via the ergodic mapping, which finds the transformation that renders the wealth process ergodic, so that time and ensemble averages coincide
‘Ergodic’ kept looking like noise to me, and this seemed to be intended as an explanation but, to me, clarified nothing. (Feldspars problem, presumably.) I’ll attempt to rephrase, reflecting my present understanding after a few minutes of research:
Most processes that vary do so differently over time than at any present moment. The ensemble average is looking at a bunch of similar processes and how they’re varying right now, like forming the S&P 500 from 500 individual stocks, essentially buying the ensemble average of their growth (now and in the future). The time average of the growth, is the long run performance of one stock; the (past and/or future) time average of any individual stock is not the same as the ensemble. This makes prediction hard.
Something is ergodic if it does not have this problem. That could mean a steady state; a dollar bill’s value, denominated in USD, is mostly ergodic, though imperfectly because we might someday go full Weimar, abolish greenbacks, or make it an old print that’s a collector’s item. But you can construct things to be ergodic on purpose, as the examples describe; Kelly-betting is a strategy whose ‘many-worlds’ possible present instances have the same bankroll growth ensemble average that, in the long run, the time average predicts.
I am unclear if ‘the ergodic mapping’ is a specific coherent concept or a more flexible process that you apply to a phenomenon in imprecise ways to arrive at a result you can then prove is ergodic.
‘Ergodic’ kept looking like noise to me, and this seemed to be intended as an explanation but, to me, clarified nothing. (Feldspars problem, presumably.) I’ll attempt to rephrase, reflecting my present understanding after a few minutes of research:
Most processes that vary do so differently over time than at any present moment. The ensemble average is looking at a bunch of similar processes and how they’re varying right now, like forming the S&P 500 from 500 individual stocks, essentially buying the ensemble average of their growth (now and in the future). The time average of the growth, is the long run performance of one stock; the (past and/or future) time average of any individual stock is not the same as the ensemble. This makes prediction hard.
Something is ergodic if it does not have this problem. That could mean a steady state; a dollar bill’s value, denominated in USD, is mostly ergodic, though imperfectly because we might someday go full Weimar, abolish greenbacks, or make it an old print that’s a collector’s item. But you can construct things to be ergodic on purpose, as the examples describe; Kelly-betting is a strategy whose ‘many-worlds’ possible present instances have the same bankroll growth ensemble average that, in the long run, the time average predicts.
I am unclear if ‘the ergodic mapping’ is a specific coherent concept or a more flexible process that you apply to a phenomenon in imprecise ways to arrive at a result you can then prove is ergodic.