We have often pondered the question: when you find that you and someone else disagree, how much weight should you give to your and their opinions in forming your new opinion? To explore this, I've worked out a simple math model of disagreement between two "Bayesian wannabes", i.e., agents who are trying to act like Bayesians, but know that they make mistakes, and try to adjust for this fact.

Consider two agents, A and B, having a conversation about a truth t = x_{1} + x_{2} + x_{3} + ... First A sees clue x_{1}, and reports r_{1}, his estimate of truth t. Next B sees report r_{1}, and also clue x_{2}, and then reports r_{2}, his estimate of truth t. A now sees report r_{2}, a new clue x_{3} and reports r_{3}. The two of them could go back and forth like this for a long time.

If A and B were perfect Bayesians (and if each x_{i} were independently and normally distributed with zero mean and a known variance V_{i}), then we would have r_{i} = x_{i} + r_{i-1}. When combining their last two expressed opinions, each agent puts *zero *weight on his own last report, and just adds his new clue to the other agent's last report!

OK, but what about imperfect agents? I assume:

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