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The data available is sporadic, meaning non-statistical or techniques involving no prior knowledge on the parameter of interest may result in very bad estimate of the parameter. For example, a Doppler frequency estimator with DFT with very sparse data in time. So, we can call the bad estimates for the parameters as bad(weak) estimates. I am currently looking at Bayesian principles with Markov Chain Monte Carlo techniques to estimate a probability distribution of parameters.

However, I came across parameter estimation with quantum weak measurement recently. I am curious how can I translate what people use in the parameter estimation for the weak measurements in a classical setting.

I found this paper for the same.

Hofmann, H. F. (2011). On the estimation of interaction parameters in weak measurements. AIP Conference Proceedings, 1363(October 2011), 125–128. https://doi.org/10.1063/1.3630162

Here, they say that the estimation problem can be defined by this following equation.

$$ \hat{E_m} = \sqrt{w_m}(\hat{1} + \epsilon k_m \hat A) $$

Where ##\hat{E_m}## is the quantum statistics of the measurements, ##w_m## are the output probabilities, ##\hat{A}## is the observable, ##k_m## is the correlation between the outcome and the effects, ##\epsilon## is an interaction parameter. How can I relate these quantities with a classical estimation problem. For example, my data are represented as ##z##, the model I have is ##s## and the parameter I want to estimate is ##\Theta##. So,

$$ x = f(\Theta) $$

$$ z = f(\Theta) + noise $$

A bad estimate of ##\Theta## with a blind estimator (without any prior knowledge) can be named as ##\Gamma##.

So, the likelihood of ##z## given ##\Theta## is represented as ##p(z|\Theta)##, the prior probability of ##\Theta## is represented as #p(\Theta)# and the joint probability of data and the parameter is

$$ p(z, \Theta) = p(\Theta) p(z|\Theta) $$

There is also a joint probability mentioned in that paper, but I am confused.

Is ##m## all the values permitted for the parameter? Or ##m## are the weak estimates for the parameters? If someone could look into the paper and give me some insight with some relationship with what I defined as a classical estimation problem, that would be great. Thanks!