Polarization and Disagreement: What Does Bayes Tell Us?

You have been called to sit on a jury for deciding whether a particular defendant in a murder case is guilty. During the trial, you are shown many different pieces of evidence. The situation is confusing, because they seem to point in different directions. You are shown, for example, a police report stating that a weapon like the one used in the murder was found in the defendant's house. Evidence of guilt, you might think. But you are also told that the DNA traces left on the body do not match the defendant’s. Evidence to the contrary, then. After hearing all the witnesses and reports, you do your utmost best to weigh all the evidence as reasonably as possible. As a result, you decide that the defendant is very likely guilty. Yet, on entering the jury deliberation room, you discover that other people think differently. In particular, after reviewing the same evidence, another juror, let’s call him ‘Juror X’, has come to the conclusion that the defendant is very unlikely to be guilty.

Surely Juror X has made a mistake, you think. They must have allowed some irrational biases to influence their judgment. After all, you feel that you yourself processed all the evidence rationally. Clearly something went wrong on their end.

On the other hand, some self-doubt might begin to creep in. Maybe it is I who made the mistake. If another person, who seems just as competent as I am to judge the trial, has come to a completely opposite conclusion, should I take this as evidence against my own judgment? Should I change my opinion? Or, at the very least, become a little less certain of it?

Disagreements like this do arise. And psychological experiments indicate that even when people have been exposed to exactly the same evidence, their opinions may move in opposite directions, leading to even greater disagreement (Lord et al. 1979; Plous 1991; McHoskey 1995; Pomerantz et al. 1995). This phenomenon is called ‘belief polarization’. Does belief polarization arise because one or more of the people involved are responding to the evidence in a way that is not entirely rational?

Psychologists have suggested that belief polarization happens because of ‘biased assimilation’ of evidence. For instance, the bias could be a form of motivated reasoning, where emotional motivations affect how the new evidence is perceived (Taber et al. 2009). A person processes the evidence in a way that supports a conclusion they want to believe. Another possibility is ‘biased interpretation’. In this case, the subject’s network of background beliefs unduly influences how they revise their opinions based on evidence.

But what does it mean to be ‘biased’? When are you, for example, ‘unduly’ influenced by your prior views? It is not that background beliefs should have no influence. Indeed, we expect that they should influence how evidence is taken up. In biased interpretation, however, they exert too much influence and get in the way of a reasonable response to the evidence.

One way to bring the question of bias into focus is to turn to Bayesian models. These are often regarded as a kind of normative standard, showing us how opinions should rationally be updated in response to new evidence. Deviating from Bayesian rules for processing evidence might then mean dealing with evidence in a biased way (Hahn and Harris 2014).

Complexes of background beliefs can be represented in a Bayesian formalism by ‘Bayesian networks’. These can represent not only how confident someone is in particular hypotheses, but how they they are thought to hang together causally. Again, our Bayesian rules tell us how to adjust our network as we gather more evidence. Now, if two people have very different background belief networks, Bayesian updating can clearly lead them to greater disagreement, even on the same evidence. This could happen, for example, if their background assumptions about what the evidence shows are different. Suppose two detectives, Alice and Bob, both think that a handwritten letter was left at the scene of a crime by the guilty party. They agree that the letter should be treated as a piece of trace evidence, bearing on the hypothesis that suspect A (rather than B) is guilty. But their background assumptions about what the cramped style of the handwriting indicates may be different. If Alice thinks such cramped writing more likely to have been produced by suspect A than B, the sample will strengthen her belief in A’s guilt. On the other hand, if Bob thinks the writing more likely characteristic of suspect B, then the very same evidence will lower his belief in A’s guilt. If Alice initially thinks A is more likely to be guilty than Bob does, their opinions will diverge. But both might well be following the normative Bayesian rules for updating their opinion.

But what happens when people do not differ in their background beliefs about what the evidence indicates, but rather are on more or less the same page about what the situation is? Given some reasonable amount of agreement between their belief networks, is it still possible for the same evidence to lead them in different directions, whilst they are both updating rationally according to Bayesian standards?

It turns out that the answer to this question is ‘yes’, at least in some cases, depending on the structure of the networks involved (Jern et al. 2014). This is a bit surprising and even troublesome. We might after all have hoped that getting more shared evidence would bring us together rather than driving us apart. It means that small differences in background beliefs can be amplified into significant differences of opinion, even without assuming anyone is dealing with the evidence in a biased fashion.

And what is more, the small differences in background beliefs may not even be about the actual topic of disagreement, but can be about how reliable people take their different sources of evidence to be. Belief polarization can happen even if the two people agree on everything other than the reliabilities and have the same initial attitude towards the hypothesis itself (Henderson and Gebharter 2021). People can be driven apart by background differences in trust, not just belief.

It looks as though we do not have to necessarily assume anyone has dealt with the evidence irrationally in a case of disagreement or belief polarization. Though of course, they might have! But then what should I do when I discover that Juror X has come to an entirely different conclusion than I have? Should I stick to my guns?

Again, Bayesian models can shed some light on what the rational reaction should be here. Think of Juror X’s opinion as just another piece of evidence which you have received. You need to process this along with your own evidence. A Bayesian model can tell you what the rational response is for a model representing a particular situation. The model helps you to evaluate how to navigate the delicate balance between your prior opinions and the pieces of evidence you have received. But there is no unique across-the-board right answer regarding the strategy you should take. In some situations, you should heavily revise your opinion in the direction of what the other person says. In other cases, the rational response is to remain fairly steadfast in your own opinion.

The advantage of the Bayesian lens is that it allows us to diagnose when certain reactions would be crossing over into irrationality. It can help to guide us in deciding when we are being unreasonably stubborn. Or when we are being too spineless in caving to another’s opinion. Disagreement and belief polarization do not have to be rooted in any irrational behavior. But sometimes they are. And Bayesian models help us to decide when.