From QMCGroup
Sub-shotnoise optical state discrimination via a real-time closed-loop quantum measurement
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Bayesian Re-Analysis of Data
A problem that arose when applying optimal control methods to hypothesis testing is that to obtain a simple decision-making procedure, it is necessary to assume that all feedback was implemented perfectly. In this case, the closed-loop procedure originally developed by Dolinar leads to a decision making procedure that selects 
when the number of photon clicks is odd and 
when the number of clicks is even. This decision scheme assumes that all feedback displacements are implemented perfectly.
Perfect implementation of an optimal control policy is rarely the case in a real lab. In our case, finite extinction of our optical intensity modulators and detector dark counts suggest that changing hypotheses with every photon arrival may not in fact be optimal. Rather, the decision making procedure should account for well-characterized flaws in our application of the feedback displacements.
The files linked on the right demonstrate how we re-interpreted our closed-loop measurement data to implement Bayesian hypothesis testing, meaning that the hypothesis 
is chosen to maximize the 'a posteriori' probability ![\mathbbm{P}( H_j | n_{[0,t]}) = \frac{ p( n_{[0,t]} | H_j) \mathbbm{P}_0( H_j )}{p( n_{[0,t]} ) }](/images/math/7/c/d/7cd3582cdd2a2c26cb8d085954147cd5.png)
. The downloadable notes define what exactly we mean by this probability, and the Matlab files compute these probabilities from laboratory APD photon counting data after imposing a physical model of the limitations of our feedback displacements.