Different internal Bayesian likelihood functions may be why we disagree. The recent shooting tragedies in Colorado and Wisconsin have set off a new round of arguments about gun control. Following the debate has made me realize that the reason the two sides can’t agree (and why different sides of almost every controversial issue can’t agree) is that their likelihood functions are completely different. The optimal way to make inferences about the world is to use Bayesian inference and there is some evidence that we are close to optimal in some circumstances. Nontechnically, Bayesian inference is a way to update the strength of your belief in something (i.e. probability) given new data. What you do is to combine your prior probability with the likelihood of the data given your internal model of the issue (and then normalize to get a posterior probability). For a more technical treatment of Bayesian inference, see here. I posted previously (see here) that I thought that drastic differences in prior probabilities is why people don’t seem to update their beliefs when faced with overwhelming evidence to the contrary. However, I’m starting to realize that the main reason might be that they have completely different models of the world, which in technical terms is their likelihood function.
Consider the issue of gun control. The anti-gun control side argue that “guns don’t kill people, people kill people’ and that restricting access to guns won’t prevent determined malcontents from coming up with some means to kill. The pro-gun control side argues that the amount of gun violence is inversely proportional to the ease of access to guns. After all, you would be hard pressed to kill twenty people in a movie theatre with a sword. The difference in these two viewpoints can be summarized by their models of the world. The anti-gun control people believe that the distribution of the will of people who would commit violence looks like this
where the horizontal line represents a level of gun restriction. In this world view, no amount of gun restriction would prevent these people from undertaking their nefarious designs. On the other hand, the pro-gun control side believes that the same distribution looks like this
in which case, the higher you set the barrier the fewer the number of crimes committed. Given these two views of the world, it is clear why new episodes of gun violence like the recent ones in Colorado and Wisconsin do not change people’s minds. What you would need to do is to teach a new likelihood function to the other side and that may take decades if at all.
4 thoughts on “What’s your likelihood?”
I think it’s even worse than that.
The posterior will be a product of the prior p(X) and the data likelihood p(d|X). But suppose X encodes not just a position on gun control, but also a set of beliefs about the authenticity of different sources of data. In that case p(d|X) might be high for two very different scenarios, one in which d comes from a trusted source and is consistent with X, and another in which d comes from an untrusted sources and is inconsistent with X. Then confirming data coming from a trusted source reinforces the gun control position and the trustworthiness of the sources; meanwhile, refuting data coming from an untrusted source just undermines the source further and possibly still confirms the gun control position.
Once most priors X are packaged in this way, encoding both the policy positions and the trust in sources, then the priors can never be moved with any amount of data. When this correlation becomes tight enough, even data against the policy position which comes from a trusted source may be more consistent with the source being untrustworthy than with the position being wrong, and then the source becomes excluded from the list of trusted sources.
This is my Bayesian theory of polarization, although I’m sure I’m not the first to describe it.
Rick: I agree with your theory but I would lump authenticity into the likelihood function.
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