Just think of the whole thing on a log scale. The error bars become epsilon close in ratio on a log scale. There’s some terms and conditions to that approximation—average expected log return must be nonzero, for instance. But it shouldn’t be terribly restrictive.
If your immediate thought is “but why consider things on a log scale in the first place?”, then remember that we’re talking about mode/order statistics, so monotonic transformations are totally fine.
(Really, though, if we want to be precise… the exact property which makes Kelly interesting is that if you take the Kelly strategy and any other strategy, and compare them to each other, then Kelly wins with probability 1 in the long run. That’s the property which tells us that Kelly should show up in evolved systems. We can get that conclusion directly from the central limit theorem argument: as long as the “average expected log return” term grows like O(n), and the error term grows like O(sqrt(n)) or slower, we get the result. In order for a non-Kelly strategy to beat Kelly in the long run with greater-than-0 probability, it would somehow have to grow the error term by O(n).)
If your immediate thought is “but why consider things on a log scale in the first place?”, then remember that we’re talking about mode/order statistics, so monotonic transformations are totally fine.
Riiight, but, “totally fine” doesn’t here mean “Kelly approximately maximizes the mode as opposed to maximizing an approximate mode”, does it?
I have approximately two problems with this:
Bounding the ratio of log wealth compared to a true mode-maximizer would be reassuring if my utility was doubly logarithmic. But if it’s approximately logarithmic, this is little comfort.
But are we even bounding the ratio of log wealth to a true mode-maximizer? As I mentioned, I’m not sure a mode-maximizer is even a constant-fraction strategy.
Actually, you’re right, I goofed. Monotonic increasing transformation respects median or order statistics, so e.g. max median F(u) = F(max median u) (since F commutes with both max and median), but mode will have an additional term contributed by any nonlinear transformation of a continuous distribution. (It will still work for discrete distributions—i.e. max mode F(u) = F(max mode u) for u discrete, and in that case F doesn’t even have to be monotonic.)
So I guess the argument for median is roughly: we have some true optimum policy θ∗ and Kelly policy θK, and medianP[u|θK]F(u)≈medianP[u|θ∗]F(u), which implies medianP[u|θK]u≈medianP[u|θ∗]u as long as F is continuous and strictly increasing.
Just think of the whole thing on a log scale. The error bars become epsilon close in ratio on a log scale. There’s some terms and conditions to that approximation—average expected log return must be nonzero, for instance. But it shouldn’t be terribly restrictive.
If your immediate thought is “but why consider things on a log scale in the first place?”, then remember that we’re talking about mode/order statistics, so monotonic transformations are totally fine.
(Really, though, if we want to be precise… the exact property which makes Kelly interesting is that if you take the Kelly strategy and any other strategy, and compare them to each other, then Kelly wins with probability 1 in the long run. That’s the property which tells us that Kelly should show up in evolved systems. We can get that conclusion directly from the central limit theorem argument: as long as the “average expected log return” term grows like O(n), and the error term grows like O(sqrt(n)) or slower, we get the result. In order for a non-Kelly strategy to beat Kelly in the long run with greater-than-0 probability, it would somehow have to grow the error term by O(n).)
Riiight, but, “totally fine” doesn’t here mean “Kelly approximately maximizes the mode as opposed to maximizing an approximate mode”, does it?
I have approximately two problems with this:
Bounding the ratio of log wealth compared to a true mode-maximizer would be reassuring if my utility was doubly logarithmic. But if it’s approximately logarithmic, this is little comfort.
But are we even bounding the ratio of log wealth to a true mode-maximizer? As I mentioned, I’m not sure a mode-maximizer is even a constant-fraction strategy.
Sorry if I’m being dense, here.
Actually, you’re right, I goofed. Monotonic increasing transformation respects median or order statistics, so e.g. max median F(u) = F(max median u) (since F commutes with both max and median), but mode will have an additional term contributed by any nonlinear transformation of a continuous distribution. (It will still work for discrete distributions—i.e. max mode F(u) = F(max mode u) for u discrete, and in that case F doesn’t even have to be monotonic.)
So I guess the argument for median is roughly: we have some true optimum policy θ∗ and Kelly policy θK, and medianP[u|θK]F(u)≈medianP[u|θ∗]F(u), which implies medianP[u|θK]u≈medianP[u|θ∗]u as long as F is continuous and strictly increasing.