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Will_Sawin comments on Searching for Bayes-Structure - Less Wrong
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So, we already have the underlying mathematical structure of the first half of cognition (determining the state of the world). What about the second half- influencing the world in a predictable way to achieve goals?
= creating mutual information between your utility function and the world. This is left as an exercise to the reader.
Isn't correlation the proper term here, not mutual information?
"Correlation" is a big old fuzzy mess, usually just defined in terms of what's not correlated. As a result it boils down to E[x]E[y] =/= E[xy], or sometimes p(x|y) =/= p(x). It can only really be made quantitative (i.e. correlation coefficients) with linear variables, rather than categories. Mutual information really captures in a quantitative way how much you can predict one from the other.
That said, they're both bad terms because a utility function is not a probability distribution.
But you can have a probability distribution of utility functions. Now that is true only in certain circumstances, but there is a simple model in which you can make a very nice probabilistic statement.
If the state of the world consists of a vector of N real variables, and a utility function is another vector, with the utility being the dot product (meaning all utility functions are linear in those variables), and the expected value of each coefficient is 0
then expected utility can be expressed as the covariance of this vector, and rational behavior maximizes that covariance. So that's something.