# Is there a way to directly compute maximum of a sum of several Gaussian functions?

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!

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You should not be adding the pdfs together; rather you should (assuming the measurements are independent) be taking the product i.e. add the pdfs on the log scale. A good starting point for any model is to write down the likelihood if you have one; in this case the observations are just a string of Gaussians. Also, you probably want $x^i \sim N(x + \mu^i, \sigma^i)$ which are conditionally independent given $x$; you should model $x$ first (or regard it as an unknown constant) and then the distribution of the $x^i$'s given $x$. – guy Sep 19 '12 at 16:44