I now like the “time vs ensemble” description better. I was trying to understand everything coming from a Bayesian frame, but actually, all of these ideas are more frequentist.
In a Bayesian frame, it’s natural to think directly in terms of a decision rule. I didn’t think time-averaging was a good description because I didn’t see a way for an agent to directly replace ensemble average with time average, in order to make decisions:
Ensemble averaging is the natural response to decision-making under uncertainty; you’re averaging over different possibilities. When you try to time-average to get rid of your uncertainty, you have to ask “time average what?”—you don’t know what specific situation you’re in.
In general, the question of how to turn your current situation into a repeated sequence for the purpose of time-averaging analysis seems under-determined (even if you are certain about your present situation). Surely Peters doesn’t want us to use actual time in the analysis; in actual time, you end up dead and lose all your money, so the time-average analysis is trivial.
Even if you settle on a way to turn the situation into an iterated sequence, the necessary limit does not necessarily exist. This is also true of the possibility-average, of course (the St Petersburg Paradox being a classic example); but it seems easier to get failure in the time-avarage case, because you just need non-convergence; ie, you don’t need any unbounded stuff to happen.
However, all of these points are also true of frequentism:
Frequentist approaches start from the objective/external perspective rather than the agent’s internal uncertainty. They don’t want to define probability as the subjective viewpoint; they want probability to be defined as limiting frequencies if you repeated an experiment over and over again. The fact that you don’t have direct access to these is a natural consequence of you not having direct access to objective truth.
Even given direct access to objective truth, frequentist probabilities are still under-defined because of the reference class problem—what infinite sequence of experiments do you conceive of your experiment as part of?
And, again, once you select a sequence, there’s no guarantee that a limit exists. Frequentism has to solve this by postulating that limits exist for the kinds of reference classes we want to talk about.
So, I now think what Ole Peters is working on is frequentist decision theory. Previously, the frequentist/Bayesian debate was about statistics and science, but decision theory was predominantly Bayesian. Ole Peters is working out the natural theory of decision making which frequentists could/should have been pursuing. (So, in that sense, it’s much more than just a new argument for kelly betting.)
Describing frequentist-vs-Bayesian as time-averaging vs possibility-averaging (aka ensemble-averaging) seems perfectly appropriate.
So, on my understanding, Ole’s response to the three difficulties could be:
We first understand the optimal response to an objectively defined scenario; then, once we’ve done that, we can concern ourselves with the question of how to actually behave given our uncertainty about what situation we’re in. This is not trying to be a universal formula for rational decision making in the same way Bayesianism attempts to be; you might have to do some hard work to figure out enough about your situation in order to apply the theory.
And when we design general-purpose techniques, much like when we design statistical tests, our question should be whether given an objective scenario the decision-making technique does well—the same as frequentists wanting estimates to be unbiased. Bayesians want decisions and estimates to be optimal given our uncertainty instead.
As for how to turn your situation into an iterated game, Ole can borrow the frequentist response of not saying much about it.
As for the existence of a limit, Ole actually says quite a bit about how to fiddle with the math until you’re dealing with a quantity for which a limit exists. See his lecture notes. On page 24 (just before section 1.3) he talks briefly about finding an appropriate function of your wealth such that you can do the analysis. Then, section 2.7 says much more about this.
The general idea is that you have to choose an analysis which is appropriate to the dynamics. Additive dynamics call for additive analysis (examining the time-average of wealth). Multiplicative dynamics call for multiplicative analysis (examining the time-average of growth, as in kelly betting and similar settings). Other settings call for other functions. Multiplicative dynamics are common in financial theory because so much financial theory is about investment, but if we examine financial decisions for those living on income, then it has to be very different.
I now like the “time vs ensemble” description better. I was trying to understand everything coming from a Bayesian frame, but actually, all of these ideas are more frequentist.
In a Bayesian frame, it’s natural to think directly in terms of a decision rule. I didn’t think time-averaging was a good description because I didn’t see a way for an agent to directly replace ensemble average with time average, in order to make decisions:
Ensemble averaging is the natural response to decision-making under uncertainty; you’re averaging over different possibilities. When you try to time-average to get rid of your uncertainty, you have to ask “time average what?”—you don’t know what specific situation you’re in.
In general, the question of how to turn your current situation into a repeated sequence for the purpose of time-averaging analysis seems under-determined (even if you are certain about your present situation). Surely Peters doesn’t want us to use actual time in the analysis; in actual time, you end up dead and lose all your money, so the time-average analysis is trivial.
Even if you settle on a way to turn the situation into an iterated sequence, the necessary limit does not necessarily exist. This is also true of the possibility-average, of course (the St Petersburg Paradox being a classic example); but it seems easier to get failure in the time-avarage case, because you just need non-convergence; ie, you don’t need any unbounded stuff to happen.
However, all of these points are also true of frequentism:
Frequentist approaches start from the objective/external perspective rather than the agent’s internal uncertainty. They don’t want to define probability as the subjective viewpoint; they want probability to be defined as limiting frequencies if you repeated an experiment over and over again. The fact that you don’t have direct access to these is a natural consequence of you not having direct access to objective truth.
Even given direct access to objective truth, frequentist probabilities are still under-defined because of the reference class problem—what infinite sequence of experiments do you conceive of your experiment as part of?
And, again, once you select a sequence, there’s no guarantee that a limit exists. Frequentism has to solve this by postulating that limits exist for the kinds of reference classes we want to talk about.
So, I now think what Ole Peters is working on is frequentist decision theory. Previously, the frequentist/Bayesian debate was about statistics and science, but decision theory was predominantly Bayesian. Ole Peters is working out the natural theory of decision making which frequentists could/should have been pursuing. (So, in that sense, it’s much more than just a new argument for kelly betting.)
Describing frequentist-vs-Bayesian as time-averaging vs possibility-averaging (aka ensemble-averaging) seems perfectly appropriate.
So, on my understanding, Ole’s response to the three difficulties could be:
We first understand the optimal response to an objectively defined scenario; then, once we’ve done that, we can concern ourselves with the question of how to actually behave given our uncertainty about what situation we’re in. This is not trying to be a universal formula for rational decision making in the same way Bayesianism attempts to be; you might have to do some hard work to figure out enough about your situation in order to apply the theory.
And when we design general-purpose techniques, much like when we design statistical tests, our question should be whether given an objective scenario the decision-making technique does well—the same as frequentists wanting estimates to be unbiased. Bayesians want decisions and estimates to be optimal given our uncertainty instead.
As for how to turn your situation into an iterated game, Ole can borrow the frequentist response of not saying much about it.
As for the existence of a limit, Ole actually says quite a bit about how to fiddle with the math until you’re dealing with a quantity for which a limit exists. See his lecture notes. On page 24 (just before section 1.3) he talks briefly about finding an appropriate function of your wealth such that you can do the analysis. Then, section 2.7 says much more about this.
The general idea is that you have to choose an analysis which is appropriate to the dynamics. Additive dynamics call for additive analysis (examining the time-average of wealth). Multiplicative dynamics call for multiplicative analysis (examining the time-average of growth, as in kelly betting and similar settings). Other settings call for other functions. Multiplicative dynamics are common in financial theory because so much financial theory is about investment, but if we examine financial decisions for those living on income, then it has to be very different.