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I am simulating (using Mathematica) a Mach-Zehnder interferometer, with photon counting measurements at the end (based on the setup described in the recent post) for the input state $|\psi\rangle:=|0,2\rangle$ (for the different arms of the Mach-Zehnder I consider angles $\theta/2$ and $-\theta/2$). I set some true value of $\theta$ (say $\theta_{true} = 1.5$).

Using a random number generator, I carry out $\nu = 300$ independent measurements, from these measurement results I construct the Likelihood function $$L(\theta):= P(0,2)^{m_{02}}P(1,1)^{m_{11}}P(2,0)^{m_{20}},$$ (as in the post but without the multinomial coefficients since I retain each measurement result) and then numerically (Mathematica) find the $\theta$ that maximizes the Likelihood function. By doing so I numerically find the Maximum Likelihood Estimator (MLE). I do this many times about $10^5$ times, so that I have $10^5$ MLE estimators of $\theta$. I am trying to saturate the well-known Cramer-Rao bound, which states $(\Delta \theta)^2 \geq 1/(\nu\cdot FI)$, where $FI$ denotes the classical Fisher information (in this case the Fisher Information is analytically found to be $FI=2$). I would expect that for large enough trials $\gtrapprox 10^5$ as stated, I would approach equality in the stated Cramer-Rao bound (for $\nu = 300$ this is $\frac{1}{(300\cdot 2)} = 0.00167$). What I find is that the mean of the estimators seem to stabilize about 1.496 with the variance of the MLEs stabilizing at $(\Delta \theta)^2 \approx 0.006$. This is about a factor 3 larger than what is required to saturate the Cramer-Rao bound, which would be $(\Delta \theta)^2 \approx 0.00167.$

Query: There seems to be an inherent under-estimation (regardless of the number of iterations), in the sense that the mean of the MLEs are always around $1.496$ but never greater than $1.5$, for large enough iterations. Could this be a result of possible MLE bias for this particular setup? Or is this more likely a numerical imprecision lost during the coding process? The estimation is close but not close enough. Thanks for any ideas and assistance.

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    $\begingroup$ Is it possible to calculate the Maximum Likelihood Estimator analytically? $\endgroup$
    – narip
    Commented Mar 9 at 8:52
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    $\begingroup$ You do 300 shots to get probabilities then repeat this 100 000 times with 100 000 separate estimates, then average the estimates? What if you put together all of the shots and do one estimate? Or, the one that's easier to compute, do an estimate for each of the shots? It's easier because you can figure out the estimate for each of the three measurement outcomes, then just average them weighted by the number of times each measurement occurs $\endgroup$ Commented Mar 9 at 17:45
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    $\begingroup$ what's the exact formula you have for the dependence of the probabilities $P(i,j)$ wrt $\theta$? At the end of the day that's the only thing that matters to try and reproduce this $\endgroup$
    – glS
    Commented Apr 5 at 16:32
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    $\begingroup$ also, the MLE saturates the CRB only asymptotically. Here, asymptotically means with respect to the number of measurements used to compute the MLE. If I understand you correctly, you're fixing that number at $\nu=300$, which might be why you're not seeing the variance approaching the FI value. The "large" $10^5$ trials you used only ensures you get an accurate estimate of the MSE with the MLE estimator you're using (which however for $\nu=300$ doesn't have the optimal variance/MSE) $\endgroup$
    – glS
    Commented Apr 6 at 14:59

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