We want to find the parameters of a multivariate normal distribution $N_d(\mu,\Sigma)$ which approximates a known vector $x\in\mathbb{R}^n$ as follows
First, sample $t$-times from a multivariate normal distribution $y_j \sim N_d(\mu,\Sigma)$ for $j\in\{1,\ldots,t\}$.
Then, apply some fixed operator $A$ (for the sake of simplicity we can assume linear at first), $A(y_j)=\hat{y}_j\in\mathbb{R}^n$.
Now, set the $i$-th index of $z_j=(0,\ldots,0)^T\in\mathbb{R}^n$ to $1$ if $\hat{y}_{ji} = \Vert \hat{y}_j\Vert_{max}$. Then, we get $x \approx \frac1t\sum_{j=0}^t z_j$, i.e. for each transformed sample we count the index with the maximal value.
For large $t$ we should then approximate $x$. Clearly, we restrict $x$ to fullfil $\sum_{i=0}^n x_i=1$.
How would you estimate the parameters $\mu$ and $\Sigma$? Since we are working with samples here, I do not know if we could even use classical optimisation tools.