I have a problem which goes as follows. I am trying to predict the value of a variable $x$. I also have a set of measurements (the actual context is an image) $x^i$. I know from some training examples that my measurements are able to predict the value of $x$ with varying accuracy; for example, I have that $x \approx N(x^i + \mu^i, \sigma^i)$ for each measurement. That is, the difference between $x$ and $x^i$ gives a normal distribution which I know the mean and variance of.
So, given that I have several measurements $x^i$ each with their own Gaussian predication of $x$, I want to find the most likely value of $x$. My questions are:
1) Am I correct in thinking that the best approach here is to sum the Gaussian PDFs and find the maximum?
And if so:
2) Is there a way to directly compute the maximum of a sum of Gaussian functions?
If I am thinking about this correctly I think it is similar to a Gaussian Mixture Model, although this has yet to help me.
Any reassurance and general advice on my approach would be much appreciated!