How can Bayes' theorem assign a probability to the existence of God?
Chris Wiggins, an associate professor of applied mathematics at Columbia University, offers this explanation:
In the 18th century Reverend Thomas Bayes first expressed the probability of any event—given that a related event has occurred—as a function of the probabilities of the two events independently and the probability of both events together.
Take the following example. A patient goes to see a doctor for a checkup. The doctor knows a test he performs has 99 percent reliability—that is, 99 percent of sick people test positive, and 99 percent of healthy people test negative. The doctor also knows that only 1 percent of the population is sick. Now the question is: If the patient tests positive, what are the chances the patient is sick? The intuitive answer is 99 percent, but the correct answer is actually 50 percent. Bayes' theorem relates the probability of being sick given a positive test result, p(s|+), to the probability of receiving a positive test result given that one is sick, p(+|s), and to the general probability of being sick, p(s), and the general probability of getting a positive result, p(+). (Calculating the last probability is left to the reader.) Bayes' theorem in this case would read p(s|+) = p(+|s) p(s) / p(+).
The importance of accurate data in quantitative modeling is central to the subject raised here: using Bayes' theorem to calculate the probability of the existence of God. (Bayes, for his part, never related his theorem to the subject.) Scientific discussion of religion is a popular topic at present, with several recent books arguing against theism. In one, The God Delusion (Houghton Mifflin, 2006), University of Oxford professor Richard Dawkins argues specifically against the use of Bayes' theorem for assigning a probability to God's existence.
Dawkins takes exception to this usage not because he doubts the veracity of Bayes' theorem but because turning human experience into numbers is, he argues, an inherently subjective process. A Bayesian approach to evaluating the likelihood of God's existence involves enumerating possible outcomes (the presence of good, evil, religious revelations, and so on) and determining their probabilities assuming the existence or nonexistence of God. One must also express 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. Dawkins notes that these figures cannot be determined quantitatively, rendering Bayes' theorem useless in this enterprise. In applications for which data are available, however, Bayes' theorem lies at the heart of almost all statistical modeling and is a critical tool for thinking concretely about uncertainty.
How do certain hairs know to grow back when you trim them?
—C. Spain, New York City
George Cotsarelis, a University of Pennsylvania dermatologist and specialist in hair and scalp disorders, replies:
Clipping hairs on the surface of the skin does not affect their growth, because the hair above the surface is dead.
The hair visible on our bodies grows out of living hair follicles within the skin. All these follicles go through the so-called hair follicle cycle, which has three stages: growth, degeneration and rest—called anagen, catagen and telogen, respectively. During anagen, the rapid proliferation of cells at a follicle's base, or bulb, results in the constant production of a hair fiber that emerges from the skin surface. (Follicles produce hairs of different lengths by staying in anagen for varying periods.) At the end of anagen, these hair-producing cells begin to die, entering into the catagen stage. After dying back for a few weeks, the follicle rests for several weeks or even months in telogen. A new lower hair follicle then regenerates from stem cells in the telogen follicle, and anagen begins anew. The old hair fiber then falls out—often while you are brushing your hair—as a new strand replaces it. The hairs that appear to “know” to grow back after being trimmed just happen to be in the anagen stage when you cut them.