Suppose you know the distribution of $(S_1,\ldots,S_{10})$ conditioned on $S_\mathrm{main}$. In the case of continuous distributions, you'll have $f(s_1,\ldots,s_{10}|s_\mathrm{main})$. This tells you how much of the probability is concentrated around $(s_1,\ldots,s_{10})$ when $S_\mathrm{main}=s_\mathrm{main}$. But you want to go the other way around: You want to be able to guess what $s_\mathrm{main}$ is once you know $(s_1,\ldots,s_{10})$. One method of estimation is maximum likelihood. Here, you define likelihood as
$$
\mathcal{L}(s_\mathrm{max};s_1,\ldots,s_{10})
:=f(s_1,\ldots,s_{10}|s_\mathrm{main}).
$$
Note that if you know the distribution, you get the likelihood function for free. For maximum likelihood estimation, all you have to do is find the $s_\mathrm{max}$ that maximizes $\mathcal{L}(s_\mathrm{max};s_1,\ldots,s_{10})$. For exponential-type distributions, it's often more convenient to take the log (and get a log-likelihood function), and maximizing this is equivalent since log is an increasing function.

If you also know the distribution $g(s_\mathrm{main})$ of $S_\mathrm{main}$, you should use a maximum a posterior probability (MAP) estimate. This method updates the distributional knowledge of $S_\mathrm{main}$ with the observation of $(s_1,\ldots,s_{10})$ according to Bayes' rule. After determining the posterior distribution of $S_\mathrm{main}$, your estimate is the $s_\mathrm{main}$ which maximizes this distribution, similar maximum likelihood estimation. This form of estimation is the bread and butter of Bayesian inference.