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What does it mean for "Generate pseudodata by Monte Carlo"?

For the text I was reading it says suppose that the real data $Z_n \sim N(\mu(\theta), \Sigma (\theta))$, then the pseudodata are generated by Monte Carlo according to a normal distribution based on a chosen initial $\theta$.

I though the Monte Carlo method is kind of generating the random number then take the average, I am not quite sure how is this method works here.

Thanks for any help!

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    $\begingroup$ "generating the random number then take the average" might be called Monte Carlo Integration. You can use inverse CDF methods to generate random numbers according to some distributions, or accept-reject algorithm when inverse CDF cannot be obtained. $\endgroup$ – Deep North Oct 31 '18 at 0:06
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pseudo-data are often used in error propagation to avoid theoretical estimation.

Theoretical estimation is usually more complicated and also might be inaccurate (because it is based on linearization: famous formula with square root of sum of squares of errors of inputs).

Pseudo-data approach is a genuine error propagation approach with the only limitation: statistics (or computer time if you want).

Imagine you have a published data with erros $(q_i,\delta q_i)$ and you want to compute something from them $R = f(q_i,\delta q_i)$. As an example: you have set of measured values of a function $(y_i,\delta y_i)$, assuming corresponding ($x_i$) are known numbers (without error) you want to compute integral $ R = \int_a^b y(x)dx$ using trapezoidal rule. You ask the question: what is error on $R$?

In pseudo-data approach you simulate. You have information about standard deviations $\delta q_i$ but (usually) you do not know the whole distribution. Therefore you assume, often Gaussian distribution (or maybe some other, you you have some reasons to believe it). Using random numbers you produce many (=$N$) clones of the initial $q_i$ data $\{q_i^{[j]}\}_{j=1}^{j=N}$. Each individual $q_i^{[j]}$ you produce as a random number from assumed distribution, the distribution having $q_i$ as mean and $\delta y_i$ as standard deviation. For each data clone you compute you $R^{[j]}$ (integral, in our example). With enough statistic this provides you full distribution for $R$, you can compute for example mean $R_{mean} = \frac{1}{N} \sum_{j=1}^{j=N}R^{[j]}$ or the error on R (which you are interested in) $$ R_{error} = \sqrt{ \frac{1}{N} \sum_{j=1}^{j=N} (R^{[j]}-R_{mean})^2 }$$.

Pseudo data fully respects all correlations among inputs ${q_i}$, as told, it's only limitation is statistics.

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