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There is a paper in Geostatistics literature by Lloret-Cabot et al. (2014) namely 'On the estimation of scale of fluctuation in geostatistics' which suggests that conditional simulation of a Gaussian process $X(t)$ can lead to improved estimates of the spatial correlation parameter $\theta$.

Let me briefly explain the technique: Consider a continuous stationary Gaussian process $X(t)$ parameterised by a mean $\mu$, standard deviation $\sigma$ and 'scale of fluctuation' $\theta$ which parameterises some correlation function e.g.

$$\rho(\tau) = \exp\left(-\frac{2\tau}{\theta}\right)$$

Where $\tau = |t_1-t_2|$. Say you are given a partially observations $x(t_1)$, $x(t_2) \dots$ of some realisations $x(t)$ of the process $X(t)$ then techniques such as the fitting the sample correlation function can be used to estimate $\theta$. The idea of Lloret-Cabot et al (2014) is that conditonal simulation of the process (i.e. simulating a process that takes the partial observations as fixed points) can be used to improve estimates of $\theta$. Essentially new data points are generated between the partial observations and fitting these in addition to the original data improves estimates of $\theta$.

I was wondering if this idea had basis outside of geostatistics literature?

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    An obvious instance where one simulates continuous realisations of a processes observed only at discrete time points is in the analysis of diffusion processes - in this instance the likelihood is unavailable without the continuous path, on the other hand in this application for a Gaussian process it is perfectly easy to write down the likelihood for $\theta$ given discrete observations so it isn't immediately obvious what the complete path gets you – 2017-02-24
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    @Nadiels Indeed. There is no new information (i.e. real data points). The argument would be that this method makes better use of the original data points. One point to note is that in the paper they do not use MLE estimators. Perhaps the benefit is only when a suboptimal estimator for $\theta$ is used. – 2017-02-24
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    Yeah I hadn't noticed that initially but it does seem in this field the method of moments/fit the variogram function is preferred to maximum likelihood, perhaps because it somehow feels like to much of a leap to specify a complete model for these fields? But then if you are just simulating Gaussian processes I'm not sure what you have gained? – 2017-02-24
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    For example when performing inference for diffusion processes where method of moments have been quite popular there is still something that might be regraded as a true likelihood and therefore a reference point on which to argue for asymptotic convergence etc. which doesn't seem present in this setting, so motivating this choice of estimator seems hard – 2017-02-24

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