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Because . They are the same. Does that help?
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The term is meant to be a posterior distribution after seeing data. If you have a good prior you could take . However, note could be high. You want trade-off between the cost of updating the prior and the loss reduction.
Example, say we have a neural network. Then our prior would be the initialization and the posterior would be the distribution of outputs from SGD.
(Btw thanks for the correction)
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Your example is interesting and clarifies exchange rates. However,
The shadow price quantifies the opportunity cost, so if I’m paid my shadow price, then that’s just barely enough to cover my opportunity cost.
This is an interpretive point I’d like to focus on. When you move a constraint, in this case with price, the underlying equilibrium of the optimization shifts. From this perspective your usage of the word ‘barely’ stops making sense to me. If you were to ‘overshoot’ you wouldn’t be optimal in the new optimization problem.
At this point I understand that the cheerful price will be equivalent to or more than the shadow price. You want to be able to shift the equilibrium point and have slack left over. It just seems obvious, to me, that shadow price isn’t an exactly measurable thing in this context and so you’d naturally be led to make a confidence interval (belief) for it. Cheerful price is just the upper estimate on that. Hence, I’m surprised why this is being treated as a new / distinct concept.
I suppose this is the most correct answer. I’m not really updating very much though. From my perspective I’ll continue to see cheerful price as a psychological/subjective reinvention of shadow price.
Edit: It seems clear in this context, shadow price isn’t exactly measurable. Cheerful price is just the upper estimate on the shadow price.
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