So the first question is: “how much should we expect the sample mean to move?”.
If the current state is μn, and we see a sample of x (where x is going to be 0 or 1 based on whether or not we have heads or tails), then the expected change is:
In these steps we are using the facts that (x is independent of the previous samples, and the distribution of x is Bernoulli with p=μn. (So E(x)=μn and Var(x)=σ2x=μn(1−μn)).
To do the proper version of this, we would be interested in how our prior changes, and our distribution for x wouldn’t purely be a function of μn. This will reduce the difference, so I have glossed over this detail.
The next question is: “given we shift the market parameter by O(1/n2), how much money (pnl) should we expect to be able to extract from the market in expectation?”
For this, I am assuming that our market is equivalent to a proper scoring rule. This duality is laid out nicely here. Expending the proper scoring rule out locally, it must be of the form f(x)=c+O(x2), since we have to be at a local minima. To use some classic examples, in a log scoring rule:
The liquidity calculation looks quite interesting but I’m not able to follow it.
I would really appreciate it if you wrote out a more detailed calculation!
what’s x ? What’s Sigma ? What’s pnl ?
So the first question is: “how much should we expect the sample mean to move?”.
If the current state is μn, and we see a sample of x (where x is going to be 0 or 1 based on whether or not we have heads or tails), then the expected change is:
E((μn−μn+1)2)=1(n+1)2E(μn−x)=1(n+1)2(μ2n−2μnE(x)+E(x2))
⋯=1(n+1)2(μ2n−2μnμn+μ2n+σ2x)=σ2x(n+1)2∼O(1n2)
In these steps we are using the facts that (x is independent of the previous samples, and the distribution of x is Bernoulli with p=μn. (So E(x)=μn and Var(x)=σ2x=μn(1−μn)).
To do the proper version of this, we would be interested in how our prior changes, and our distribution for x wouldn’t purely be a function of μn. This will reduce the difference, so I have glossed over this detail.
The next question is: “given we shift the market parameter by O(1/n2), how much money (pnl) should we expect to be able to extract from the market in expectation?”
For this, I am assuming that our market is equivalent to a proper scoring rule. This duality is laid out nicely here. Expending the proper scoring rule out locally, it must be of the form f(x)=c+O(x2), since we have to be at a local minima. To use some classic examples, in a log scoring rule:
qlog(p)+(1−q)log(1−p)=(qlog(q)−(q−1)log(1−q))+(p−q)2(2(q−1)q)+O((p−q)3)
in a brier scoring rule:
q(1−p)2+(1−q)p2=q(1−q)2+(1−q)q2+(q−p)2
x is the result of the (n+1)th draw sigma is the standard deviation after the first n draws pnl is the profit and loss the bettor can expect to earn
Thank you!
I’m still not following the entire derivation if I’m honest. Would you be able to share a more detailed derivation ? :)
Then imo the n+1 factor inside the expected value should be deleted. (n+1)(1n∑ni=1ai−1n+1∑n+1i=1ai)=(1+1n)∑ni=1ai−∑n+1i=1ai=1n∑ni=1ai−an+1
Whoops. Good catch. Fixing