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I have simulated a set of probabilities $\{p_n\}$ by the hit-or-miss Monte Carlo method. Specifically, my program checks whether or not an $n\times n$ matrix has a certain feature or not; say that, out of $N$ trials, $k_n$ have this feature (success). Then $p_n=k_n/N.$

Question: Now, I want to find the uncertainty (or standard error) $\sigma_{p_n}$ on $p_n$. I'd like to do this so that I can weigh my probabilities when fitting these probabilities as a function of $n.$ My weights are defined as $w_n=\sigma_{p_n}^{-2}.$ This definition seems to be completely standard, but it doesn't go well with my intuition, as explained below. I'd like to know if my intuition is simply wrong (and why), or if there are more suitable weights I could be using.

My problem: As far as I understand, $k_n$ is distributed binomially, so $$\sigma_{p_n}=N^{-1/2}\sqrt{p_n(1-p_n)}$$

This means that a $p_n,$ smaller than some $p_m\leq1/2,$ has a smaller uncertainty than $p_m.$ This makes sense to me (a random walk will not go very far from its starting point if it has a very low probability of taking a step). However, it seems very counter-intuitive to me to weigh a very low probability $p_n$ (where my program perhaps only had $10^2$ successes out of $10^9$ trials) higher than a high probability $p_m$ (which had success, say, half of the time). Using a "relative fluctuation" away from some $p_n$, defined as $$\sigma_{p_n}/p_n=N^{-1/2}\sqrt{\frac{1}{p_n}-1},$$ as my effective uncertainty seems closer to what my intuition tells me is important when choosing my weights.

I'm having a hard time formulating why this is, but it is like this: Flip a coin $100$ times: It comes up heads $54$ times. Flip another coin $100$ times: It comes up heads a single time. Now estimate the bias of each coin. Which estimate are you most confident in? I would say that I'm much more confident in my estimate for the first coin. Should I be? This is the essence of my question.

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  • $\begingroup$ I'm probably misunderstanding something, but what if you do something like bootstrapping to approximate the sampling distribution of $p_n$ and measure the variance from there? Also, for the coin analogy, if you have no prior knowledge of how a coin works, why would you trust the first estimate more? Consider an alien machine that outputs $a$ or $b$. You press it 100 times and get 54 $a$s, then your friend presses it 100 times and gets 1 $a$. I'd not trust one more than the other. $\endgroup$ – user3658307 May 17 '17 at 18:47
  • $\begingroup$ It is not so much the variance I'm worried about. Rather, it is the weights that I assign to each of my found $p_n$ when fitting $p(n)$ to my points $\{p_n\},$ and these weights are functions of the variance. I feel like the standard definition of weights $w=\sigma^{-2}$ favor the wrong points. Regarding the coin analogy, I was talking about two different coins, so in your analogy, it should be two different machines. Perhaps you are still right. $\endgroup$ – Bobson Dugnutt May 17 '17 at 19:33

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