Hi everybody
There are 11 signals:
S_main : The original signal
S1 ~ S10 : 10 signals that are correlated to S_main with different correlation coefficients (coeff1 ~ coeff10)
Now here's the question:
How can I combine signals S1, S2, ..., S10 in order to regenerate S_main !!?
Thanks a billion for your upcoming help.
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.
