A probability threshold belief set is deductively closed exactly when the distribution has a step, and the new revision rule is the minimal Kullback-Leibler change that raises the new statement to the threshold.
Jeffrey's update rule as a minimizer of Kullback-Leibler divergence
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abstract
In this paper, we show a more concise and high level proof than the original one, derived by researcher Bart Jacobs, for the following theorem: in the context of Bayesian update rules for learning or updating internal states that produce predictions, the relative entropy between the observations and the predictions is reduced when applying Jeffrey's update rule to update the internal state.
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On Lockean beliefs that are deductively closed and minimal change
A probability threshold belief set is deductively closed exactly when the distribution has a step, and the new revision rule is the minimal Kullback-Leibler change that raises the new statement to the threshold.