Suppose I have a mixture of Gaussians and I know the mean and variance of each separate Gaussian. How can I tell whether or not the resulting distribution will be multimodal or, more specifically, that there will be be a number of modes equal to the number of Gaussians in the mixture? I see that there are some formulas that can answer this question in the case of a mixture of two Gaussians, but I cannot find any literature related to a more general case. I only need to deal with a one dimensional Gaussian if this is of any relevance.
I doubt you will find an explicit solution. The number of modes is the number of roots of the first derivative of the density function - which is quite messy - I am quite sure you can't find an explicit solution.
Miguel Carrera-Perpinan has a webpage on this topic with associated software. This does not directly solve your question, but indicates that
- unidimensional Gaussian mixtures with $k$ components have at most $k$ modes;
- unidimensional non-Gaussian mixtures with $k$ components may have more than $k$ modes;
- multidimensional mixtures with $k$ components may have more than $k$ modes.
In the unidimensional Gaussian, a component will induce a mode near its mean if the variance is small enough (relative to the other variances) and if the weight is large enough (relative to the other weights). This may suggest a specific numerical way to check for modes by changing the variance of one component until a mode appears or disappears near a component mean.
Obviously, running EM algorithms starting from each normal mean should indicate whether or not this component induces a local mode.
Else, you may always follow Larry Wasserman's advice: "mixtures, like tequila, are inherently evil and should be avoided at all costs."