I am trying to predict the outcome of a random variable x, which is a real-valued number. In some cases I can observe another variable y1, which should approximate x. I model y1 as a Gaussian distribution with mean of 0 and an empirically estimated standard deviation, and use the probability density function of that Gaussian to predict x.
In some cases I also have a second observation y2, which I can model similarly as a Gaussian.
What is the appropriate way to combine y1 and y2 into an estimate on x? Should I add the distributions and model x as a mixture of Gaussians, or should I multiply them? Or, should I try both and pick the answer that maximizes the probability of sampling an independent set of data?