Saturday, December 8, 2012

Defining faith via Bayesian reasoning

As you have hopefully noticed, I've been writing quite a lot about Bayes's theorem lately, since discussing it with John Loftus and applying it to his Outsider Test for Faith. My thinking has distilled a little further on this, and I think it could be possible to define the word faith in terms of this kind of analysis, i.e. in terms of the kind of effects it has on the reasoning process, given that a huge swath of reasoning processes are based upon Bayesian analysis.

First, consider the common definition for "faith." Though they vary, perhaps the best one I've seen is
Faith (n.): Belief without evidence or in spite of contradictory evidence.
Of course, this "or" is inclusive, as in it could account for both at the same time, and it provides for us a very clear understanding of what faith is doing when it comes to the (Bayesian) reasoning process. Indeed, it does exactly what I claimed it does (by claiming that the OTF has the goal of undoing these effects) in my previous posts about Bayes's theorem and the OTF.

So this article can stand alone, allow me to provide a very quick recap of the meaningful numbers in Bayesian analysis.
  1. P(h|b): the probability that the hypothesis h is true given background knowledge b, but not including the role of evidence. This is called the "prior probability," because it is an evaluation or guess made prior to examining the evidence obtained by testing the hypothesis against some body of evidence, e.
  2. P(e|h,b): the probability that the body of evidence e is explained by the combination of our background knowledge b on the assumption that our hypothesis h is true, called a "consequent." I will call this the "positive consequent," for convenience. In a sense, this is an estimate of how well the body of evidence is actually explained by the hypothesis.
  3. P(e|~h,b): the probability that the body of evidence e is explained by the combination of our background knowledge b on the assumption that our hypothesis h is false (the tilde ~ means "not" in logical shorthand), also called a "consequent." I will call this the "negative consequent." In a sense, this is an estimate of how well the body of evidence is explained by other hypotheses than the one we are testing.
  4. P(h|e,b): the probability that our hypothesis is true after weighing background knowledge b and the evidence e, called the "posterior probability." The goal of most Bayesian analysis is to obtain this number (so it is, usually, "the answer").
Bayes's theorem is a mathematical statement containing a formula involving these numbers that allows us to determine how likely a hypothesis is in light of the evidence, depending, of course, upon how good these three estimates are.


I want to define faith according to these numbers, then, using the definition I gave above:
Faith (n.): A cognitive bias in which a person overestimates the prior probability, overestimates the positive consequent, and/or underestimates the negative consequent in a Bayesian analysis.
Indeed, I argue that "overestimating the positive consequent" and "overestimating the prior probability" are technical ways to say "believe without evidence" and that "underestimating the negative consequent" is likewise a technical way of saying "believe in the face of contradictory evidence," although it's certainly not so cleanly cut since these categories overlap somewhat.

(Though the overestimate of the prior is more obviously the way to go with "believe without evidence," overestimating the positive consequent means "believing on false evidence," which is seeing evidence that isn't actually evidence, which implies of believing without actually having evidence since belief follows what is believed to be evidence.)

The effect is pretty straightforward:
  1. Overestimates to the positive consequent will have the effect of overestimating the posterior probability, which measures the likelihood that the hypothesis is true after evaluating the body of evidence.
  2. Underestimates to the negative consequent will also have the effect of overestimating the posterior probability.
  3. (Incidentally, overestimating the prior probability also overestimates the posterior probability.)
If someone were to approach a hypothesis entirely without bias, the posterior probability should tell them the actual degree of confidence upon which they can proceed with in terms of accepting the hypothesis. Faith, by this definition, then, overestimates the actual degree of confidence that one can have in a particular hypothesis.

The advantage of this definition is that it puts the cognitive bias of faith into clear contrast in terms of how it impacts the reasoning capacity of a person that holds it. In this light, it makes it very difficult to accept, at least in general, that this cognitive bias can be considered a "virtue."

N.b.: Skepticism is often attacked at this point as being a particular perspective that engages in its own cognitive biases (specifically to lower the posterior probability by whatever means necessary), but indeed, skepticism is functionally the attempt to remove all sources of cognitive bias from these sorts of analysis and to accept the (unbiased) posterior probability for what it is. Specifically and importantly, skepticism does not attempt to raise or lower the "real" posterior probability, but rather it seeks to come as close to that number as possible and then accept it (though new efforts at refinement are always welcomed). In that sense, one could say that skepticism is a bias against biases, but ample evidence (e.g. all of the accomplishments of science in terms of predictive power) support the claim that a tendency to avoid biases provides accurate, trustworthy information vastly more often than does biased information. Thus, calling it a "bias" really misses the meaning of the word "bias" and comes off, at best, as prevaricating.

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2 comments:

  1. Googling faith and Bayes theorem proves quite fruitful.

    Here are a couple interesting sites:

    http://blog.massimofuggetta.com/2012/07/18/blinded-by-evidence/: Just found this one today. Several good articles relating Bayes to faith. This is one of the early ones, to provide context.

    http://lesswrong.com/lw/na/trust_in_bayes/: This is an extremely good site/resource to get up to speed on Bayesian reasoning. Tons of good articles. This one I picked just because it seemed related to 'faith'.

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  2. "First, consider the common definition for "faith." Though they vary, perhaps the best one I've seen is

    "Faith (n.): Belief without evidence or in spite of contradictory evidence."


    Hi Dr. Lindsay

    I have a couple of questions about what I've copied above:

    (1) You call your definition of faith "the common definition"; my first question is, common to whom, or among what specific group(s)?

    (2) You say that the common definition you've offered is "the best" among the definitions you've seen; my second question is, best in what sense?

    I ask these questions because the definition you've offered is decidedly at odds not only with the dominant strands of the Christian theological and philosophical tradition, but also with the meanings of the Greek and Hebrew terms commonly translated into English as "faith." Faith is most commonly understood as trust and commitment, and these notions are quite consistent with beliefs held on poor grounds, decent grounds or strong grounds. Now you might often hear your conception of 'faith' among televangelists or among people who aren't familiar with the intellectual underpinnings of their religious beliefs -- I concede that. But then perhaps you should make it clear in your post that you're attacking not the best common definition of faith, but the most common misunderstanding of faith.

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