Wednesday, December 7, 2011

A Bayesian Look at Belief and Mind-Changing Argumentation

This posting presents a model of belief and argumentation based on the idea that people are rational Bayesian agents, ie. that they reevaluate beliefs on the basis of Bayes' theorem. It was prompted by a posting on Daniel Midgley's blog Good Reason.

Bayes' Theorem for Beliefs and Arguments

We start with defining some notation and some probability distributions. This will look a little mathematical, but there's really very little maths involved. I'll use B for the belief we're discussing, and Z for the person we're discussing. When we talk about Z's idea of someone else's mental state, we'll refer to that other person as Y. Finally, we'll use A for the argument that Z is considering.
Now here are some basic terms in Bayes' Theorem.
  • P(B;Z) - this is the probability Z assigns to B among all possible competing beliefs. For example, maybe there are only 2 possible beliefs: B that the earth is round, and B' that the earth is flat. In that case P(B;Z)+P(B';Z)=1, as any probability that Z assigns to B can't be assigned to B' and vice versa: if Z thinks that there is a 60% chance that the earth is flat, then s/he thinks that there is a 40% chance that it is round.
  • P(A|B;Z) - this is the probability that Z would find argument A convincing if s/he believed B to be 100% sure.
  • P(A;Z) - this is the probability that Z would find argument A convincing regardless of what they believed.
  • P(B|A;Z) - this is the probability that Z would apply to B if they were convinced by argument A.
Now we get to the little bit of maths which is Bayes' Theorem. Don't be frightened.
P(B|A;Z) = P(A|B;Z) P(B;Z) / P(A;Z)
In words, the theorem is saying that the probability you'd assign to a potential belief B if you were convinced by argument A depends on how likely you would be to be convinced by A if you accepted B 100%, multiplied by the probability of believing B, and divided by the probability of being convinced by A no matter what you believed.

The rest of this posting is going to explore some implications of this theorem.

The Bayesian Agent Assumption

In exploring belief and mind-changing in a Bayesian model of this kind, I am making an assumption about the behaviour of the (potential) believer. This assumption is that they adapt their belief to the evidence they see in a rational, Bayesian way. This might sound to be asking to much of people, but there does seem to be some evidence in its favour (eg, see Bayesian Rationality by Oaksford and Chater, 2007).

The Bayesian Theory of Mind

But not only do humans act as rational Bayesian agents, but they also presume others are as well. So Z might have a model of beliefs in another person Y. I'll write Y@Z for Y as imagined by Z. Z will impute Bayes' Theorem to Y.
P(B|A;Y@Z) = P(A|B;Y@Z) P(B;Y@Z) / P(A;Y@Z)
This gives us a theory of mind. We can use it to talk about what Z will be trying to do in changing the belief states of Y.

Certainty and Uncertainty

Now, let's look at the 3 fundamental relationships that Z can have with a belief B.
  1. Certainty - P(B;Z) = 1. In this case, Z is certain that B is true in an absolute sense.
  2. Uncertainty - 0 < P(B;Z) < 1. In this case Z is neither certain that B is true, or that it is false.
  3. Certainty that not - P(B;Z) = 0. In this case, Z is absolutely certain that B is not true.
There are slightly weaker versions of certainty and certainty that not. These are versions I'd call asymptotic certainty and asymptotic certainty that not, with probabilities for P(B;Z) of 1-ε and ε respectively, where ε is sufficiently small that it might as well be 1. (I may expand on this notion in a later blog post.)

Separate Magisteria

If Z is certain of B, P(B;Z) = 1, and so if C is any alternative belief to B, then P(C;Z) = 0. In this situation, P(A|B;Z)=P(A;Z), that is, the likelihood of any argument being judged convincing by Z will not change whether or not Z is asked to take B into account. It's easy to see why: because Z believes B with certainty, asking them to assume that B is true when evaluating A makes no difference; they will assume it anyway.

