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.