Timeline for How to play golf in one dimension?
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
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| Jan 22 at 11:57 | comment | added | Nate River | @James Martin Yes your interpretation is correct. With $X$ as you’ve defined, set $g(x) = \inf \mathbb E^x[ \tau]$, where $\tau := \inf\{n \geq 0 \, | \, |X_n| \leq 1\}$, $\mathbb E^x$ denotes the expectation operator under a probability measure such that $X_0 = x$, and the infimum is taken over all sequential choices of $\mu$ at each step. Then $g$ satisfies the given recursion by the dynamic programming principle argument. | |
| Jan 22 at 11:43 | comment | added | domotorp | @James Sorry, $f$ is the same as $g$, and the expectation sign was also missing - I fixed both. | |
| Jan 22 at 11:40 | comment | added | domotorp | @Nate I don't see any direct relation. It's more like the opposite of your question: we pick the direction, but we have no control over the step size. | |
| Jan 22 at 11:39 | history | edited | domotorp | CC BY-SA 4.0 |
changed f to g in the question and added the missing expectation E
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| Jan 22 at 11:19 | comment | added | James Martin | Based on the comments above, other people already seem to understand what the model is... :) Can someone write the description more clearly? What is a "step"? What are "$f$" and "$g$"? I'm guessing that you have a position $X_n$ at step $n$, then you choose a $\mu$, and you get a new position $X_{n+1}\sim N(X_n+\mu, c\mu^2)$. And you're trying to minimise (the expectation of?) something - the number of steps until you achieve $|X_n|\leq 1$? The recursion for $g$ seems to have a min over uncountably many r.vs. - maybe it needs something more like $g(x)=1+\min_\mu E g(x+N(\mu, c\mu^2))$? | |
| Jan 22 at 9:43 | comment | added | Nate River | Haha, calling this process golf is so apt. Is there any direct relation to the planar survival problem I have in the linked post? | |
| Jan 22 at 8:16 | comment | added | domotorp | @Dror Great link, thank you! I have been trying to find other models of golf, but I couldn't. They also study one-dimensional golf, but only consider the probability of a putt, so ending the game from a given distance in one move. Their data seems to agree with the normal distribution model I proposed. | |
| Jan 22 at 8:04 | comment | added | Dror Speiser | You will probably enjoy Gelman's and Nolan's foray into (2-d) golf. They wrote up the first part here stat.columbia.edu/~gelman/research/published/golf.pdf and added data here statmodeling.stat.columbia.edu/2019/03/21/… with further links there. | |
| Jan 22 at 7:53 | history | asked | domotorp | CC BY-SA 4.0 |