Note that Kelly is valid under the assumption that you know the true probabilities. II do not know whether it is still valid when all you know is a noisy estimate of true probabilities—is it? It definitely gets more complicated when you are betting against somebody with a similarly noisy estimate of the same probably, as at some level you now need to take their willingness to bet into account when estimating the true probability—and the higher they are willing to go, the stronger the evidence that your estimate may be off. At the very least, that means that the uncertainty of your estimate also becomes the factor (the less certain you are, the more attention you should pay to the fact that somebody is willing to bet against you). Then the fact that sometimes you need to spend money on things, rather than just investing/betting/etc, and that you may have other sources of income, also complicates the calculus.
Note that Kelly is valid under the assumption that you know the true probabilities.
I object to the concept of “know the true probabilities”; probabilities are in the map, not the territory.
II do not know whether it is still valid when all you know is a noisy estimate of true probabilities—is it?
Is Kelly really valid even when you “know the true probabilities”, though?
From a Bayesian perspective, there’s something very weird about the usual justification of Kelly, as I discuss (somewhat messily) here. It shares more in common with frequentist thinking than with Bayesian. (Hence why it’s tempting to say things like “it’s valid when you know the true probabilities”—but as a Bayesian, I can’t really make sense of this, and would prefer to state the actual technical result, that it’s guaranteed to win in the long run with arbitrarily high probability in comparison with other methods, if you know the true frequencies.)
Ultimately, I think Kelly is a pretty good heuristic; but that’s all. If you want to do better, you should think more carefully about your utility curve for money-in-hand. If your utility is approximately logarithmic in money, then Kelly will be a pretty good strategy for you.
There is a sense in which fractional Kelly is combining two noisy estimates of the probabilities. (Market probability and your probability). (I say more here).
Note that Kelly is valid under the assumption that you know the true probabilities. II do not know whether it is still valid when all you know is a noisy estimate of true probabilities—is it? It definitely gets more complicated when you are betting against somebody with a similarly noisy estimate of the same probably, as at some level you now need to take their willingness to bet into account when estimating the true probability—and the higher they are willing to go, the stronger the evidence that your estimate may be off. At the very least, that means that the uncertainty of your estimate also becomes the factor (the less certain you are, the more attention you should pay to the fact that somebody is willing to bet against you). Then the fact that sometimes you need to spend money on things, rather than just investing/betting/etc, and that you may have other sources of income, also complicates the calculus.
I object to the concept of “know the true probabilities”; probabilities are in the map, not the territory.
Is Kelly really valid even when you “know the true probabilities”, though?
From a Bayesian perspective, there’s something very weird about the usual justification of Kelly, as I discuss (somewhat messily) here. It shares more in common with frequentist thinking than with Bayesian. (Hence why it’s tempting to say things like “it’s valid when you know the true probabilities”—but as a Bayesian, I can’t really make sense of this, and would prefer to state the actual technical result, that it’s guaranteed to win in the long run with arbitrarily high probability in comparison with other methods, if you know the true frequencies.)
Ultimately, I think Kelly is a pretty good heuristic; but that’s all. If you want to do better, you should think more carefully about your utility curve for money-in-hand. If your utility is approximately logarithmic in money, then Kelly will be a pretty good strategy for you.
There is a sense in which fractional Kelly is combining two noisy estimates of the probabilities. (Market probability and your probability). (I say more here).