The separate magisteria arises where agent Z debates with Y, attempting to convert Y to his/her belief B. Argument A is irrelevant for Z, because it cannot affect the probability of B: P(B|A;Z) = P(A|B;Z) P(B;Z) / P(A;Z) = P(B;Z), because P(A|B;Z)/P(A;Z) = 1. It cannot touch their certain worldview. However, if they think they have any chance of converting Y, it must mean that in their view, Y assigns a non-zero probability to B a priori: P(B;Y@Z) > 0. Their goal, then, is to construct arguments A which will likely convince you, but which are particularly likely to be convincing only if belief B is true. In symbols, P(A|B;Y@Z) / P(A;Y@Z) >> 0. For example, argument A might be that ships leaving port disappear hull-first then sail. If the world is round (B), then this is much more likely P(A|B;Y@Z) than if we have no idea P(A;Y@Z).

Why do I say that there are separate magisteria at play here? Because as a round-earth fundamentalist, Z's acceptance of B cannot be influence by argument A, or any other kind of evidence. The evidence is only there to convince the open-minded. So the two magisteria are: the realm of certainties, and the realm of arguables. This accords with Gould's non-overlapping magisteria: religion is the magisterium of non-evidential, a priori certainties, while science is the magisterium of a priori uncertainties that allow experience-based re-evaluation.

"You Haven't Proven B to be False"

One common discourse pattern among the certain proceeds thus:
  • interlocutor presents argument A for non-B alternative C,
  • arguer says (correctly) but you haven't proven that B is false,
  • and this is enough for them to retain their assuredness of B.
Once again, this is a rational attitude, given particular probability assignments in Bayes' theorem. Let Z be the arguer, and Y the interlocutor. To make this work, we need Z to view the prior probability of B as infinitesimally close to 1, without actually being there (Z has an infinitesimally open mind): P(B;Z) = 1-ε, where ε is very, very small but not quite zero. For convenience, we'll abbreviate by a whatever probability P(A|B;Z) would be assigned by Z to the argument A given their belief. Then P(A;Z) = (1-ε)a+εk ≅ a, where k is the probability that Z would assign to A if B wasn't true ... an estimation of no practical consequence, unless a is also infinitesimally small, ie. unless argument A disproves belief B. Unless a is that small, P(A|B;Z)/P(A;Z) ≅ a/a = 1. Consequently, no argument that doesn't conclusively eliminate B will not be sufficient to weaken Z's belief by any noticeable amount.

Hope

I promised some hope for those arguing against beliefs held with certainty. The hope I offer is this: sometimes people lose their faith (in whatever it is they believe in). This may only mean that their belief drops by, say, 1%. Look at the two options:
  • Z has never met strong arguments against B, so that (using A! for the combination of all arguments put together that pertain to B) the arguments for and against are evenly weighted P(A!|B;Z)/P(A!;Z) ≅1. In this case, B still comes out the most likely hypothesis after considering the arguments, and self-reinforcement might push that prior probability P(B;Z) back towards 1.
  • In the 2nd scenario, all sorts of input has been filling Z's head to the point that P(A!|B;Z) ≅ 0.01, but for the alternative hypothesis C, P(A!|C;Z) ≅ 0.99, in other words, Z has met lots of convincing arguments against his/her views, they just haven't been able to change his/her absolute belief in B. While Z's mind is only infinitesimally open, the arguments are of no consequence, certainty is retained. But with a 10% loss of faith from B to C, (P(B;Z)=0.9, P(C;Z)=0.1) suddenly the probability of the data P(A!;Z) at 0.108 becomes significantly larger than P(A!|B;Z) at 0.01, and so the posterior probability of B, ie P(B|A!;Z), drops to around 0.1, while the posterior of C rises to 0.9. At this point, Z undergoes a conversion experience.
My take-away point, if this model can be believed, is as follows. It is worthwhile challenging even those who seem untouchable by reasoned, probabilistic argument. Because one day, they may step away from their certainty, just a little bit, and if your arguments have been strong, even a little step will be enough.

Conclusion

The times of problems I'm dealing with here are the clash between holding beliefs as a priori certainties that are therefore impervious to evidence, and holding them as a priori uncertainties whose fortunes rise and fall as evidence comes to light.

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