Effect of noise on an MLE estimate Question : In general, what is the effect of noise on an estimate obtained from Maximum Likelihood Estimation technique?
 A: For future reference, the paper related to this question is: 

Jaap C. Schouten, Floris Takens, and Cor M. van den Bleek,
  "Maximum-likelihood estimation of the entropy of an attractor", Phys.
  Rev. E 49, 126 – Published 1 January 1994.

In this paper, the authors want to obtain the Maximum Likelihood (ML) estimate of the Kolmogorov entropy, denoted with $K$. In order to do so, as a preliminary step, a quantity called $k=K \tau_s$, where $\tau_s = 1/f_s$ and $f_s$ is the sampling frequency, is estimated via ML techniques. This answer will focus on the statistical aspects of the question.
In this case, the pdf (probability density function) is given by eq. (17) of the paper, i.e.
$$
p(b_1,b_2,\dots,b_M; k) = (e^k -1)^M \exp \left\{ -k \sum_{i=1}^M b_i \right\}
$$
where $M$ indipendent realizations of $b \in \mathbb{N}_0$, which is a random variable (RV) whose pdf is a geometric distribution, are observed. Reformulating it directly in function of $K$, the actual problem to be solved is
$$
\hat{K}_{ML} = \arg \max_{K} \ (e^{K \tau_s} -1)^M \exp \left\{ -K \tau_s \sum_{i=1}^M b_i \right\}
$$
and the closed-form solution is given by eq. (20), which can be rewritten as
$$
\hat{K}_{ML} = - f_s \left| 1 - \frac{1}{\bar{b}} \right|
$$
where $\bar{b}$ is the sample mean of the $b_i$. Observe that, since the $b_i$ are RVs, $\bar{b}$ is itself a RV. Clearly, $\hat{K}_{ML}$ maximizes the likelihood function. $\hat{K}_{ML}$ is an estimator, thus, by definition, it is a random variable. More precisely, its randomness comes from the fact that $\bar{b}$ is random. In principle, if the distribution of $\bar{b}$ could be calculated, then also the distribution of $\hat{K}_{ML}$ could be derived.
In general, if an estimator $\hat{x}$ is an absolutely continuous RV, denoting two estimates (i.e., realizations of the estimator) with $\hat{x}_1$ and $\hat{x}_2$, the condition $\hat{x}_1 = \hat{x}_2$ happens wp0 (with probability zero). This does not mean it can never happen, but only that it is "negligible" (using measure theory, a clear analogy is a zero-measure set). In fact, an event with probability zero is called negligible (and not "impossible") and this is the true subtility.
This does lead to somewhat counter-intuitive results, such as the fact that negligible events can happen even an infinite amount of times (!), given an infinitely long observation period.

SNR influences the behaviour of the system and plays an important role. I will explain it with an analogy. Consider the following simplified statement
$$
(1) \quad \hat{K}_{ML} \sim \mathcal{N}\left( K, \frac{1}{SNR} \right)
$$
If SNR changes, the parameters of the distribution of $\hat{K}_{ML}$ change too and this directly affects the estimates. To see this more clearly, note that
$$
(2) \quad \lim_{SNR \to +\infty} \hat{K}_{ML} = K
$$
exactly as expected. Equivalently, if you observe that
$$
(3) \quad \hat{K}_{ML} = K + n, \quad n \sim \mathcal{N}(0, \sigma^2)
$$
where $\sigma^2$ is related to SNR, the gaussian distribution degenerates in the limit $\sigma^2 \to 0$ and $n$ collapses in its mean, i.e. zero. This is a very simple example: however, many important practical problems exhibit similar behaviour (and one should be wary of problems that do not!).


Summarizing, everytime the system is simulated, you obtain an estimate. This will be "almost surely" different from previous and future estimates, since every change in the parameters which govern the distribution of the estimator $\hat{K}_{ML}$ (be it SNR, guessed parameter, $M$, $f_s$, ...) directly affect the estimates. This should not be thought as a negative behaviour, since what actually matters are the statistical properties of the estimator, such as being unbiased, consistent, efficient, etc..., and this is also the reason why bounds such as the CRLB are useful and frequently used.
