (7). However, as has been observed, the inference to the best explanation isn’t immune to error (8). Just because a proposed explanation seems to be able to best explain phenomena, it doesn’t guarantee that it is true. After all, it could possibly be the case that the options themselves aren’t exhaustive and that the true explanation remains unknown (perhaps a 4th option to account for Jill’s dilemma is that the wind blew the money off of the windowsill and under her table, a hypothesis she hadn’t entertained that might very well be true). However, as C. Behan McCullagh explained in his book, Justifying Historical Descriptions, if an explanation, or historical reconstruction (if we are looking to infer the best explanation for an event in the past), outmatches other competing explanations then it is likely to be true, and that we should accept it as true unless new evidence suggests otherwise (9).

References.

1. Stuart, A. 1994. Kendall’s Advanced Theory of Statistics: Volume I—Distribution Theory. p. 294, 300.

2. Wiggins, C. What is Bayes’s theoremAvailable.

3. Otte, R. The Resurrection of God Incarnate Review. Available.

4. Swinburne, R. 2003. The Resurrection of God Incarnate. p. 3.

5. Otte, R. Ibid.

6. Gilbert Harman quoted by The Information Philosopher. Available.

7. Craig, W. 2012. Stephen Law on the Non-existence of Jesus of Nazareth. Available.

8. Craig, W. 2008. Reasonable Faith (3rd ed.). p. 92 (Scribd ebook format)

9. McCullagh, B. 1984. Justifying Historical Descriptions. p. 19.

 

Advertisements

4 responses to “Probability Theory: Bayes’ Theorem & the Inference to the Best Explanation.

  1. You need to proofread

    4th paragraph- ways (weighs) heavily on how Jesus lived, a key piece of evidence he argues is
    2nd paragraph- is best to use phrases such as “highly improbable” (>>.5) or “highly improbable (probable)” (<<.5)
    6th paragraph- Jill puts her money on the windowsill, leaves her room, and then come(s) back to

Let me know your thoughts!

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s