For quite a while I've been interested in Bayesian reasoning/statistics, even though I've never understood this subject very well.
Now I'm reading Steven Pinker's new book, "Rationality." It has a chapter on Beliefs and Evidence that focuses on Bayesian reasoning. Which is, basically (this is an introduction to an online tutorial):
Bayes' rule or Bayes' theorem is the law of probability governing the strength of evidence -- the rule saying how much to revise our probabilities (change our minds) when we learn a new fact or observe new evidence.
"Prior probability" in the Bayesian perspective is our credence in an idea before looking at the evidence for something. "Posterior probability" is our credence in an idea after we've examined the evidence.
How this is calculated is kind of complex. But even without knowing anything more about Bayesian reasoning, the following excerpt from the Beliefs and Evidence chapter should be mostly understandable, with the exception of the paragraph in parentheses, which is quite technical.
Pinker explains why believing in miracles doesn't make sense.
Our neglect of base rates is a special case of our neglect of priors: the vital, albeit more nebulous, concept of how much credence we should give a hypothesis before we look at the evidence.
Now, believing in something before you look at the evidence may seem like the epitome of irrationality. Isn't that what we disdain as prejudice, bias, dogma, orthodoxy, preconceived notions? But prior credence is simply the fallible knowledge accumulated from all our experience in the past.
Indeed, the posterior probability from one round of looking at evidence can supply the prior probability for the next round, a cycle called Bayesian updating.
It's simply the mindset of someone who wasn't born yesterday. For fallible knowers in a chancy world, justified belief cannot be equated with the last fact you came across.
As Francis Crick liked to say, "Any theory that can account for all the facts is wrong, because some of the facts are wrong." This is why it is reasonable to be skeptical of claims for miracles, astrology, homeopathy, telepathy, and other paranormal phenomena, even when some eyewitness or laboratory study claims to show it.
Why isn't that dogmatic and pigheaded?
The reasons were laid out by that hero of reason, David Hume. Hume and Bayes were contemporaries, and though neither read the other, word of the other's ideas may have passed between them through a mutual colleague, and Hume's famous argument against miracles is thoroughly Bayesian.
Nothing is esteemed a miracle, if it ever happen in the common course of nature. It is no miracle that a man, seemingly in good health, should die on a sudden: because such a kind of death, though more unusual than any other, has yet been frequently observed to happen. But it is a miracle, that a dead man should come to life, because that has never been observed in any age or country.
In other words, miracles such as resurrection must be given a low prior probability. Here is the zinger:
No testimony is sufficient to establish a miracle, unless the testimony be of such a kind, that its falsehood would be more miraculous, than the fact, which it endeavors to establish.
In Bayesian terms, we are interested in the posterior probability that miracles exist, given the testimony. Let's contrast it with the posterior probability that no miracles exist given the testimony.
(In Bayesian reasoning, it's often handy to look at the odds, that is, the ratio of the credence of a hypothesis to the credence of the alternative, because it spares us the tedium of calculating the marginal probability of the data in the denominator, which is the same for both posteriors and conveniently cancels out.)
The "fact which it endeavors to establish" is the miracle, with its low prior, dragging down the posterior. The testimony "of such a kind" is the likelihood of the data given the miracle, and it's "falsehood" is the likelihood of the data given no miracle: the possibility that the witness lied, misperceived, misremembered, embellished, or passed along a tale tale he heard from someone else.
Given everything we know about human behavior, that's far from miraculous! Which is to say, its likelihood is higher than the prior probability of a miracle.
That moderately high likelihood boosts the posterior probability of no miracle, and lowers the overall odds of a miracle compared to no miracle.
Another way of putting it is this: Which is more likely -- that the laws of the universe as we understand them are false, or that some guy got something wrong?