klderivation

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klderivation [2024/12/24 12:55] – [Cost of deliberation] pedroortegaklderivation [2024/12/24 12:58] (current) – [Connecting to the free energy objective] pedroortega
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 We can transform the above equality into a variational principle by replacing the individual choice costs $C(x \cap d|x \cap c)$ with arbitrary numbers. The resulting expression is convex in the posterior choice probabilities $P(x|d)$, so we get a nice and clean objective function with a unique minimum. We can transform the above equality into a variational principle by replacing the individual choice costs $C(x \cap d|x \cap c)$ with arbitrary numbers. The resulting expression is convex in the posterior choice probabilities $P(x|d)$, so we get a nice and clean objective function with a unique minimum.
  
-We can even go a step further: by multiplying the expression by $-1$, we can treat the costs as utilities, obtaining+We can even go a step further: noticing that the variational problem is translationally invariant in the costs, and multiplying the expression by $-1$, we can treat the resulting "negative costs plus a constant" as utilities, obtaining
 \[ \[
   \sum_x P(x|d) U(x) - \frac{1}{\beta} \sum_x P(x|d) \log \frac{ P(x|d) }{ P(x|c) }.   \sum_x P(x|d) U(x) - \frac{1}{\beta} \sum_x P(x|d) \log \frac{ P(x|d) }{ P(x|c) }.
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