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Estimation of the underlying probabilities from measurements is a standard problem in statistics. The estimation of the probability from the measured histogram is known as the maximum likelihood estimator (MLE). As the number of samples goes to infinity, it can be shown that this estimation converges to the underlying (which follows from the definition of the probability). When the number of samples is small (relative to the range of possible values) MLE suffers because it assigns a probability of zero to any value not observed in the observed data. This problem can be addressed by using prior information about our expectations about the data. From this Bayesian perspective, the estimated probability density becomes a function of the prior expectation. This is frequently implemented by assuming that every possible observation has a uniform probability of being observed; i.e. we add a small amount of probability to each possible observation at the start (the add-$\lambda$ technique). In this technique the probability of observing $x$ is calculated as  \[  p_\lambda(x) = \frac{h(x,b)+\lambda}{N + b*\lambda} b\lambda}  \]  where $h(x,b)$ is the histogram of $x$ over $b$ bins. The denominator is the value needed to ensure that the sum of probabilities is equal to one. The joint probability is calculated in the same way  \[  p_\lambda(x,y) = \frac{h(x,y,b)+\lambda}{N + b^2*\lambda} b^2\lambda}  \]  where $h(x,y,b)$ is the 2D histogram calculated for $b$ bins for both $x$ and $y$. If $\lambda = 0$ we obtain the MLE, if $\lambda = 1$ it is called Laplace estimation and if $\lambda = 0.5$ it is called the expected likelihood estimator. The argument leading to the ELE is that an observation that is not observed in our measurements might be observed once if we increased the number of measurements and therefore the unbiased estimator of its probability should be 1/2 rather than 0.