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An obvious special case is odds in sports betting updating rapidly up to the start of a sporting event, but this is more about new information coming in (more bettors) rather than more bits being computed from the *same* information.
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We need a quantitative concept of the time value of increasing precision. Like, the idea that a 70% right answer today is more valuable than the 99% right answer a week from now. Can we turn this into a marginal value vs time graph (set of graphs actually, 1 per bit)